INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES C 0 U R S E S AN D LECTURES • No. 70 CLIFFORD TRUESDELL UNIVERSITY OF BALTIMORE THE TRAGICOMEDY OF CLASSICAL THERMODYNAMICS COURSE HELD AT THE DEPARTMENT OF MECHANICS OF SOLIDS (JULY 1971) UDINE 1971 Springer-Verlag Wien GmbH ISBN 978-3-211-81114-6 ISBN 978-3-7091-2393-5 (eBook) DOI 10.1007/978-3-7091-2393-5 Copyright by Springer-Verlag Wien 1971 Originally published by CISM, Udine in 1971. 3 PREFACE The early history of thermodynamics is troced. The indicationa given in Fourier's work on heat conduction, the elaborotion of the caloric theory by Car not, and the introduction of internal energy by Clausiw are given major attention. The historical analysis searches for the formation of precise concepts and for their analysis by clear mathematics. 5 Introduction. We are not such optimists as were our teachers and parents. We do > not have to equate "progress" with every M(t) if l>t 0, t being the time. In stead of a field of brilliant success like hydrodynamics or electromagnetism, I shall discuss one accursed by misunderstanding, irrelevance, retreat, and failure : ther modynamics in the 19th century. You will not be surprised, consequently, by my use of a delta to define progress, since thermodynamics is the kingdom of deltas. However, in this lecture I will not use a delta again. Thermodynamics is the kingdom also of running current history as well as polemics. In no other discipline have the same equations been published so many times by different authors in different notations and therefore claimed as his own by each ; in no other has a single author seen fit to publish essentially the same thing over and over again within a period of twenty years ; and nowhere else is the ratio of words to equations so high. In no other part of mathematical phys ics have so many claims and counterclaims of priority been issued by the leading creators of the subject itself, and in no other have these same men set their hands to writing the history of the subject within a decade or two of their own first paper on the theory itself. In truth, only now could a real history of thermodynamics be written, since only in the last ten years have the aims of the creators of thermodynamics been achieved. 1. Fourier. Thermodynamics opens with FOURIER's Analytical Theory of Heat, published in 1822. FOURIER inferred a partial differential equation for the tem perature corresponding to the conduction of heat in a rigid body. His theory rests on four ideas about heat, which I here expressed in modern form as follows : E(R, t) = f R p e(8) dV, (1.1) 6 Fourier where E is the total internal energy in the region R at time t, p is the mass density, e is the specific internal energy or energetic. + (1.2) Q(U,t) = JU g(~_, S, t) dA, where Q(U,t) is the conduction of heat (rate of increase of non-mechanical ener gy) through the subsurface U of S and where g is the heating influx through S. (1.3) aE(R, t) = Q(aR, t); at that is, the rate of increase of total energy in R is exactly the conduction of heat through the boundary aR. (1.4) g(., S, .) =h-. -n ; that is, there exists a vector field~(~_, t) such that the heating influx at a point of - S is the component of h in the direction of the outer normal n- of S, with an appropriate convention of orientation. FOURIER stated this last assumption only in connection with the linear constitutive relation - (1.5) h = K(O) grad 0, which is suitable only for· bodies later defined clearly and called isotropic. The scalar K is the conductivity. Routine assumptions of smoothness and calculations which were long but routine in 1800 and are short and routine today lead to a partial differential - equation for the temperature field O(x, t) : , ao (1.6) pe (O)at = div (K(O) grad 0). 7 Fourier FOURIER, guided by the continuum mechanics of the eighteenth century, took the first principle for granted, and it was an easy step from hydro dynamics for him to express in the form of the balance law the then common as sumption that heat was indestructible. For the rest he had no specific earlier pat tern to imitate. He chose to adopt in modified form a law of cooling often now adays attributed to NEWTON: The flux of heat from a surface is proportional to the difference between the temperature of the surface and the temperature of the surroundings. FOURIER stated expressly that this same law should hold both at a bounding surface and for "infinitely near molecules". He did not see that the two statements strictly contradict one another. He may have thought he was fol lowing a classical precedent, since his predecessors had regularly derived partial differential equations by applying a general principle to a small element. This method works for a general law, which is a statement about the body as a whole, but not for a constitutive relation, which, since it defines a special material, is a local statement from the outset. Throughout FOURIER's book, the general prin ciples governing the transfer of heat are jumbled with the constitutive relation for an isotropic linear conductor. If we regard the constitutive relation ~ = K grad 0 as an assumption, which it is, rather than as a consequence of an empirical law of cooling, which it is not, we can derive FOURIER's differential equation clearly and correctly. If we do not adopt the constitutive relation, we can still derive the flux principle. That is, the balance law for energy (heat) and the flux principle imply the existence of the heating-flux vector field ~(~,t). Since the principle is one of the cornerstones of modern thermodynamics, we naturally search for it in FOURIER's book, and we can find a near miss. In § § 146-153 he applied the balance law to a small element on an exterior boundary which radiates according to the law of cooling, and by its aid he did reach the flux principle. The argument 8 Fourier makes no use of the law of cooling and applies to interior surfaces as well. Thus FOURIER did not need all the assumptions he made. In this sense we may attri bute to him the existence of the heating-flux vector, hut he himself seems not to have noticed it, and he obviously failed to see its importance. He spoke the pro logue to the tragicomedy of classical thermodynamics. We shall soon encounter in thermodynamics mathematicians in con trast with whom FOURIER, modest though his talents were by the standards of predecessors like EULER and contemporaries like CAUCHY and POISSON and successors like MAXWELL and KIRCHHOFF, will seem a giant. The moment we step outside thermodynamics, however, a different scale of competence rules. From the beginnings of mathematical science mathematicians have weighed as sumptions and have discarded the unnecessary ones. This is the very essence of mathematical thought. In thermodynamics, on the contrary, the unnecessary as sumptions have been treasured, repeated, and inflated to the point that they came to conceal the whole conceptual structure of the science. This unhappy quality, the feeble thread of the plot of the tragicomedy, hegins with FOURIER himself. To see the sorry difference between the standards of thermodynamics and its con temporary mathematical sciences we need only look at continuum mechanics, where the counterpart of FOURIER's unstated and half-implied flux principle is CAUCHY's theorem of the existence of the stress tensor, published in 1823. CAUCHY, who knew full well the difference between a balance principle and a constitutive relation, stated the result clearly and proudly; he gave a splendid proof of it, which has been reproduced in every hook on continuum mechanics from that day to this ; and he recognized the theorem as being the foundation stone it still is. The theorem of existence of a vector field, which FOURIER h.ad in his hands hut did not see, was first published by STOKES in 1851. By then it was too late. Thermodynamics had gone another way, excluding heat conduction by what seems to have been a tacit agreement to forget about it. Heat conduction, Fourier 9 likewise, went its own way and either tacitly excluded cases in which mechanical work was done or considered only situations in which FOURIER's theory could be superimposed without modification upon some simple and equally special the ory of continuum mechanics. The influence of FOURIER, and it was vast, was upon mechanics, acoustics, electromagnetism, the theory of approximation, probability, and func tions of a real or complex variable - in two words, upon pure mathematics and upon methods for solving linear partial differential equations- but not upon ther modynamics. Until about 1965 there was no general yet definite field theory of the interaction of heat and work ; thermodynamics had no partial differential e quations and hence no boundary-value or initial-value problems to solve. Interest in the theory of conduction slowly waned, until today its main use is to enable the student to illustrate and visualize the properties of parabolic differential equa tions. FOURIER's title is the first in a long sequence to be belied by the contents of the work. A theory of temperature he did give, but not a theory of heat. Alas, FOURIER's brilliance in his own bailiwick had, indirectly, a negative effect upon thermodynamics. Glorying in his ultra-linear theory, he taught a century of physicists that their ideal should be to spew out explicit and detailed solutions of boundary-value and initial-value problems. Indeed, the simpler the constitutive relation, the vaster the class of easy problems amenable to essentially routine mathematics. By grossly oversimplifying the physical model, the theorist may extract incredibly precise predictions about the most intimate detail in bodies of the most complicated shapes. In FOURIER's theory the student must pay as the price for such detail and precision a willingness to admit that differences of temperature propagate at infinite speed through material bodies and that heating a body does not change its size or shape. The constitutive relations natural and useful in thermodynamics are not linear. The kind of problem FOURIER deftly disposed of does not exist, the 10 Camot moment one generalizes his theory so as to include work done. 2. Camot Two years after FOURIER's treatise, in 1824, appeared CARNOT's Reflections on the Motive of Fire, and on Machines Fitted to Develop that Power. In the opening pages CARNOT stated that "the phenomenon of the production of motion by heat has not been studied from a sufficiently general point of view." It was necessary, he wrote, to "establish principles applicable not only to steam engines hut also to all imaginable heat-engines ... " Purely mechanical machines, CARNOT recalled, could he "studied even in their smallest details" by means of "the mechanical theory. All cases are foreseen, all imaginable movements are re ferred to these general principles, firmly established and applicable under all cir cumstances. This is the character of a complete theory. A similar theory is evident ly needed for heat-engines. We shall have it only when the laws of physics shall he extended enough, generalized enough, to make known beforehand all the effects of heat acting in a determined manner on any body". CARNOT did not follow the tradition of eighteenth-century rational mechanics he had just praised for its generality and inclusiveness. Instead, he chose to present his arguments in words. Now anything that can he said in mathematics can he said also in ordinary words. The converse, however, is false, and, as we may easily verify by simple domestic experiments, most of what words can and all too often do say must he draconically revised if not altogether expunged if even mini mal standards of logic are to he met. In CARNOT's treatise we encounter that fuzzine.ss which was to become and remain a distinguishing feature of thermody namics for puzzled outsiders, and from the day CARNOT's words were first read there has been disagreement as to what he really meant.