Table Of ContentINTERNATIONAL 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.
