The Thermodynamics of Fluid Systems (The Oxford engineering science series)

186 Pages·1975·3.47 MB·english

by  L. C. Woods

OXFORD ENGINEERING SCIENCE SERIES
THE THERMODYNAMICS
OF FLUID SYSTEMS
WOODS
L.C.
OXFORD
THE
OXFORD ENGINEERING SCIENCE
SERIES
GeneralEditors
L. C. WOODS, W. H. WITTRICK, A. L. CULLEN

THE
THERMODYNAMICS
OF FLUID
SYSTEMS
BY
WOODS
L. C.
CLARENDON PRESS OXFORD
1975
OxfordUniversityPress,ElyHouse, London W.l
GLASGOW NEWYORK TORONTO MELBOURNE WELLINGTON
CAPETOWN IBADAN NAIROBI DARESSALAAM LUSAKA ADDISABABA
DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA
KUALALUMPUR SINGAPORE HONG K^ONG TOKYO
ISBN 198561253
• OXFORD UNIVERSITY PRESS 1975
All rights reserved. No part of this publication may be reproduced,
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PRINTED IN GREAT BRITAIN BY
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PREFACE
A suitable sub-title for this book would have been 'The time-variable in
thermodynamics',forthroughouttheemphasisis placedontherole oftime
scales, in determining both the state coordinates of equilibrium thermo-
dynamics (Part I) and the affinitiesor thermodynamic forces ofirreversible
processes(PartII).Thenotionthattheentropyofasystemdependsinteralia
on thetime scaleonwhich it is observed (orequivalently on howdetailed is
theobserver'sinformationaboutit)isemployedinbothpartstomakeclear
the unity ofclassical, kinetic, statistical, and process thermodynamics. This
practical approach also enables us to resolve the occasional paradoxes that
arise withentropy and its increase in isolated systems, for thesecan usually
be traced to the false beliefin the existence in an absolute entropy, a belief
unfortunately encouraged by thecurrently-popular axiomatictreatmentsof
thermodynamics.Thermodynamicsis,aboveall,anapproximatingsciencein
which it is necessary not only to understand the various limit forms it can
takebutalso thephysics oftheasymptotics involved.
My aim initially was to write a concise account of the thermodynamics
ofirreversibleprocessesingases,plasmas,andotherfluids,placingagreater
emphasis on the underlying assumptions and less on chemical applications
thanthefewreliabletextspresentlyavailable. Butitbecameclearatanearly
stage ofwriting that the key and unavoidable hypothesis of'local thermo-
dynamicequilibrium' needs muchmoreelaboration thanitusuallyreceives.
The concept of thermodynamic equilibrium is not nearly as simple and
preciseasmostaccountsimply;andwithoutaclearideaofitsmeaning,itis
pointlessforthestudenttoadvancetoprocessthermodynamics. Thus PartI
(onequilibriumthermodynamics)waswrittentoservebothasasurveyofthe
principal ideas of the subject and, because of its emphasis on the relative
natureof'equilibrium',as a suitable introduction to PartII.
PartIstartswithwhatmaybecalled 'engineering'thermodynamics,forit
deals with long-time-scale external processes; it is based on the first three
laws of thermodynamics, which combine to define an entropy S via the
fundamental formula TdS = dU+pdV. The observer's time scale is then
progressively shortened, and at each step terms added to the fundamental
equation to represent the increasing knowledge assumed of the system;
andsoweproceedthrough'chemical'thermodynamicsand'kinetic'thermo-
dynamicstotheultimatedescriptioncontainedinstatisticalthermodynamics.
By introducing an internal coordinate ^ defined over an 'interior' space y,
we are able to transform the fundamental expression for TdS into the pre-
scription S = -k£ £,yIn £y,which,unconstrained,holdsonalltimescales.
It is the constraints'that are added to turn the prescription into a genuine
PREFACE
vi
definition thataretime-scaledependent, and theshorter theobserver's time
scale thegreater the numberofconstraints that mustbe imposed.
I hope this novel approach will provide a useful skeletal survey ofbasic
thermodynamics suitable forfinal-year students in the physical sciences.
In Part II the concepts of process thermodynamics (often confusingly
termed 'non-equilibrium' thermodynamics, despite the fact that local
thermodynamic equilibrium is a sine qua non ingredient) are explained and
applied to a variety offluid phenomena. There are four basic ideas.
First there is the separation ofprocesses into reversible and irreversible
elements,adivisionthatdependsonthephysicalmodelassumedtorepresent
thesystem. Andwhen the reversible changes ofentropy are known, the rate
<7atwhichentropyisirreversiblycreatedcanbeexpressedasabilinearform,
say, a = JX, where J and X are vectors representing the various processes
and thermodynamic forces in the system. In all this time scales play an
importantpart.Asaisnon-negative,wehavethebasicinequalitya =JX 3*0,
which has much the same role in process thermodynamics as does the
fundamental equation dS = T_1 dV+T~lpdV+ ... = Z AY, say, for
equilibrium thermodynamics. From the latter we use the fact that AS is the
exactdifferentialofafunctionS = S(Y)toinfertheexistenceofstateequations
Z = Z(Y); and likewise as a ^ requires J and X to vanish together, we
may infer constitutive relations J = J(X) from the bilinear form.
With the assumption oflinear constitutive relations, say J = LX, where
Lisaphenomenologicalmatrix,wecome to thesecondbasicideaofprocess
thermodynamics, namely, the existence ofreciprocal relations between the
coefficientsLuofL,whichinthesimplestcaseareLtj = LJt. Inessence,these
relationsareduetoW.Thomson(LordKelvin),althoughitisusualtodismiss
hisargumentsasbeingtoointuitivetoofferaconvincingproof. Ontheother
hand, the accepted treatment, due to L. Onsager, relies too much on the
analogy taken to exist between the regression of large thermodynamic
fluctuations andmacroscopic processes like heat orcurrent flux that persist
not only in stationary flow conditions but, more importantly, in systems
pressed to the (thermodynamic) limit at which fluctuations strictly vanish.
Thenewapproachofferedinthis book (alongwithaccounts oftheworks of
Thomson and Onsager) resembles Thomson's in being genuinely 'macro-
scopic'.Bythetime-scaleargumentsrequiredtodrawthedistinctionbetween
reversibility and irreversibility, it is shown that the essential nature of the
reciprocal relations is that they ensure that the entropy production rate
a = XLX is due only to irreversible processes.
The two remaining basicconcepts are common to all branches ofmacro-
scopic physics. They are 'reference-frame indifference', which requires the
constitutive relations to be expressible in covariant form, and 'material
invariance',bywhichspatialsymmetriesknowntoexistinthesystemenable
us to reducesubstantially thenumberofindependentelements appearingin
PREFACE
L. Particular attention has been given to the lateral isotropy of magneto-
plasmas,inwhichthemagneticfieldimposesacharacteristicdirectiononthe
medium.
Theuseofinternalcoordinates £ permitsagreatextensioninthescopeot
processthermodynamics. Andwhenconstitutiverelationsforftt)areknown
or assumed, some or all of these coordinates may be eliminated with the
eifnfteecgtraolf rreelpaltaicoinngJ(tth)e=inJsLt,anWta-neWou)sdcofns,tituwthievererefltatt)ioinsJa=phLeXnobmyentoh-e
logical response function. If,further, owing to thermodynamicfluctuations,
the <f(t) are assumed to be stochastic processes, the theory leads on to the
importantfluctuation-dissipationtheorem,ofwhichtheearliestexamplesare
containedinEinstein'sworkonBrownianmotionandNyquist'sonJohnson
noise.
Fluid mixtures are systems for which the systematic approach ot process
thermodynamics is particularly useful,forthenumber ofpossibleprocesses
and cross-couplings between them may be quitelarge. A general account of
mixture theory, which includes the effects of mass transfer between the
components,isfollowedbyapplicationstotwoimportantexamples,namely,
magneto-plasmas and helium II. A magneto-plasma can be described ade-
quatelybyatwo-fluidtheoryonacertaintimescaleandisagoodexampleof
an anisotropic medium. Helium II, on the other hand, has the interesting
feature (rather neglected in standard treatments)that its normal and super-
fluid components can exchange mass elements. These aspects have been
pursuedtothestageofobtainingsomenewresultsforeachmixture.Rotating
heliumII,whichisan anisotropic medium,isalso given someattention.
Kinetic constitutive relations are described in the final chapter. The set
(SFx,Sdi2s,t..r.iSbrutoifoenntfruonpciteisonfsoryitehledosnae-speaqrutiecnlcee(oFfx)n,otnw-o-npeagrattiicvleee(nFt2)r,o..p.yr-ppraortdiuccl-e
tiro)n rates S,,S2,... 5r from which may be inferred constitutive relations for
F F ofincreasingaccuracy.Thisapproachyieldsthekineticequationsof
Boxl,tz2m..a.nn,Bogoliubov,andothers,andmayhavesomemeritovertheusual
methodsofclosingtheBBGKYhierarchyofequations. Inanycaseitreveals
the close similaritybetween kinetic equations and the constitutiverelations
ofmacroscopicfluid dynamics.
Part II will, I hope, be of interest to graduates working in continuum
mechanics and otherbranches ofmacroscopic physics.
AmongthemanyrelevanttextsandarticlesIhavestudied—ofwhichonly
those essential to the textual argument have been cited—I am particularly
indebted to the work of S. R. de Groot and P. Mazur (1962), which first
engaged my attention in the thermodynamics ofirreversible processes. And
perhapsitisrighttoadmittoothatagrowingaversiontoaxiomaticthermo-
dynamics,withitspenchantforsubstitutingdeductivemathematicalconstruc-
tions in place ofphysics, has stimulated me to write this book, and in it to
PREFACE
viii
emphasizetheapproximationsinherentinthermodynamics,oratleastthose
approximations associated with time scales. It is, ofcourse, the existence of
such approximations that makes thermodynamics such a rich and diverse
branch ofphysics.
I am grateful to both H. Troughton and Stanley Morris for help with
checking the manuscript and proofs, to Mrs. Ina Godwin for her excellent
typing of rather complicated material, and finally to the Delegates of the
Oxford University Press for accepting my biased recommendation as a
serieseditorthattheworkwas suitableforpublicationin the Oxford Engin-
eering Science Series.
Mathematical Institute L.C.W.
Oxford University
August, 1974
CONTENTS
PART EQUILIBRIUM THERMODYNAMICS
I. 1
THERMODYNAMIC VARIABLES AND PROCESSES 3
1.
1. Thermodynamicsystems 3
2. Theempiricaltemperature 6
3. Workprocesses '
THE BASIC VARIABLES OF THERMODYNAMICS 15
2.
4. Internalenergy 15
5. Heat 18
6. Theempiricalentropy 20
7. Absolutetemperatureandentropy 22
8. Degeneracyofsimplesystems 26
9. Thesecondlawofthermodynamics 28
10. Heatcapacities,compressibility,andotherderivatives 30
11. Systemsinelectromagneticfields 38
SYSTEMS INVOLVING INTERNAL VARIABLES 44
3.
12. Compoundsystemsnotinthermalequilibrium 44
13. Thelawofentropyincrease 47
14. Homogeneousopensystems 50
53
15. Characteristicfunctions
16. Thermodynamicinequalities 57
17. Equilibriumandstability 59
18. Chemicalequilibriaandstability 63
19. Thetimevariableinthermodynamics 66
70
20. Irreversibility
SYSTEMS POSSESSING INTERNAL DEGREES OF
4.
FREEDOM 73
21. Internaldegreesoffreedom 74
22. Kineticdefinitionsfortheindependentvariables 78
23. Thestateequations 79
24. TheMaxwelliandistributionfunction 84
25. Kineticentropy 87
STATISTICAL THERMODYNAMICS 95
5.
26. Entropyasastatisticalaverage 96
27. Theprincipalensemblesofstatisticalthermodynamics 101
28. Otherensembles 105
29. Fluctuationtheory 109
30. Ensemblesinphase-space 112
31. SomeapplicationsofHamiltonians 116
32. Thedistributionsofquantumstatisticalmechanics 118
33. Stateequationsforperfectgases 122
34. Crystallinesolids 128
35. Thethermodynamicsofradiation 132
36. Informationtheoryandthermodynamics 135