Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 19
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Thus the early researchers invariably do not make itclear that the heat dQ and the work dW are applied to the surface of thebody. Nor do they state clearly that the T and the p occurring in theirequations, or inequalities, are the homogeneous temperature and the homogeneous pressure on the surface which may or may not be equal to those inthe interior of the body; they are equal in equilibrium or in reversibleprocesses, i.e. slow processes, but not otherwise.The kinetic energy of the flow field inside the body is never mentionedby either Carnot or Clausius although, of course, its conversion into heatwas paramount in the minds of Mayer, Joule and Helmholtz.All this had to be cleaned up and incorporated into a systematic theory.That was a somewhat thankless task, taken on by scientists like Duhem, and50According to S.G. Brush: ‘‘The Temperature of History. Phases of Science and Culture inthe Nineteenth century.” Burt Franklin & Co.
New York (1978).51 N. Georgescu-Roegen: ‘‘The Entropy Law and the Economic Process.” HarvardUniversity Press, Cambridge, Mass (1971).52 I. Müller, W. Weiss: ‘‘Entropy and Energy – A Universal Competition, Chap. 20: Sociothermodynamics.” Springer, Heidelberg, (2005).A simplified version of socio-thermodynamics is presented at the end of Chap. 5.743 EntropyJaumann 53 and Lohr.
54,55 These people recognized the first and second lawsfor what they are: Balance equations, or conservation laws on a par – formally – with the balance equations of mass and momentum.Generically an equation of balance for some quantity Ψ Ô ρψ dV in aVvolume V, whose surface V – with the outer normal ni – moves with thevelocity ui, has the formdU\d8dV 8³ ³ ( U\ (X K W K ) )K )PK d# ³ V d8 .w88ȡ is the mass density and ȥ is the specific value of Ȍ , such that ȡȥ is thedensity of Ȍ.56 The velocity of the body, a fluid (say), is X i.
In the surfaceintegral, ȡȥ(X i – ui)ni is the convective flux of Ȍ through the surfaceelement dA and ĭini is the non-convective flux. ı is the source density of Ȍ;it vanishes for conservation laws.For mass, momentum, energy, and entropy the generic quantities in theequation of balance have values that may be read off from Table 3.1.tli is called the stress tensor, whose leading term is the pressure –pįli ; thatis the only term in tli, if viscous stresses are ignored. Ekin is the kinetic energyof the flow field and qi is the heat flux.
Mass, momentum and energy areconserved, so that their source-densities vanish.57 Note that the internalenergy is not conserved, because it may be converted into kinetic energy.The entropy source is assumed non-negative which represents the growthproperty of entropy.53G. Jaumann: ‘‘Geschlossenes System physikalischer und chemischer Differentialgesetze”[Closed system of physical and chemical differential laws] Sitzungsbericht Akademie derWissenschaften Wien, 12 (IIa) (1911).54 E. Lohr: ‘‘Entropie und geschlossenes Gleichungssystem” [Entropy and closed system]Denkschrift der Akademie der Wissenschaften, 93 (1926).55 While Lohr is largely forgotten, Gustav Jaumann (1863–1924) lives on in the memory ofmechanicians as the author of the Jaumann derivative, a ‘‘co-rotational” time derivative,i.e.
the rate of change of some quantity – like density or velocity – as seen by an observerlocally moving and rotating with the body; that derivative plays an important role inrheology and in theories of plasticity. Jaumann was a student of Ernst Mach and carriedMach’s prejudice against atoms far into the 20th century, thus making himself an outsiderof any serious scientific circle. He died in a mountaineering accident.56 It has become customary in thermodynamics to denote global quantities – those referringto the whole body – by capital letters, and specific quantities – referred to the mass – bythe corresponding minuscules.57 We ignore gravitation and radiation. See, however, Chap. 7, where radiation is treated.Gravitation changes thermodynamic in some subtle and, indeed, interesting ways, sincethe pressure field cannot be homogeneous in equilibrium, – neither on V, nor in V.However, here is not the place to treat gravitational effects, because we do not wish toencumber our arguments.
Let is suffice to say that in gases and vapours the gravitationaleffects are usually so small as to be negligible.Modern Version of Zeroth , First and Second Laws75Table 3.1. Canonical notation for specific values of mass, momentum, energy and entropyand their fluxes and sourcesȥĭiȈmass m100momentum PlXl–tli0–tliXN+qi01energy U + Ekinu+ /2 Xinternal energyUuentropy Ss2VNKqiqiT58XȌw NwZ Kı 0In order to clarify the special status of Clausius’s first law, the equationfor dU, we first observe that viscous forces did not enter Clausius’s mind inconnection with the first law. Also he considered closed systems, whosesurfaces move with the velocity of the body on the surface so that noconvective flux appears.
Therefore Clausius would have written theequation of balance of energy in the formd(U E kin )dtxxQ W , wherexQ ³ qi ni dA is the heating, andWXwVx ³ p i ni dA is the working of pressure.wVThe balance of internal energy should then have the formwX ldV is the internal working.wxlVIf we assume that the pressure is homogeneous on wV, the first equationxdUdtxx³ pQ W int where W intbecomes5958This form of the entropy flux is nearly universally accepted, although the kinetic theory ofgases furnishes a different form; the difference is small and we ignore it for the timebeing. See, however, Chap.
4.59Note that³8X N PN d# 8³wwX NwZNd8d8dV.763 Entropyd (U Edtkin)xQ pdV ;dtand if we assume that the pressure is homogeneous throughout V, thesecond equation becomesxdUdtQ pdV.dtBy comparison it follows that, for a homogeneous pressure p in V, thereis no change of kinetic energy of the flow field which, of course, isreasonable. Indeed, according to the momentum balance, there is noacceleration in this case.
Thus now, under all these restrictive assumptions –and with Q dt dQ – we have obtained the Clausius form of the first law.All these assumptions were tacitly made by Clausius, and hisforerunners, and the majority of his followers to this very day. Indeed,among students thermodynamics has acquired the reputation of a difficultsubject just because of the many tacit assumptions. The difficulty is notinherent in the field, however; it is due to sloppy teaching.According to Table 3.1, the entropy balance contains a non-negativesource density and a non-convective flux which is assumed to be given byqiT, so that we may writedSdt³wVqinidA t 0.TThis inequality is known as the Clausius-Duhem inequality. If T is homogeneous on wV, we may writexdSQt 0,dtTxwhereQ³ q n dAiiwVxand that is – again with Q dt = dQ – the form obtained by Clausius.
Heconsidered only this case. If T is not homogeneous on wV , the naturalextension of his inequality was conceived by Pierre Maurice Marie Duhem(1861–1916).Duhem was professor of theoretical physics in Bordeaux. He worked successfullyin thermodynamics at the time when Gibbs was still unknown in Europe. However,he is also known as a philosopher of science, who expressed the view that the lawsof physics are but symbolic constructions, neither true nor wrong representations ofreality.
He advocated metaphysical hypotheses for a provisional understanding ofWhat is Entropy ?77nature. Somehow Duhem’s ideas found their way from Bordeaux to Vienna, wherethey were welcomed by Ernst Mach who thought that science should concentrateexclusively on finding relations between observed phenomena, see Chap. 4.Duhem’s thoughts helped to underpin this kind of positivistic thinking in whatbecame known as the Vienna circle, a niche for philosophers belly-aching abouttruth in the laws of natural science. A latter-day representative of the school wasKarl Raimund Popper (1902–1994) – Sir Karl since 1964 – in whose writings thedilemma is largely reduced to the question of how, or whether, and why we knowthat the sun will rise tomorrow, after approximately 90,000 pulse beats, – or will iteverywhere and always? Popper wrote a book about this important problem.60The energy balance implies that the normal component of the heat flux qiis continuous at a diathermic wall, i.e.
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