Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 14
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Law: ‘‘The Steam Engine” loc.cit. p. 13.503 EntropyFig. 3.2. Watt’s steam enginepayment, they claimed one third of the saving in coal over the old[Newcomen] engines.This was enough to make Watt a rich man, because, indeed, Watt’sengine was three to four times more efficient than Newcomen’s.6 Wattretired in 1800. He was famous by then, and much honoured for his life’swork. Thus he was elected to membership of the Royal Society of Londonand he received an honorary doctorate from the University of Glasgowwhere he had previously served in the lowly position of a laboratoryassistant.Liquid water and steam are particularly well-suited for the conversion ofheat into work, because the heat absorbed and emitted – by boiler andcooler respectively – is exchanged isobarically.
And a large portion of thoseisobars are also isotherms, because they lie in the two-phase region of wetsteam, where boiling liquid and saturated vapour coexist. This makes theprocess somewhat similar to a Carnot process, which has maximum efficiency, see below.6Actually, the efficiencies were all quite low: In Newcomen’s case about 2% and 5–7% inWatt’s case. A modern power station reaches between 45% and 50%. The engineers havedone a good job indeed over the past 200 years.Heat Engines51In 1783 he tested a strong horse anddecided that it could raise a 150pound weight nearly four feet in asecond.
He therefore defined a‘‘horsepower” as 550 foot-pounds persecond. This unit of power is stillused, particularly for automobiles.However, the unit of power inthe metric system is called 1 Watt, inhonour of the Scottish engineer. Onehorsepower equals 746 Watt.Fig. 3.3. James Watt. A quote from Asimov 7Heat can also be converted into work by an air engine or, more generally,a gas engine. Figure 3.4 shows the prototypical Joule processschematically, where an adiabatic compressor furnishes hot air which isthen further heated by isobarically absorbing the heat Q+. Afterwards thegas cools by adiabatic expansion in the working cylinder which pushes itinto a heat exchanger, where it gives off the heat Q- isobarically.
In itsalternation between adiabatic and isobaric steps the process is much like theprocess in the steam engine. However, in the Joule process the isobars arein no way similar to isotherms, since no phase transition occurs.Fig. 3.4. The Joule process and a (pressure,volume)-diagram of the Joule process in an idealgasNone of the engineers who invented or improved the steam engine – orthe air engine – was in any way distracted by any soul-searching about thenature of heat, or whether or not there was a caloric. They proved that heatcould produce work by doing it, – and doing it better and better as timewent on.The efficiency of the engines climbed up slowly but surely through manyingenious improvement and in the 1820s it had arrived at 18%.
At that time7I. Asimov: ‘‘Biographies…” loc.cit. p. 187.523 EntropySadi Carnot, a physicist educated at the École Polytechnique in Paris, posedhimself the question, how far this improvement could possibly go and heattempted to find an answer.Nicolas Léonard Sadi Carnot (1796–1832)Sadi Carnot was named after the 13th century Persian poet Saadi Musharifed Din who was en vogue in the France of the directorate. His father LazareCarnot was one of the directors, and later he became one of Napoléon’sloyal and efficient generals. The father was also an accomplished mathematician who published a book on mechanical machines in 1803: ‘‘Fundamental principles on equilibrium and movement.” In that book LazareCarnot strongly supported the view that a perpetuum mobile wasimpossible.By hereditary taint, perhaps, the son picked up the question whether thepossible improvements [of heat engines] might have an assignable limit.And, in 1824, Sadi Carnot published a book in which he addressed theproblem: ‘‘Réflexions sur la puissance motrice du feu et sur les machinespropres à déveloper cette puissance”8 Everything seemed conceivable at thetime:x The process in which heating and cooling occurred at constantpressures might be improved by letting the heat exchange occur atconstant volumes or constant temperatures, andx perhaps working agents like sulphur or mercury might have anadvantage over water.Carnot came to correct conclusions concerning both propositions.
Aboutthe first one he says:The best manner to employ a heat engine, whose working agent assumestemperatures between TLow and THigh in the process, is the engine – which wenow call a Carnot engine – which exchanges heat only at thosetemperatures.Because, so Carnot, [that process is] …le plus avantageux possible, car ilne s’est fait aucun rétablissment inutile d´équilibre dans la calorique.989S. Carnot: [Reflections on the motive power of fire and on machines fitted to develop thatpower] à Paris chez Bachelier, Libraire.
Quai des Augustin, No. 55 (1824). Englishtranslation by R.H. Thurston: ‘‘Reflections on the motive power of fire by Sadi Carnot andother papers on the second low of thermodynamics by É. Clapeyron and R. Clausius.”E. Mendoza (ed.) Dover Publ. New York (1960). pp. 1–59.S. Carnot: ‘‘Réflexions…” loc.cit. p. 35.Nicolas Léonard Sadi Carnot (1796–1832)53The argument goes like this: Carnot plausibly postulates that a machineis optimal when the temperature of the working agent is alwayshomogeneous and, if it changes in time, that change must be connected witha change in volume.10 Other changes in temperature are useless, and evendetrimental.
It is clear that the steam engine does not satisfy that optimalitycondition, since the cold feed-water from the condenser enters the hotboiler, so that a rétablissement inutile must occur. Actually Carnot shows alot of insight and ingenuity here, because in a lengthy footnote he proposesto preheat the feed-water by condensing a part of the vapour after partialexpansion and at a temperature intermediate between boiler and theprincipal condenser.11 This kind of feed-water preheating – actually inseveral steps – is done routinely in modern power stations; it is known asCarnotization of the steam engine process. To be sure, in order to bepractical, the procedure requires expansion in a turbine, not in a steamcylinder, but the principle was recognized by Carnot.Concerning Carnot’s second proposition, – the one on the potentialadvantage of using an agent other than water – he comes to the conclusionthatWhen a Carnot engine is used, all agents provide the same work.In Carnot’s words: La puissance motrice de la chaleur est indépendentedes agens mis en oeuvre pour la réaliser ; sa quantité est fixée uniquementpar les temperatures entre lesquels se fait en dernier résultat le transportdu calorique.12This statement is proved by letting two Carnot engines – with differentagents, but the same heat exchanges, and in the same temperature range –work against each other, one as a heat engine and one as a refrigerator, orheat pump.
If one engine requires more work than the other one produces,we should be able to create motive power without consumption either ofcaloric or of any other agent whatever. Such a creation is entirely contraryto ideas now accepted, to the laws of mechanics and of sound physics. It isinadmissible. It would be perpetual motion.13It was his insight into the working of heat engines that permitted Carnotto come to these conclusions. For the above arguments it was quiteirrelevant, whether he knew what heat was, – and he didn’t! Indeed, Carnotbelieved in the caloric theory of heat and he thought that the caloricentering the boiler came out of the cooler unchanged in amount. Thereforeit was natural for Carnot to draw an analogy between the motive power ofheat and that of a waterfall, – une chute d’eau, see Fig. 3.5.10S.
Carnot: ibidem p. 23.S. Carnot: ibidem p. 26.12 S. Carnot: ibidem p. 38.13 S. Carnot: ibidem p. 21.11543 Entropy… on peut comparer avec assez de justesse lapuissance motrice de la chaleur à celle d´unechute d´eau : toutes deux ont un maximum quel´on ne peut pas dépasser, quelle que soit d´unepart la machine employée à recevoir l´actionde l´eau, et quelle que soit de l´autre la substanceemployée à recevoir l´action de la chaleur.La puissance motrice d´une chute d´eau dépendde sa hauteur et de la quantité du liquide;la puissance motrice de la chaleur dépend aussi dela quantité de calorique employé, et de ce… que nous appellerons en effet la hauteurde sa chute, c´est-à-dire de la différencede température…Fig.
3.5. Sadi Carnot. His reflections about the fall of heat14This misconception, and the false information, which Carnot had aboutthe specific heat of gases, and the latent heat of water vapour, invalidatesmuch of the second half of his paper.15 He tied himself into knots over thespecific heats of gases, which he thinks he can prove to be logarithmicfunctions of the density when in reality they are constants, independent ofboth density and temperature.However, Carnot did ask the right questions. Thus he was interested toknow how the location of the temperature range of the Carnot engineaffected the efficiency.
He states that a given fall of the caloric [a giventemperature difference] produces more motive power at inferior than atsuperior temperatures.16 This is true, but unfortunately Carnot invalidatesthe statement in his marginal analysis,17 where he proves that – for temperature-independent specific heats – the efficiency is independent of thetemperature range. The whole argument is a mess.The best concrete result, which Carnot reached, concerned a Carnotengine working in the infinitesimal temperature range dt at t.
In his notationthe efficiency e is given by e = Fƍ(t)dt, where Fƍ(t) is a universal function,sometimes called the Carnot function. Carnot could not determine thatfunction. Thus, although he proved that the efficiency of a Carnot engine ismaximal, he did not know the value of the maximum, – not even for aninfinitesimal cycle. The Carnot function, however, partly because of itsuniversal character, provided a strong stimulus for further research on the14S.
Carnot: ‘‘Réflexions…” loc.cit. p. 28.Carnot refers repeatedly to the experimental results of MM. Delaroche and Bérard, whothought that they had measured the specific heat of air to be dependent on pressure. Werecall that Mayer was led to a wrong value of the mechanical equivalent of heat bymeasurements of the same two men, see Chap. 2.16 S. Carnot: ‘‘Réflexions…” loc.cit.
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