AXIS OF VOLUME
F to. 5.—Elementary Carnot Cycle for Gas.
16. Experimental Verification • of Carnet's Principle.—Carnot endeavoured to test his result by the following simple calculations. Suppose that we have a cylinder fitted with a frictionless piston, containing 1 gram of water at too° C., and that the pressure of the steam, namely 76o mm., is in equilibrium with the external pressure on the piston at this temperature. Place the cylinder in connexion with a boiler or hot body at lot° C. The water will then acquire the temperature of 101 ° C., and will absorb I gramcalorie of heat. Some waste of motive power occurs here because heat is allowed to pass from one body to another at a different temperature, but the waste in this case is so small as to be immaterial. Keep the cylinder in contact with the hot body at 101° C. and allow the piston to rise. It may be made to perform useful work as the pressure is now 27.7 mm. (or 37.7 grams per sq. cm.) in excess of the external pressure. Continue the process till all the water is converted into steam. The heat absorbed from the hot body will be nearly 540 gramcalories, the latent heat of steam at this temperature. The increase of volume will be approximately 1620 c.c., the volume of 1 gram of steam at this pressure and temperature. The work done by the excess pressure will be 37.7X1620=61,000 gramcentimetres or 0.61 of a kilogrammetre. Remove the hot body, and allow the steam to expand further till its pressure Is 76o mm. and its temperature has fallen to too° C. The work which might be done in this expansion is less than 10100th part of a kilogrammetre, and may be neglected for the present purpose. Place the cylinder in contact with the cold body at loo C., and allow the steam to condense at this temperature. No work is done on the piston, because there is equilibrium of pressure. but a quantity of heat equal to the latent heat of steam at too° C. is given to the cold body. The water is now in its initial condition, and the result of the process has been to gain o•61 of a kilogrammetre of work by allowing 540 gramcalories of heat to pass from a body at sot ° C. to a body at too° C. by means of an ideally simple steamengine. The work obtainable in this way from woo gramcalories of heat, or t kilocalorie, would evidently be 1.13 kilogrammetre (=o•61 X i a ).
Taking the same range of temperature, namely toe to too° C., we may perform a similar series of operations with air in the cylinder, instead of water and steam. Suppose the cylinder to contain t gramme of air at toe C. and 76o mm. pressure instead of water. Compress it without loss of heat (adiabatically), so as to raise its temperature to tee C. Place it in contact with the hot body at tot C., and allow it to expand at this temperature, absorbing heat from the hot body, until its volume is increased by 374th part (the expansion per degree at constant pressure). The quantity of heat absorbed in this expansion, as explained in § 14, will be the difference of the specific heats or the latent heat of expansion R' = .o69 calorie.
Remove the hot body, and allow the gas to expand further without of the form gain of heat till its temperature falls to too° C. Compress it at too° C. to its original volume, abstracting the heat of compression by i contact with the cold body at too° C. The air is now in its original state, and the process has been carried out in strict accordance with Carnot's rule. The quantity of external work done in the cycle is easily obtained by the aid of the indicator diagram ABCD (fig. 5), which is approximately a parallelogram in this instance. The area of the diagram is equal to that of the rectangle BEHG, being the product of the vertical height BE, namely, the increase of pressure per 1° at constant volume, by the increase of volume BG, which is s41rdof the volume at o° C. and 76o mm., or 2.83 c.c. The increase of pressure BE is P$, or 2.03 mm., which is equivalent to 2.76 gm. per sq. cm. The work done in the cycle is 2.76X2.83=7.82 gm. cm., or .0782 grammetre. The heat absorbed at tot° C. was .069 gramcalorie, so that the work obtained is .0782/.069 or 1.13 grammetre per gramcalorie, or 1.13 kilogrammetre per kilogramcalorie. This result is precisely the same as that obtained by using
F't=R/C(t±to) . (3) where C and to are unknown constants. A similar result follows from his expression for the difference of the specific heats. If this is assumed to be constant and equal to C, the expression for F't becomes R/CT, which is the same as the above if to=273. Assuming.' the specific heat to be also independent of the volume, he shows that the function F't should be constant. But this assumption is inconsistent with the caloric theory of latent heat of expansion, which requires the specific heat to be a function of the volume. It appears in fact impossible to reconcile Carnot's principle with the caloric theory on any simple assumptions. As Carnot remarks: " The main principles on which the theory of heat rests require most careful examination. Many experimental facts appear almost inexplicable in the present state of this theory."
Carnot's work was subsequently put in a more complete analytical form by B. P E. Clapeyron (J own. de''6t; polytechn.,
Paris, 1832, 14, p. 153), who also made use of Watt's indicator diagram for the first time in discussing physical problems. Clapeyron gave the general expressions for the latent heat of a vapour, and for the latent heat of isothermal expansion of any substance, in terms of Carnot's function, employing the notation of the calculus. The expressions he gave are the same in form as those in use at the present day. He also gave the general expression for Carnot's function; and endeavoured to find its variation with temperature; but having no better data, he succeeded no better than Carnot. Unfortunately, in describing Carnot's cycle, he assumed the caloric theory of heat, and made some unnecessary mistakes, which Carnot (who, we now know, was a believer in the mechanical theory) had been very careful to avoid. Clapeyron directs one to compress the gas at the lower temperature in contact with the body B until the heat disengaged is equal to that which has been absorbed at the higher temperature.' He assumes that the gas at this point contains the same quantity of heat as it contained in its original state at the higher temperature, and that, when the body B is removed, the gas will be restored to its original temperature, when compressed to its initial volume. This mistake is'still attributed to Carnot, and regarded as a fatal objection to his reasoning by nearly all writers at the present day.
18. Mechanical Theory of Heat.—Accordingto the caloric theory, the heat absorbed in the expansion of a gas became latent, like the latent heat of vaporization of a liquid, but remained in the gas and was again evolved on compressing the gas. This theory gave no explanation of the source of the motive power produced by expansion. The mechanical theory had explained the production of heat by friction as being due to transformation of visible motion into a brisk agitation of the ultimate molecules, but it had not so far given any definite explanation of the converse production of motive power at the expense of heat. The theory could not be regarded as complete until it had been shown that in the production of work from heat, a certain quantity of heat disappeared, and ceased to exist as heat; and that this quantity was the same as that which could be generated by the expenditure of the work produced. The earliest complete statement of the mechanical theory from this point of view is contained in some notes written by Carnot, about 1830, but published by his brother (Life of Sadi Carnot, Paris, 1878). Taking the difference of the specific heats to be •o78, he estimated the mechanical equivalent at 370 kilogrammetres. But he fully recognized that there were no experimental data at that time available for a quantitative test of the theory, although it appeared to afford a good qualitative explanation of the phenomena. He therefore planned a number of crucial experiments such as the " porous plug " experiment, to test the equivalence of heat and motive power. His early death in 1836 put a stop to these experiments, but many of them have since been independently carried out by other observers.
The most obvious case of the production of work from heat 23, p. 263). In this paper he showed that the heat produced
is in the expansion of a gas or vapour, which served in the first by currents generated by magnetoelectric induction followed
instance as a means of calculating the ratio of equivalence, on the same law as voltaic currents. By a simple and ingenious
the assumption that all the heat which disappeared had been arrangement he succeeded in measuring the mechanical power
expended in producing the currents, and deduced the mechanical
equivalent of heat. and of electrical energy. The amount of
mechanical work required to raise 1 lb of water 10 F. (I
B.Th.U.), as found by this method, was 838 footpounds. In
a note added to the paper he states that he found the value
770 footpounds by the more direct method of forcing water_
through fine tubes. In a paper " On the Changes of Tempera
ture produced by the Rarefaction and Condensation. of Air" (Phil.
Mag., May 1845), he made the first direct measurements of
the quantity of heat disengaged by compressing air, and also
of the heat absorbed when the air was allowed to expand against
atmospheric pressure; as the result he deduced the value 798
footpounds for the mechanical equivalent of 1 B.Th.U. He also
showed that there was no appreciable absorption of heat when
definitely, deducing it from the old principle, causa aequat effectum. Assuming that the sinking of a mercury column by which a gas was compressed was equivalent to the heat set free by the compression, he deduced that the warming of a kilogramme of water 1° C. would correspond to the fall of a weight of one kilogramme from a height of about 365 metres. But Mayer did not adduce any fresh experimental evidence, and made no attempt to apply his theory to the fundamental equations of thermodynamics. It has since been urged that the experiment of GayLussac (1807), on the expansion of gas from one globe to another (see above, § 11), was sufficient justification for the assumption tacitly involved in Mayer's calculation. But Joule was the first to supply the correct interpretation of this experiment, and to repeat it on an adequate scale with suitable precautions. Joule was also the first to measure directly the amount of heat liberated by the compression of a gas, and to prove that heat was not merely rendered latent, but disappeared altogether as heat, when a gas did work in expansion.
19. Joule's Determinations of the Mechanical Equivalent.—The honour of placing the mechanical theory of heat on a sound experimental basis belongs almost exclusively to J. P. Joule, who showed by direct experiment that in all the most important cases in which heat was generated by the expenditure of mechanical work, or mechanical work was produced at the expense of heat, there was a constant ratio of equivalence between the heat generated and the work expended and vice versa. His first experiments were on the relation of the chemical and electric energy expended to the heat produced in metallic conductors and voltaic and electrolytic cells; these experiments were described in a series of papers published in the Phil. Mag., 18401843. He first proved the relation, known as Joule's law, that the heat produced in a conductor of resistance R by a current C is proportional to OR per second. He went on to show that the total heat produced in any voltaic circuit was proportional to the electromotive force E of the battery arid to the number of equivalents electrolysed in it. Faraday had shown that electromotive force depends on chemical affinity. Joule measured the corresponding heats of combustion, and showed that the electromotive force corresponding to a chemical reaction is proportional to the heat of combustion of the electrochemical equivalent. He also measured the E.M.F. required to decompose water, and showed that when part of the electric energy EC is thus expended in a voltameter, the heat generated is less than the heat of combustion corresponding to EC by a quantity representing the heat of combustion of the decomposed gases. His papers so far had been concerned with the relations between electrical energy, chemical energy and heat which he showed to be mutually equivalent. The first paper in which he discussed the relation of heat to mechanical power was entitled " On the Calorific Effects of MagnetoElectricity, and on the Mechanical Value of Heat " (Brit. Assoc., 1843; Phil. Mag.,
transformed into work and had not merely become latent. Marc Seguin, in his De l'influence des Chemins de fer (Paris, 1839), made a rough estimate in this manner of the mechanical equivalent of heat, assuming that the loss of heat represented by the fall of temperature of steam on expanding was equivalent to the mechanical effect produced by the expansion. He also remarks (loc. cit. p. 382) that.it was absurd to suppose that " a finite quantity of heat could produce an indefinite quantity of mechanical action, and that it was more natural to assume that a certain quantity of heat disappeared in the very act of producing motive power." J. R. Mayer (Liebig's Annalen, 1842, 42, p. 233) stated the equivalence of heat and work more
1 It was for this reason that Professor W. Thomson (Lord Kelvin)
stated (Phil. Mag., 1852, 4) that " Carnot's original demonstration air was allowed to expand in such a manner as not to develop
utterly fails," and that he introduced the " corrections " attributed
to James Thomson and Clerk Maxwell respectively. In reality 1 mechanical power, and he pointed out that the mechanical Carnot's original demonstration requires no correction. 1 equivalent of heat could not be satisfactorily deduced from
the relations of the specific heats, because the knowledge of the specific heats of gases at that time was of so uncertain a character. He attributed most weight to his later determinations of the mechanical equivalent made by the direct method of friction of liquids. He showed that the results obtained with different liquids, water, mercury and sperm oil, were the same, namely, 782 footpounds; and finally repeating the method with water, using all the precautions and improvements which his experience had suggested, he obtained the value 772 footpounds, which was accepted universally for many years, and has only recently required alteration on account of the more exact definition of the heat unit, and the standard scale of temperature (see CALORIMETRY). The great value of Joule's work for the general establishment of the principle of the conservation of energy lay in the variety and completeness of the experimental evidence he adduced. It was not sufficient to find the relation between heat and mechanical work or other forms of energy in one particular case. It was necessary to show that the same relation held in all cases which could be examined experimentally, and that the ratio of equivalence of the different forms of energy, measured in different ways, was independent of the manner in which the conversion was effected and of the material or working
substance employed.
As the result of Joule's experiments, we are justified in con
cluding that heat is a form of energy, and that all its transformations are subject to the general principle of the conservation of energy. As applied to heat, the principle is called the first law of thermodynamics, and may be stated as follows: When heat is transformed into any other kind of energy, or vice versa, the total quantity of energy remains invariable; that is to say, the quantity of heat which disappears is equivalent to the
quantity of the other kind of energy produced and vice versa.
The number of units of mechanical work equivalent to one unit
of heat is generally called the mechanical equivalent of heat, or Joule's equivalent, and is denoted by the letter J. Its numerical value depends on the units employed for heat and mechanical energy respectively. The values of the equivalent in terms of the units most commonly employed at the present time are as follows:
777 footpounds (Lat. 45°)are equivalent tot B. Th. U. (lb deg.Fahr.) 1399 footpounds „ „ t lb deg. C.
426.3 kilogrammetres t kilogramdeg.C. or kilocalorie.
426.3 grammetres t gramdeg. C. or calorie.
4.18o joules t gramdeg. C. or calorie.
The water for the heat units is supposed to be taken at 20° C. or 68° F., and the degree of temperature is supposed to be measured by the hydrogen thermometer. The acceleration of gravity in latitude 45° is taken as 980.7 C.G.S. For details of more recent and accurate methods of determination, the reader should refer to the article CALORIMETRY, where tables of the variation of the specific heat of water with temperature are also
given.
The second law of thermodynamics is a title often used to
denote Carnot's principle or some equivalent mathematical expression. In some cases this title is not conferred on Carnot's principle itself, but on some axiom from which the principle may be indirectly deduced. These axioms, however, cannot as a rule be directly applied, so that it would appear preferable to take Carnot's principle itself as the second law. It may be observed that, as a matter of history, Carnot's principle was established and generally admitted before the principle of the conservation of energy as applied to heat, and that from this point of view the titles, first and second laws, are not particularly
appropriate.
20. Combination of Carnot's Principle with the Mechanical
Theory.—A very instructive paper, as showing the state of the science of heat about this time, is that of C. H. A. Holtzmann, " On the Heat and Elasticity of Gases and Vapours " (Mannheim, 1845; Taylor's Scientific Memoirs, iv. 189). He points out that the theory of Laplace and Poisson does not agree with facts when applied to vapours, and that Clapeyron's formulae,though probably correct, contain an undetermined function (Carnot's F't, Clapeyron's i/C) of the temperature. He determines the value of this function to be J/T by assuming, with Seguin and Mayer, that the work done in the isothermal expansion of a gas is a measure of the heat absorbed. From the then accepted value •078 of the difference of the specific heats of air, he finds the numerical value of J to be 374 kilogrammetres per kilocalorie. Assuming the heat equivalent of the work to remain in the gas, he obtains expressions similar to Clapeyron's for the total heat and the specific heats. In consequence of this assumption, the formulae he obtained for adiabatic expansion were necessarily wrong, but no data existed at that time for testing them. In applying his formulae to vapours, he obtained an expression for the saturationpressure of steam, which agreed with the empirical formula of Roche, and satisfied other experimental data on the supposition that the coefficient of expansion of steam was •00423, and its specific heat 1.69values which are now known to be impossible, but which appeared at the time to give a very satisfactory explanation of the phenomena.
The essay of Hermann Helmholtz, On the Conservation of Force (Berlin, 1847), discusses all the known cases of the transformation of energy, and is justly regarded as one of the chief landmarks in the establishment of the energyprinciple. Helmholtz gives an admirable statement of the fundamental principle as applied to heat, but makes no attempt to formulate the correct equations of thermodynamics on the mechanical theory. He points out the fallacy of Holtzmann's (and Mayer's) calculation of the equivalent, but admits that it is supported by Joule's experiments, though he does not seem to appreciate the true value of Joule's work. He considers that Holtzmann's formulae are well supported by experiment, and are much preferable to Clapeyron's, because the value of the undetermined function F't is found. But he fails to notice that Holtzmann's equations are fundamentally inconsistent with the conservation of energy, because the heat equivalent of the external work done is supposed to remain in the gas.
That a quantity of heat equivalent to the work performed actually disappears when a gas does work in expansion, was first shown by Joule in the paper on condensation and rarefaction of air (1845) already referred to. At the conclusion of this paper he felt justified by direct experimental evidence in reasserting definitely the hypothesis of Seguin (loc. cit. p. 383) that " the steam while expanding in the cylinder loses heat in quantity exactly proportional to the mechanical force developed, and that on the condensation of the steam the heat thus converted into power is not given back.” He did not see his way to reconcile this conclusion with Clapeyron's description of Carnot's cycle. At a later date, in a letter to Professor W. Thomson (Lord Kelvin) (1848), he pointed out that, since, according to his own experiments, the work done in the expansion of a gas at constant temperature is equivalent to the heat absorbed, by equating Carnot's expressions (given in § 17) for the work done and the heat absorbed, the value of Carnot's function F't must be equal to J/T, in order to reconcile his principle with the mechanical theory.
Professor W. Thomson gave an account of Carnot's theory (Trans. Roy. Soc. Edin., Jan. 1849), in which he recognized the discrepancy between Clapeyron's statement and Joule's experiments, but did not see his way out of the difficulty. He therefore adopted Carnot's principle provisionally, and proceeded to calculate a table of values of Carnot's function F't, from the values of the totalheat and vapourpressure of steamthen recently determined by Regnault (Memoires de l'Institut de Paris, 1847). In making the calculation, he assumed that the specific volume v of saturated steam at any temperature T and pressure p is that given by the gaseous laws, pv=RT. The results are otherwise correct so far as Regnault's data are accurate, because the values of the efficiency per degree F't are not affected by any assumption with regard to the nature of heat. He obtained the values of the efficiency F't over a finite range from t to o° C., by adding up the values of F't for the separate degrees. This latter proceeding is inconsistent with the mechanical theory, but is the
correct method on the assumption that the heat given up to the condenser is equal to that taken from the source. The values he obtained for F't agreed very well with those previously given by Carnot and Clapeyron, and showed that this function diminishes with rise of temperature roughly in the inverse ratio of T, as suggested by Joule.
R. J•. E. Clausius (Pogg. Ann., 185o, 79, p. 369) and W. J. M. Rankine (Trans. Roy. Soc. Edin., 185o) were the first to develop the correct equations of thermodynamics on the mechanical theory. When heat was supplied to a body to change its temperature or state, part remained in the body as intrinsic heat energy E, but part was converted into external work of expansion W and ceased to exist as heat. The part remaining in the body was always the same for the same change of state, however performed, as required by Carnot's fundamental axiom, but the part corresponding to the external work was necessarily different for different values of the work done. Thus in any cycle in which the body was exactly restored to its initial state, the heat remaining in the body would always be the same, or as Carnot puts it, the quantities of heat absorbed and given out in its diverse transformations are exactly " compensated," so far as the body is concerned. But the quantities of heat absorbed and given out are not necessarily equal. On the contrary, they differ by the equivalent of the external work done in the cycle. Applyingtthis principle to the case of steam, Clausius deduced a fact previously unknown, that the specific heat of steam maintained in a state of saturation is negative, which was also deduced by Rankine (loc. cit.) about the same time. In applying the principle to gases Clausius assumes (with Mayer and Holtzmann) that the heat absorbed by a gas in isothermal expansion is equivalent to the work done, but he does not appear to be acquainted with Joule's experiment, and the reasons he adduces in support of this assumption are not conclusive. This being admitted, he deduces from the energy principle alone the propositions already given by Carnot with reference to gases, and shows in addition that the specific heat of a perfect gas must be independent of the density. In the second part of his paper he introduces Carnot's principle, which he quotes as follows: " The performance of work is equivalent to a transference of heat from a hot to a cold body without the quantity of heat being thereby diminished." This is not Carnot's way of stating his principle (see § 15), but has the effect of exaggerating the importance of Clapeyron's unnecessary assumption. By equating the expressions given by Carnot for the work done and the heat absorbed in the expansion of a gas, he deduces (following Holtzmann) the value J/T for Carnot's function F't (which Clapeyron denotes by 1/C). He shows that this assumption gives values of Carnot's function which agree fairly well with those calculated by Clapeyron and Thomson, and that it leads to values of the mechanical equivalent not differing greatly from those of Joule. Substituting the value JIT for C in the analytical expressions given by Clapeyron for the latent heat of expansion and vaporization, these relations are immediately reduced to their modern
form (see THERMODYNAMICS, § 4). Being unacquainted with
Carnot's original work, but recognizing the invalidity of
Clapeyron's description of Carnot's cycle, Clausius substituted
a proof consistent with the mechanical theory, which he based
on the axiom that " heat cannot of itself pass from cold to hot."
The proof on this basis involves the application of the energy
principle, which does not appear to be necessary, and the axiom
to which final appeal is made does not appear more convincing
than Carnot's. Strange to say, Clausius did not in this paper
give the expression for the efficiency in a Carnot cycle of finite
range (Carnot's Ft) which follows immediately from the value
J/T assumed for the efficiency I't of a cycle of infinitesimal range
at the temperature t C or T Abs.
Rankine did not make the same assumption as Clausius explicitly, but applied the mechanical theory of heat to the development of his hypothesis of molecular vortices, and deduced from it a number of results similar to those obtained by Clausius. Unfortunately the paper (loc. cit.) was not published till some time later, out in a summary given in the Phil. Mag. (July 1851)
the principal results were detailed. Assuming the value of Joule's equivalent, Rankine deduced the value o2404 for the specific heat of air at constant pressure, in place of 0.267 as found by Delaroche and Berard. The subsequent verification of this value by Regnault (Comptes rendus, 1853) afforded strong confirmation of the accuracy of Joule's work. In a note appended to the abstract in the Phil. Mag. Rankine states that he has succeeded in proving that the maximum efficiency of an engine working in a Carnot cycle of finite range 11 to to is of the form (tlto)/(t1—k), where k is a constant, the same for all substances. This is correct if t represents temperature Centigrade, and k=—273.
Professor W. Thomson (Lord Kelvin) in a paper " On the Dynamical theory of Heat " (Trans. Roy. Soc. Edin., 1851, first published in the Phil. Mag., 1852) gave a very clear statement of the position of the theory at that time. He showed that the value F't= J/T, assumed for Carnot's function by Clausius without any experimental justification, rested solely on the evidence of Joule's experiment, and might possibly not be true at all temperatures. Assuming the value J/T with this reservation, he gave as the expression for the efficiency over a finite range t, to to C., or Tl to To Abs., the result,
W/H=(ti—to)/(t'+273) =(T1—To)/Ti . (4)
which, he observed, agrees in form with that found by Rankine.
21. The Absolute Scale of Temperature.—Since Carnot's function is the same for all substances at the same temperature, and is a function of the temperature only, it supplies a means of measuring temperature independently of the properties of any particular substance. This proposal was first made by Lord Kelvin (Phil. Mag., 1848), who suggested that the degree of temperature should be chosen so that the efficiency of a perfect engine at any point of the scale should be the same, or that Carnot's function F't should be constant. This would give the simplest expression for the efficiency on the caloric theory, but the scale so obtained, when the values of Carnot's function were calculated from Regnault's observations on steam, was found to differ considerably from the scale of the mercury or airthermometer. At a later date, when it became clear that the value of Carnot's function was very nearly proportional to the reciprocal of the temperature T measured from the absolute zero of the gas thermometer, he proposed a simpler method (Phis. Trans., 1854), namely, to define absolute temperature 0 as proportional to the reciprocal of Carnot's function. On this definition of absolute temperature, the expression (0 0 )/O for the efficiency of a Cannot cycle with limits BI and 0o would be exact, and it became a most important problem to determine how far the temperature T by gas thermometer differed from the absolute temperature O. With this object he devised a very delicate method, known as the " porous plug experiment " (see THERMODYNAMICS) of testing the deviation of the gas thermometer from the absolute scale. The experiments were carried out in conjunction with Joule, and finally resulted in showing (Phil. Trans., 1862, " On the Thermal Effects of Fluids in Motion ") that the deviations of the air thermometer from the absolute scale as above defined are almost negligible, and that in the case of the gas hydrogen the deviations are so small that a thermometer containing this gas may be taken for all practical purposes as agreeing exactly with the absolute scale at all ordinary temperatures. For this reason the hydrogen thermometer has since been generally adc.pted as the standard.
22. Availability of Heat of Combustion.—Taking the value 1.13 kilogrammetres per kilocalorie for 1° C. fall of temperature at loo° C., Carnot attempted to estimate the possible performance of a steamengine receiving heat at 16o° C. and rejecting it at 4o° C. Assuming the performance to be simply proportional to the temperature fall, the work done for 12o° fall would be 134 kilogrammetres per kilocalorie. To make an accurate calculation required a knowledge of the variation of the function F't with temperature. Taking the accurate formula of § 20, the work obtainable is 118 kilogrammetres per kilocalorie, which is
28% of 426, the mechanical equivalent of the kilocalorie in kilogrammetres. Carnot pointed out that the fall of 12o° C. utilized in the steamengine was only a small fraction of the whole temperature fall obtainable by combustion, and made an estimate of the total power available if the whole fall could be utilized, allowing for the probable diminution of the function F't with rise of temperature. His estimate was 3.9 million kilogrammetres per kilogramme of coal. This was certainly an overestimate, but was surprisingly close, considering the scanty data at his disposal.
In reality the fraction of the heat of combustion available, even in an ideal engine and apart from practical limitations, is much less than might be inferred from the efficiency formula of the Carnot cycle. In applying this formula to estimate the availability of the heat it is usual to take the temperature obtainable by the combustion of the fuel as the upper limit of temperature in the formula. For carbon burnt in air at constant pressure without any loss of heat, the products of combustion might be raised 2300° C. in temperature, assuming that the specific heats of the products were constant and that there was no dissociation. If all the heat could be supplied to the working fluid at this temperature, that of the condenser being 4o° C., the possible efficiency by the formula of § 20 would be 89%. But the combustion obviously cannot maintain so high a temperature if heat is being continuously abstracted by a boiler. Suppose that 0' is the maximum temperature of combustion as above estimated, 0" the temperature of the boiler, and 9° that of the condenser. Of the whole heat supplied by combustion represented by the rise of temperature 8'6°, the fraction (0'O")/(O'00) is the maximum that could be supplied to the boiler, the fraction (B"O°)/(9'B°) being carried away with the waste gases. Of the heat supplied to the boiler, the fraction (0"0°)/0" might theoretically be converted into work. The problem in the case of an engine using a separate working fluid, like a steamengine, is to find what must be the temperature 0" of the boiler in order to obtain the largest possible fraction of the heat of combustion in the form of work. It is easy to show that 0" must be the geometric mean of 0' and 0°, or 6"= jO'Oo. Taking 6'9°=2300° C., and 0°=313° Abs. as before, we find 0"= 903° Abs. or 63o° C. The heat supplied to the boiler is then 74.4% of the heat of combustion, and of this 65.3 % is converted into work, giving a maximum possible efficiency of 49% in place of 8g %. With the boiler at 16o° C., the possible efficiency, calculated in a similar manner, would be 26.3%, which shows that the possible increase of efficiency by increasing the temperature range is not so great as is usually supposed. If the temperature of the boiler were raised to 300° C., corresponding to a pressure of 126o lb per sq. in., which is occasionally surpassed in modern flashboilers, the possible efficiency would be 40%. The waste heat from the boiler, supposed perfectly efficient, would be in this case r 1 %, of which less than a quarter could be utilized in the form of work. Carnot foresaw that in order to utilize a larger percentage of the heat of combustion it would be necessary to employ a series of working fluids, the waste heat from one boiler and condenser serving to supply the next in the series. This has actually been effected in a few cases, e.g. steam and SO2, when special circumstances exist to compensate for the extra complication. Improvements in the steamengine since Carnot's time have been mainly in the direction of reducing waste due to condensation and leakage by multiple expansion, superheating, &c. The gain by increased temperature range has been comparatively small owing to limitations of pressure, and the best modern steamengines do not utilize more than 20% of the heat of combustion. This is in reality a very respectable fraction of .the ideal limit of 40% above calculated on the assumption of 126o lb initial pressure, with a perfectly efficient boiler and complete expansion, and with an ideal engine which does not waste available motive power by complete condensation of the steam before it is returned to the boiler.
23. Advantages of Internal Combustion.—As Carnot pointed out, the chief advantage of using atmospheric air as a working fluid in a heatengine lies in the possibility of imparting heat toit directly by internal combustion. This avoids the limitation imposed by the use of a separate boiler, which as we have seen reduces the possible efficiency at least 50%. Even with internal combustion, however, the full range of temperature is not available, because the heat cannot conveniently in practice be communicated to the working fluid at constant temperature, owing to the large range of expansion at constant temperature required for the absorption of a sufficient quantity of heat. Airengines of this type, such as Stirling's or Ericsson's, taking in heat at constant temperature, though theoretically the most perfect, are bulky and mechanically inefficient. In practical engines the heat is generated by the combustion of an explosive mixture at constant volume or at constant pressure. The heat is not all communicated at the highest temperature, but over a range of temperature from that of the mixture at the beginning of combustion to the maximum temperature. The earliest instance of this type of engine is the lycopodium engine of M.M. Niepce, discussed by Carnot, in which a combustible mixture of air and lycopodium powder at atmospheric pressure was ignited in a cylinder, and did work on a piston. The early gasengines of E. Lenoir (186o) and N. Otto and E. Langen (1866), operated in a similar manner with illuminating gas in place of lycopodium. Combustion in this case is effected practically at constant volume, and the maximum efficiency theoretically obtainable is 1log,r/(r1), where r is the ratio of the maximum temperature 0' to the initial temperature 6°. In order to obtain this efficiency it would be necessary to follow Carnot's rule, and expand the gas after ignition without loss or gain of heat from 0' down to 6°, and then to compress it at 0° to its initial volume. If the rise of temperature in combustion were 2300° C., and the initial temperature were o° C. or 273° Abs., the theoretical efficiency would be 73'3%, which is much greater than that obtainable with a boiler. But in order to reach this value, it would be necessary to expand the mixture to about 270 times its initial volume, which is obviously impracticable. Owing to incomplete expansion and rapid cooling of the heated gases by the large surface exposed, the actual efficiency of the Lenoir engine was less than 5%, and of the Otto and Langen, with more rapid expansion, about 1o%. Carnot foresaw that in order to render an engine of this type practically efficient, it would be necessary to compress the mixture before ignition. Compression is beneficial in three ways: (r) it permits a greater range of expansion after ignition; (2) it raises the mean effective pressure, and thus improves the mechanical efficiency and the power in proportion to size and weight; (3) it reduces the loss of heat during ignition by reducing the surface exposed to the hot gases. In the modern gas or petrol motor, compression is employed as in Carnot's cycle, but the efficiency attainable is limited not so much by considerations of temperature as by limitations of volume. It is impracticable before combustion at constant volume to compress a rich mixture to much less than kth of its initial volume, and, for mechanical simplicity, the range of expansion is made equal to that of compression. The cycle employed was patented in 1862 by Beau de Rochas (d. 1892), but was first successfully carried out by Otto (1876). It differs from the Carnot cycle in employing reception and rejection of heat at constant volume instead of at constant temperature. This cycle is not so efficient as the Carnot cycle for given limits of temperature, but, for the given limits of volume imposed, it gives a much higher efficiency than the Carnot cycle. The efficiency depends only on the range of temperature in expansion and compression, and is given by the formula (0' 0")/O', where 0' is the maximum temperature, and 0" the temperature at the end of expansion. The formula is. the same as that for the Carnot cycle with the same range of temperature in expansion. The ratio 9'/0" is ry1, where r is the given ratio of expansion or compression, and y is the ratio of the specific heats of the working fluid. Assuming the working fluid to be a perfect gas with the same properties as air, we should have 7=1'41. Taking r = 5, the formula gives 48% for the maximum possible efficiency. The actual products of combustion vary with the nature of the fuel
148
employed, and have different properties from air, but the efficiency is found to vary with compression in the same manner as for air. For this reason a committee of the Institution of Civil Engineers in 1905 recommended the adoption of the airstandard for estimating the effects of varying the compression ratio, and defined the relative efficiency of an internal combustion engine as the ratio of its observed efficiency to that of a perfect airengine with the same compression.
24. Effect of Dissociation, and Increase of Specific Heat.One of the most important effects of heat is the decomposition or dissociation of compound molecules. Just as the molecules of a vapour combine with evolution of heat to form the more complicated molecules of the liquid, and as the liquid molecules require the addition of heat to effect their separation into molecules of vapour; so in the case of molecules of different kinds which combine with evolution of heat, the reversal of the process can be effected either by the agency of heat, or indirectly by supplying the requisite amount of energy by electrical or other methods. Just a s the latent heat of vaporization diminishes with rise of temperature, and the pressure of the dissociated vapour molecules increases, so in the case of compound molecules in general the heat of combination diminishes with rise of temperature, and the pressure of the products of dissociation increases. There is evidence that the compound carbon dioxide, CO2, is partly dissociated into carbon monoxide and oxygen at high temperatures, and that the proportion dissociated increases with rise of temperature. There is a very close analogy between these phenomena and the vaporization of a liquid. The laws which govern dissociation are the same fundamental laws of thermodynamics, but the relations involved are necessarily more complex on account of the presence of different kinds of molecules, and present special difficulties for accurate investigation in the case where dissociation does not begin to be appreciable until a high temperature is reached. It is easy, however, to see that the general effect of dissociation must be to diminish the available temperature of combustion, and all experiments go to show that in ordinary combustible mixtures the rise of temperature actually attained is much less than that calculated as in § 22, on the assumption that the whole heat of combustion is developed and communicated to products of constant specific heat. The defect of temperature observed can be represented by supposing that the specific heat of the products of combustion increases with rise of temperature. This is the case for CO2 even at ordinary temperatures, according to Regnault, and probably also for air and steam at higher temperatures. Increase of specific heat is a necessary accompaniment of dissociation, and from some points of view may be regarded as merely another way of stating the facts. It is the most convenient method to adopt in the case of products of combustion consisting of a mixture of CO2 and steam with a large excess of inert gases, because the relations of equilibrium of dissociated molecules of so many different kinds would be too complex to permit of any other method of expression. It appears from the researches of Dugald Clerk, H. le Chatelier and others that the apparent specific heat of the products of combustion in a gasengine may be taken as approximately •34 to •33 in place of •24 at working temperatures between r000° C. and 17o0° C., and that the ratio of the specific heats is about 1•2g in place of 1•41. This limits the availability of the heat of combustion by reducing the rise of temperature actually obtainable in combustion at constant volume by 30 or 40%, and also by reducing the range of temperature 0','B" for a given ratio of expansion r from r'41 to r•29. The formula given in § 22 is no longer quite exact, because the ratio of the specific heats of the mixture during compression is not the same as that of the products of combustion during expansion. But since the work done depends principally on the expansion curve, the ratio of the range of temperature in expansion (B'B") to the maximum temperature 0' will still give a very good approximation to the possible efficiency. Taking r= 5, as before, for the compression ratio, the possible efficiency is reduced from 48% to 38%, if 7=1.29 instead of 1.41. A large gasengine of the present day with r = 5 may actually[TRANSFERENCE OF HEAT
realize as much as 34% indicated efficiency, which is go% of the maximum possible, showing how perfectly all avoidable heat losses have been minimized.
It is often urged that the gasengine is relatively less efficient than the steamengine, because, although it has a much higher absolute efficiency, it does not utilize so large a fraction of its temperature range, reckoning that of the steamengine from the temperature of the boiler to that of the condenser, and that of the gasengine from the maximum temperature of combustion to that of the air. This is not quite fair, and has given rise to the mistaken notion that " there is an immense margin for improvement in the gasengine," which is not the case if the practical limitations of volume are rightly considered. If expansion could be carried out in accordance with Carnot's principle of maximum efficiency, down to the lower limit of temperature 0o, with rejection of heat at Bo during compression to the original volume vo, it would no doubt be possible to obtain an ideal efficiency of nearly 8o%. But this would be quite impracticable, as it would require expansion to about ioo times vo, or 500 times the compression volume. Some advantage no doubt might be obtained by carrying the expansion beyond the original volume. This has been done, but is not found to be worth the extra complication. A more practical method, which has been applied by Diesel for liquid fuel, is to introduce the fuel at the end of compression, and adjust the supply in such a manner as to give combustion at nearly constant pressure. This makes it possible to employ higher compression, with a corresponding increase in the ratio of expansion and the theoretical efficiency. With a compression ratio of 14, an indicated efficiency of 40% has been obtained in this way, but owing to additional complications the brake efficiency was only 31%, which is hardly any improvement on the brake efficiency of 30% obtained with the ordinary type of gasengine. Although Carnot's principle makes it possible to calculate in every case what the limiting possible efficiency would be for any kind of cycle if all heat losses were abolished, it is very necessary, in applying the principle to practical cases, to take account of the possibility of avoiding the heat losses which are supposed to be absent, and of other practical limitations in the working of the actual engine. An immense amount of time and ingenuity has been wasted in striving to realize impossible margins of ideal efficiency, which a close study of the practical conditions would have shown to be illusory. As Carnot remarks at the conclusion of his essay: " Economy of fuel is only one of the conditions a heatengine must satisfy; in many cases it is only secondary, and must often give way to considerations of safety, strength and wearing qualities of the machine, of smallness of space occupied, or of expense in erecting. To know how to appreciate justly in each case the considerations of convenience and economy, to be able to distinguish the essential from the accessory, to balance all fairly, and finally to arrive at the best result by the simplest means, such must be the principal talent of the man called on to direct and coordinate the work of his fellows for the attainment of a useful object of any kind."
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