reversible, that there is not, for instance, any communication between reservoirs of gas or vapour at sensibly different pressures, and that there is no waste of power in friction. If there is equilibrium both mechanical and thermal at every stage of the cycle, the ideal engine will be perfectly reversible. That is to say, all its operations will be exactly reversed as regards transfer of heat and work, when the operations are performed in the reverse order and direction. On this understanding Carnot's principle may be put in a different way, which is often adopted, but is really only the same thing put in different words: The efficiency of a perfectly reversible engine is the maximum possible, and is a function solely of the limits of temperature between which it works. This result depends essentially on the existence of a state of thermal equilibrium defined by equality of temperature, and independent, in the majority of cases, of the state of a body in other respects. In order to apply the principle to the calculation and prediction of results, it is sufficient to determine the manner in which the efficiency depends on the temperature for one particular case, since the efficiency must be the same for all reversible engines.

steam with the same range of temperature, but a very different kind
of cycle. Carnot in making the same calculation did not obtain quite
available were not so accurate. He used the value for the
so good an agreement, because the experimental data at that time
coefficient of expansion, and 267 for the specific heat of air. More-
over, he did not feel justified in assuming, as above, that the difference
of the specific heats was the
same at 100° C. as at the
ordinary temperature of
15° to 20°C., at which it had
been experimentally deter-
mined. He made similar
calculations for the vapour

of

alcohol, which differed slightly from the vapour of water. But the agreement he found was close enough to satisfy him that his theoretical deductions were correct, and that the resulting ratio of work to heat should be the same for all substances at the same temperature.

AXIS OF PRESSURE

17. Carnot's Function. Variation of Efficiency with Temperature. By means of calculations, similar to those given above, Carnot endeavoured to find the amount of motive power obtainable from one unit of heat per degree fall at various temperatures with various substances. The value found above, namely 1.13 kilogrammetre per kilo-calorie per 1° fall, is the value of the efficiency per 1° fall at 100° C. He was able to show that the efficiency per degree fall probably diminished with rise of temperature, but the experimental data at that time were too inconsistent to suggest the true relation. He took as the analytical expression of his principle that the efficiency W/H of a perfect engine taking in heat H at a temperature to C., and rejecting heat at the temperature o° C., must be some function Ft of the temperature t, which would be the same for all substances. The efficiency per degree fall at a temperature t he represented by F't, the derived function of Ft. The function F't would be the same for all substances at the same temperature, but would have different values at different temperatures. In terms of this function, which is generally known as Carnot's function, the results obtained in the previous section might be expressed as follows:

FIG. 5.-Elementary Carnot Cycle for Gas.

16. Experimental Verification of Carnot'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 I gram of water at 100° C., and that the pressure of the steam, namely 760 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 101° C. The water will then acquire the temperature of 101° C., and will absorb 1 gram-calorie of heat. Some waste of motive power occurs here because heat is allowed to pass from ore 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 gram-centimetres or 0.61 of a kilogrammetre. Remove the hot body, and allow the steam to expand further till its pressure is 760 mm. and its temperature has fallen to 100° C. The work which might be done in this "The difference of the specific heats, or the latent heat of exexpansion is less than rooth part of a kilogrammetre, and may be pansion for any substance multiplied by the function F't, is equal neglected for the present purpose. Place the cylinder in contact to the product of the expansion per degree at constant pressure by with the cold body at 100° C., and allow the steam to condense at the increase of pressure per degree at constant volume." this temperature. No work is done on the piston, because there is Since the last two coefficients are the same for all gases if equal equilibrium of pressure, but a quantity of heat equal to the latent volumes are taken, Carnot concluded that: "The difference of the heat of steam at 100° C. is given to the cold body. The water is specific heats at constant pressure and volume is the same for equal now in its initial condition, and the result of the process has been to volumes of all gases at the same temperature and pressure." gain 0.61 of a kilogrammetre of work by allowing 540 gram-calories Taking the expression WRT loger for the whole work done by a of heat to pass from a body at 101° C. to a body at 100° C. by means gas obeying the gaseous laws pv RT in expanding at a temperature of an ideally simple steam-engine. The work obtainable in this T from a volume I (unity) to a volume r, or for a ratio of expansion way from 1000 gram-calories of heat, or I kilo-calorie, would evidently, and putting W'R loger for the work done in a cycle of range 10. be 1.13 kilogrammetre (=0.61 X1000). Carnot obtained the expression for the heat absorbed by a gas in isothermal expansion

Taking the same range of temperature, namely 101° to 100° C., we may perform a similar series of operations with air in the cylinder, instead of water and steam. Suppose the cylinder to contain gramme of air at 100° C. and 760 mm. pressure instead of water. Compress it without loss of heat (adiabatically), so as to raise its temperature to 101° C. Place it in contact with the hot body at 101° C., and allow it to expand at this temperature, absorbing heat from the hot body, until its volume is increased by 4th 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' =.069 calorie. Remove the hot body, and allow the gas to expand further without gain of heat till its temperature falls to 100° C. Compress it at 100° C. to its original volume, abstracting the heat of compression by contact with the cold body at 100° 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 rd of the volume at o° C. and 760 mm., or 2.83 c.c. The increase of pressure BE is H, 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 gram-metre. The heat absorbed at 101° C. was 069 gram-calorie, so that the work obtained is 0782/069 or 1.13 gram-metre per gram-calorie, or 1.13 kilogrammetre per kilogramcalorie. This result is precisely the same as that obtained by using

"The increase of volume of a mixture of liquid and vapour per unit-mass vaporized at any temperature, multiplied by the increase of vapour-pressure per degree, is equal to the product of the function F't by the latent heat of vaporization.

F't R/C (+1)

(3)

A similar result follows

where C and to are unknown constants.
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 prin-
ciples on which the theory of heat rests require most careful examina-
tion. 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 (Journ. de l'éc. polytechn.,

Paris, 1832, 14, p. 153), who also made use of Watt's indicator | definitely, deducing it from the old principle, causa aequat diagram for the first time in discussing physical problems. effectum. Assuming that the sinking of a mercury column by Clapeyron gave the general expressions for the latent heat of a which a gas was compressed was equivalent to the heat set free vapour, and for the latent heat of isothermal expansion of any by the compression, he deduced that the warming of a kilosubstance, in terms of Carnot's function, employing the notation gramme of water 1° C. would correspond to the fall of a weight of the calculus. The expressions he gave are the same in form as of one kilogramme from a height of about 365 metres. But those in use at the present day. He also gave the general Mayer did not adduce any fresh experimental evidence, and expression for Carnot's function, and endeavoured to find its made no attempt to apply his theory to the fundamental variation with temperature; but having no better data, he equations of thermodynamics. It has since been urged that the succeeded no better than Carnot. Unfortunately, in describing experiment of Gay-Lussac (1807), on the expansion of gas from Carnot's cycle, he assumed the caloric theory of heat, and made one globe to another (see above, § 11), was sufficient justification some unnecessary mistakes, which Carnot (who, we now know, for the assumption tacitly involved in Mayer's calculation. was a believer in the mechanical theory) had been very careful But Joule was the first to supply the correct interpretation of to avoid. Clapeyron directs one to compress the gas at the lower this experiment, and to repeat it on an adequate scale with suittemperature in contact with the body B until the heat disengaged able precautions. Joule was also the first to measure directly is equal to that which has been absorbed at the higher temperature.1 the amount of heat liberated by the compression of a gas, and to He assumes that the gas at this point contains the same quantity prove that heat was not merely rendered latent, but disappeared of heat as it contained in its original state at the higher tempera- altogether as heat, when a gas did work in expansion. ture, 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.-According to 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 078, 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.

66

The most obvious case of the production of work from heat is in the expansion of a gas or vapour, which served in the first instance as a means of calculating the ratio of equivalence, on the assumption that all the heat which disappeared had been transformed into work and had not merely become latent. Marc Séguin, 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 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

a

1 It was for this reason that Professor W. Thomson (Lord Kelvin) stated (Phil. Mag., 1852, 4) that "Carnot's original demonstration utterly fails," and that he introduced the "corrections" attributed to James Thomson and Clerk Maxwell respectively. In reality Carnot's original demonstration requires no correction.

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., 1840-1843. 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 C2R 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 and 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 Magneto-Electricity, and on the Mechanical Value of Heat" (Brit. Assoc., 1843; Phil. Mag., 23, p. 263). In this paper he showed that the heat produced by currents generated by magneto-electric induction followed the same law as voltaic currents. By a simple and ingenious 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 1° F. (1 B.Th.U.), as found by this method, was 838 foot-pounds. In a note added to the paper he states that he found the value 770 foot-pounds by the more direct method of forcing water through fine tubes. In a paper On the Changes of Temperature 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 foot-pounds for the mechanical equivalent of 1 B.Th.U. He also showed that there was no appreciable absorption of heat when air was allowed to expand in such a manner as not to develop mechanical power, and he pointed out that the mechanical equivalent of heat could not be satisfactorily deduced from

the relations of the specific heats, because the knowledge of though probably correct, contain an undetermined function the specific heats of gases at that time was of so uncertain a (Carnot's F't, Clapeyron's 1/C) of the temperature. He detercharacter. He attributed most weight to his later determina- mines the value of this function to be J/T by assuming, with tions of the mechanical equivalent made by the direct method Séguin and Mayer, that the work done in the isothermal expanof friction of liquids. He showed that the results obtained with sion of a gas is a measure of the heat absorbed. From the then different liquids, water, mercury and sperm oil, were the same, accepted value 078 of the difference of the specific heats of air, namely, 782 foot-pounds; and finally repeating the method with | he finds the numerical value of J to be 374 kilogrammetres per water, using all the precautions and improvements which his ex-kilo-calorie. Assuming the heat equivalent of the work to remain perience had suggested, he obtained the value 772 foot-pounds, in the gas, he obtains expressions similar to Clapeyron's for the which was accepted universally for many years, and has only total heat and the specific heats. In consequence of this assumprecently required alteration on account of the more exact defini- tion, the formulae he obtained for adiabatic expansion were tion of the heat unit, and the standard scale of temperature (see necessarily wrong, but no data existed at that time for testing CALORIMETRY). The great value of Joule's work for the general them. In applying his formulae to vapours, he obtained an establishment of the principle of the conservation of energy expression for the saturation-pressure of steam, which agreed with lay in the variety and completeness of the experimental evidence the empirical formula of Roche, and satisfied other experimental he adduced. It was not sufficient to find the relation between data on the supposition that the co-efficient of expansion of steam heat and mechanical work or other forms of energy in one was 00423, and its specific heat 1.69-values which are now particular case. It was necessary to show that the same relation known to be impossible, but which appeared at the time to give held in all cases which could be examined experimentally, and a very satisfactory explanation of the phenomena. 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 concluding 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 thermo-dynamics, 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:

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,

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 energy-principle. 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 he felt justified by direct experimental evidence in reasserting of air (1845) already referred to. At the conclusion of this paper definitely the hypothesis of Séguin (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 experi ments, but did not see his way out of the difficulty. He there fore adopted Carnot's principle provisionally, and proceeded to calculate a table of values of Carnot's function F't, from the values of the total-heat and vapour-pressure of steam then recently determined by Regnault (Mémoires 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 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

28% of 426, the mechanical equivalent of the kilo-calorie in | it directly by internal combustion. This avoids the limitation kilogrammetres. Carnot pointed out that the fall of 120° C. utilized in the steam-engine 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 over-estimate, but was surprisingly close, considering the scanty data at his disposal.

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.
Air-engines 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 gas-engines of E. Lenoir (1860) 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 1-loger/(r-1), where r is the ratio
of the maximum temperature ' to the initial temperature 0o.
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 ' down to 0°, and then to compress it
at 0° to its initial volume. If the rise of temperature in com-
bustion were 2300° C., and the initial temperature were 。° 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 10%.
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: (1) 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 considera-
tions of temperature as by limitations of volume. It is impractic-
able before combustion at constant volume to compress a rich
mixture to much less than th of its initial volume, and, for
mechanical simplicity, the range of expansion is made equal
to that of compression.
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′′)/0′, 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 0'0" is

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 40° 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 ' is the maximum temperature of combustion as above estimated, 0" the temperature of the boiler, and 0° that of the condenser. Of the whole heat supplied by combustion | represented by the rise of temperature 0'-0o, the fraction (0′-0′′)/(0′−0o) is the maximum that could be supplied to the boiler, the fraction (0′′ −0o)/(0′ −0o) 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 steam-engine, is to find what must be the temperature @" 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 " must be the geometric mean of ' and 0o, or 0′′ = √0'00. Taking '-00=2300° C., and 00313° Abs. as before, we find @"= 903° Abs. or 630° 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 89%. With the boiler at 160° 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 1260 lb per sq. in., which is occasionally surpassed in modern flash-boilers, the possible efficiency would be 40%. The waste heat from the boiler, supposed perfectly efficient, would be in this case 11%, 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 steam-engine 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 steam-engines 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 1260 lb initial pressure, with a perfectly efficient boiler and complete expansion, and with an ideal engine which, where r is the given ratio of expansion or compression, 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 heat-engine lies in the possibility of imparting heat to

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 y=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