Reflections on the Motive Power of Heat/Appendix B

 

 

APPENDIX B.

CARNOT'S FOOT-NOTES.

 

Note A.—The objection may perhaps be raised here, that perpetual motion, demonstrated to be impossible by mechanical action alone, may possibly not be so if the power either of heat or electricity be exerted; but is it possible to conceive the phenomena of heat and electricity as due to anything else than some kind of motion of the body, and as such should they not be subjected to the general laws of mechanics? Do we not know besides, à posteriori, that all the attempts made to produce perpetual motion by any means whatever have been fruitless?—that we have never succeeded in producing a motion veritably perpetual, that is, a motion which will continue forever without alteration in the bodies set to work to accomplish it? The electromotor apparatus (the pile of Volta) has sometimes been regarded as capable of producing perpetual motion; attempts have been made to realize this idea by constructing dry piles said to be unchangeable; but however it has been done, the apparatus has always exhibited sensible deteriorations when its action has been sustained for a time with any energy.

The general and philosophic acceptation of the words perpetual motion should include not only a motion susceptible of indefinitely continuing itself after a first impulse received, but the action of an apparatus, of any construction whatever, capable of creating motive power in unlimited quantity, capable of starting from rest all the bodies of nature if they should be found in that condition, of overcoming their inertia; capable, finally, of finding in itself the forces necessary to move the whole universe, to prolong, to accelerate incessantly, its motion. Such would be a veritable creation of motive power. If this were a possibility, it would be useless to seek in currents of air and water or in combustibles this motive power. We should have at our disposal an inexhaustible source upon which we could draw at will.

Note B.—The experimental facts which best prove the change of temperature of gases by compression or dilatation are the following:

(1) The fall of the thermometer placed under the receiver of a pneumatic machine in which a vacuum has been produced. This fall is very sensible on the Bréguet thermometer: it may exceed 40° or 50°. The mist which forms in this case seems to be due to the condensation of the watery vapor caused by the cooling of the air.

(2) The inflammation of German tinder in the so-called pneumatic tinder-boxes; which are, as we know, little pump-chambers in which the air is rapidly compressed.

(3) The fall of a thermometer placed in a space where the air has been first compressed and then allowed to escape by the opening of a cock.

(4) The results of experiments on the velocity of sound. M. de Laplace has shown that, in order to secure results accurately by theory and computation, it is necessary to assume the heating of the air by sudden compression.

The only fact which may be adduced in opposition to the above is an experiment of MM. Gay-Lussac and Welter, described in the Annales de Chimie et de Physique. A small opening having been made in a large reservoir of compressed air, and the ball of a thermometer having been introduced into the current of air which passes out through this opening, no sensible fall of the temperature denoted by the thermometer has been observed.

Two explanations of this fact may be given: (1) The striking of the air against the walls of the opening by which it escapes may develop heat in observable quantity. (2) The air which has just touched the bowl of the thermometer possibly takes again by its collision with this bowl, or rather by the effect of the détour which it is forced to make by its rencounter, a density equal to that which it had in the receiver,—much as the water of a current rises against a fixed obstacle, above its level.

The change of temperature occasioned in the gas by the change of volume may be regarded as one of the most important facts of Physics, because of the numerous consequences which it entails, and at the same time as one of the most difficult to illustrate, and to measure by decisive experiments. It seems to present in some respects singular anomalies.

Is it not to the cooling of the air by dilatation that the cold of the higher regions of the atmosphere must be attributed? The reasons given heretofore as an explanation of this cold are entirely insufficient; it has been said that the air of the elevated regions receiving little reflected heat from the earth, and radiating towards celestial space, would lose caloric, and that this is the cause of its cooling; but this explanation is refuted by the fact that, at an equal height, cold reigns with equal and even more intensity on the elevated plains than on the summit of the mountains, or in those portions of the atmosphere distant from the sun.

Note C.—We see no reason for admitting, à priori the constancy of the specific heat of bodies at different temperatures, that is, to admit that equal quantities of heat will produce equal increments of temperature, when this body changes neither its state nor its density; when, for example, it is an elastic fluid enclosed in a fixed space. Direct experiments on solid and liquid bodies have proved that between zero and 100°, equal increments in the quantities of heat would produce nearly equal increments of temperature. But the more recent experiments of MM. Dulong and Petit (see Annales de Chimie et de Physique, February, March, and April, 1818) have shown that this correspondence no longer continues at temperatures much above 100°, whether these temperatures be measured on the mercury thermometer or on the air thermometer.

Not only do the specific heats not remain the same at different temperatures, but, also, they do not preserve the same ratios among themselves, so that no thermometric scale could establish the constancy of all the specific heats. It would have been interesting to prove whether the same irregularities exist for gaseous substances, but such experiments presented almost insurmountable difficulties.

The irregularities of specific heats of solid bodies might have been attributed, it would seem, to the latent heat employed to produce a beginning of fusion—a softening which occurs in most bodies long before complete fusion. We might support this opinion by the following statement: According to the experiments of MM. Dulong and Petit, the increase of specific heat with the temperature is more rapid in solids than in liquids, although the latter possess considerably more dilatability. The cause of irregularity just referred to, if it is real, would disappear entirely in gases.

Note D.—In order to determine the arbitrary constants A, B, A', B', in accordance with the results in M. Dalton's table, we must begin by computing the volume of the vapor as determined by its pressure and temperature,—a result which is easily accomplished by reference to the laws of Mariotte and Gay-Lussac, the weight of the vapor being fixed.

The volume will be given by the equation

$v=c{\frac {267+t}{p}}$,

in which v is this volume, t the temperature, p the pressure, and c a constant quantity depending on the weight of the vapor and on the units chosen. We give here the table of the volumes occupied by a gramme of vapor formed at different temperatures, and consequently under different pressures.

t or degrees Centigrade.	p or tension of the vapor expressed in millimetres of mercury.	v or volume of a gramme of vapor expressed in litres.
°	mm.	lit.
0	5.060	185.0
20	17.32	58.2
40	53.00	20.4
60	144.6	7.96
80	352.1	3.47
100	760.0	1.70

The first two columns of this table are taken from the Traité de Physique of M. Biot (vol. i., p. 272 and 531). The third is calculated by means of the above formula, and in accordance with the result of experiment, indicating that water vaporized under atmospheric pressure occupies a space 1700 times as great as in the liquid state.

By using three numbers of the first column and three corresponding numbers of the third column, we can easily determine the constants of our equation

$t={\frac {A+B\log v}{A'+B'\log v}}.$

We will not enter into the details of the calculation necessary to determine these quantities. It is sufficient to say that the following values,

A = 2268,	A' = 19.64,
B = -1000,	B' = 3.30,

satisfy fairly well the prescribed conditions, so that the equation

$t={\frac {2268-1000\log v}{19.64+3.30\log v}}$

expresses very nearly the relation which exists between the volume of the vapor and its temperature. We may remark here that the quantity B' is positive and very small, which tends to confirm this proposition—that the specific heat of an elastic fluid increases with the volume, but follows a slow progression.

Note E.—Were we to admit the constancy of the specific heat of a gas when its volume does not change, but when its temperature varies, analysis would show a relation between the motive power and the thermometric degree. We will show how this is, and this will also give us occasion to show how some of the propositions established above should be expressed in algebraic language.

Let r be the quantity of motive power produced by the expansion of a given quantity of air passing from the volume of one litre to the volume of v litres under constant temperature. If v increases by the infinitely small quantity dv, r will increase by the quantity dr, which, according to the nature of motive power, will be equal to the increase dv of volume multiplied by the expansive force which the elastic fluid then possesses; let p be this expansive force. We should have the equation

$dr=pdv$

(1)

Let us suppose the constant temperature under which the dilatation takes place equal to t degrees Centigrade. If we call q the elastic force of the air occupying the volume 1 litre at the same temperature t we shall have, according to the law of Mariotte,

${\frac {v}{1}}={\frac {q}{p}},$ whence $p={\frac {q}{v}}.$

If now P is the elastic force of this same air at the constant volume 1, but at the temperature zero, we shall have, according to the rule of M. Gay-Lussac,

$q=P+P{\frac {t}{267}}={\frac {P}{267}}{(267+t)};$

whence

$q=p={\frac {P}{267}}{\frac {267+t}{v}}.$

If, to abridge, we call N the quantity ${\frac {P}{267}}$, the equation would become

$p=N{\frac {t+267}{v}};$

whence we deduce, according to equation (1),

$dr=N{\frac {t+267}{v}}dv.$

Regarding t as constant, and taking the integral of the two numbers, we shall have

$r=N(t+267)\log v+C.$

If we suppose r = 0 when v = 1, we shall have C = 0; whence

$r=N(t+267)\log v.$

(2)

This is the motive power produced by the expansion of the air which, under the temperature t, has passed from the volume 1 to the volume v. If instead of working at the temperature t we work in precisely the same manner at the temperature t + dt, the power developed will be

$r+\delta r=N(t+dt+267)\log v.$

Subtracting equation (2), we have

$\delta r=N\log vdt.$

(3)

Let e be the quantity of heat employed to maintain the temperature of the gas constant during its dilatation. According to the reasoning of page 69, δr will be the power developed by the fall of the quantity e of heat from the degree t + td to the degree t. If we call u the motive power developed by the fall of unity of heat from the degree t to the degree zero, as, according to the general principle established page 68, this quantity u ought to depend solely on t, it could be represented by the function Ft, whence u = Ft.

When t is increased it becomes t + td, u becomes u + du; whence

$u+du=F(t+dt).$

Subtracting the preceding equation, we have

$du=F(t+dt)-Ft=F'tdt.$

This is evidently the quantity of motive power produced by the fall of unity of heat from the temperature t + dt to the temperature t.

If the quantity of heat instead of being a unit had been e, its motive power produced would have had for its value

$edu=eF'tdt.$

(4)

But edu is the same thing as δr; both are the power developed by the fall of the quantity e of heat from the temperature t + dt to the temperature t; consequently,

$edu=\delta r,$

and from equations (3), (4),

$eF'tdt=N\log vdt;$

or, dividing by F'tdt,

$e={\frac {N}{F't}}\log v=T\log v.$

Calling T the fraction ${\frac {N}{F't}}$ which is a function of t only, the equation

$e=T\log v$

is the analytical expression of the law stated pp. 80, 81. It is common to all gases, since the laws of which we have made use are common to all.

If we call s the quantity of heat necessary to change the air that we have employed from the volume 1 and from the temperature zero to the volume v and to the temperature t, the difference between s and e will be the quantity of heat required to bring the air at the volume 1 from zero to t. This quantity depends on t alone; we will call it U. It will be any function whatever of t. We shall have

$s=e+U=T\log v+U.$

If we differentiate this equation with relation to t alone, and if we represent it by T' and U', the differential coefficients of T and U, we shall get

${\frac {ds}{dt}}=T'\log v+U';$

(5)

${\frac {ds}{dt}}$ is simply the specific heat of the gas under constant volume, and our equation (1) is the analytical expression of the law stated on page 86.

If we suppose the specific heat constant at all temperatures (hypothesis discussed above, page 92), the quantity ${\frac {ds}{dt}}$ will be independent of t; and in order to satisfy equation (5) for two particular values of v, it will be necessary that T' and U' be independent of t; we shall then have $T'=C$, a constant quantity. Multiplying T' and C by dt, and taking the integral of both, we find

$T=Ct+C_{1};$

but as $T={\frac {N}{F't}}$, we have

$F't={\frac {N}{T}}={\frac {N}{Ct+C_{1}}}.$

Multiplying both by dt and integrating, we have

$Ft={\frac {N}{C}}\log(Ct+C_{1})+C_{2};$

or changing arbitrary constants, and remarking further that Ft is 0 when t = 0°,

$Ft=A\log {\bigg (}1+{\frac {t}{B}}{\bigg )}.$

(6)

The nature of the function Ft would be thus determined, and we would thus be able to estimate the motive power developed by any fall of heat. But this latter conclusion is founded on the hypothesis of the constancy of the specific heat of a gas which does not change in volume—an hypothesis which has not yet been sufficiently verified by experiment. Until there is fresh proof, our equation (6) can be admitted only throughout a limited portion of the thermometric scale.

In equation (5), the first member represents, as we have remarked, the specific heat of the air occupying the volume v. Experiment having shown that this heat varies little in spite of the quite considerable changes of volume, it is necessary that the coefficient T' of log v should be a very small quantity. If we consider it nothing, and, after having multiplied by dt the equation

$T'=0,$

we take the integral of it, we find

constant quantity;  $T=C,$   constant quantity;

but

$T={\frac {N}{F't}},$

whence

$F't={\frac {N}{T}}={\frac {N}{C}}=A;$

whence we deduce finally, by a second integration,

$Ft=At+B.$

As Ft = 0 when t = 0, B is 0; thus

$Ft=At;$

that is, the motive power produced would be found to be exactly proportional to the fall of the caloric. This is the analytical translation of what was stated on page 98.

Note F.—M. Dalton believed that he had discovered that the vapors of different liquids at equal thermometric distances from the boiling-point possess equal tensions; but this law is not precisely exact; it is only approximate. It is the same with the law of the proportionality of the latent heat of vapors with their densities (see Extracts from a Mémoire of M. C. Despretz, Annales de Chimie et de Physique, t. xvi. p. 105, and t. xxiv. p. 323). Questions of this nature are closely connected with those of the motive power of heat. Quite recently MM. H. Davy and Faraday, after having conducted a series of elegant experiments on the liquefaction of gases by means of considerable pressure, have tried to observe the changes of tension of these liquefied gases on account of slight changes of temperature. They have in view the application of the new liquids to the production of motive power (see Annales de Chimie et de Physique, January, 1824, p. 80).

According to the above-mentioned theory, we can foresee that the use of these liquids would present no advantages relatively to the economy of heat. The advantages would be found only in the lower temperature at which it would be possible to work, and in the sources whence, for this reason, it would become possible to obtain caloric.

Note G.—This principle, the real foundation of the theory of steam-engines, was very clearly developed by M. Clement in a memoir presented to the Academy of Sciences several years ago. This Memoir has never been printed, and I owe the knowledge of it to the kindness of the author. Not only is the principle established therein, but it is applied to the different systems of steam-engines actually in use. The motive power of each of them is estimated therein by the aid of the law cited page 92, and compared with the results of experiment.

The principle in question is so little known or so poorly appreciated, that recently Mr. Perkins, a celebrated mechanician of London, constructed a machine in which steam produced under the pressure of 35 atmospheres—a pressure never before used—is subjected to very little expansion of volume, as any one with the least knowledge of this machine can understand. It consists of a single cylinder of very small dimensions, which at each stroke is entirely filled with steam, formed under the pressure of 35 atmospheres. The steam produces no effect by the expansion of its volume, for no space is provided in which the expansion can take place. It is condensed as soon as it leaves the small cylinder. It works therefore only under a pressure of 35 atmospheres, and not, as its useful employment would require, under progressively decreasing pressures. The machine of Mr. Perkins seems not to realize the hopes which it at first awakened. It has been asserted that the economy of coal in this engine was 9/10 above the best engines of Watt, and that it possessed still other advantages (see Annales de Chimie et de Physique, April, 1823, p. 429). These assertions have not been verified. The engine of Mr. Perkins is nevertheless a valuable invention, in that it has proved the possibility of making use of steam under much higher pressure than previously, and because, being easily modified, it may lead to very useful results.

Watt, to whom we owe almost all the great improvements in steam-engines, and who brought these engines to a state of perfection difficult even now to surpass, was also the first who employed steam under progressively decreasing pressures. In many cases he suppressed the introduction of the steam into the cylinder at a half, a third, or a quarter of the stroke. The piston completes its stroke, therefore, under a constantly diminishing pressure. The first engines working on this principle date from 1778. Watt conceived the idea of them in 1769, and took out a patent in 1782.

We give here the Table appended to Watt's patent. He supposed the steam introduced into the cylinder during the first quarter of the stroke of the piston; then, dividing this stroke into twenty parts, he calculated the mean pressure as follows:

Portions of the descent from the top of the cylinder.				Decreasing pressure of the steam, the entire pressure being 1.
	0.05	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$	Steam arriving freely from the boiler.	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$	1.000	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$	Total pressure.
	0.10				1.000
	0.15				1.000
	0.20				1.000
Quarter.....	0.25				1.000
	0.30	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$	The steam being cut off and the descent taking place only by expansion	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$	0.830
	0.35				0.714
	0.40				0.625
	0.45				0.555	$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	Half original pressure.
Half........	0.50				0.500
	0.55				0.454
	0.60				0.417
	0.65				0.385
	0.70				0.375
	0.75				0.333		One third.
	0.80				0.312
	0.85				0.294
	0.90				0.277
	0.95				0.262
Bottom of cylinder...	1.00				0.250		Quarter.
			Total,		11.583
Mean pressure ${\dfrac {11.583}{20}}$ = 0.579.

 

On which he remarked, that the mean pressure is more than half the original pressure; also that in employing a quantity of steam equal to a quarter, it would produce an effect more than half.

Watt here supposed that steam follows in its expansion the law of Mariotte, which should not be considered exact, because, in the first place, the elastic fluid in dilating falls in temperature, and in the second place there is nothing to prove that a part of this fluid is not condensed by its expansion. Watt should also have taken into consideration the force necessary to expel the steam which remains after condensation, and which is found in quantity as much greater as the expansion of the volume has been carried further. Dr. Robinson has supplemented the work of Watt by a simple formula to calculate the effect of the expansion of steam, but this formula is found to have the same faults that we have just noticed. It has nevertheless been useful to constructors by furnishing them approximate data practically quite satisfactory. We have considered it useful to recall these facts because they are little known, especially in France. These engines have been built after the models of the inventors, but the ideas by which the inventors were originally influenced have been but little understood. Ignorance of these ideas has often led to grave errors. Engines originally well conceived have deteriorated in the hands of unskilful builders, who, wishing to introduce in them improvements of little value, have neglected the capital considerations which they did not know enough to appreciate.

Note H.—The advantage in substituting two cylinders for one is evident. In a single cylinder the impulsion of the piston would be extremely variable from the beginning to the end of the stroke. It would be necessary for all the parts by which the motion is transmitted to be of sufficient strength to resist the first impulsion, and perfectly fitted to avoid the abrupt movements which would greatly injure and soon destroy them. It would be especially on the working beam, on the supports, on the crank, on the connecting-rod, and on the first gear-wheels that the unequal effort would be felt, and would produce the most injurious effects. It would be necessary that the steam-cylinder should be both sufficiently strong to sustain the highest pressure, and with a large enough capacity to contain the steam after its expansion of volume, while in using two successive cylinders it is only necessary to have the first sufficiently strong and of medium capacity,—which is not at all difficult,—and to have the second of ample dimensions, with moderate strength.

Double-cylinder engines, although founded on correct principles, often fail to secure the advantages expected from them. This is due principally to the fact that the dimensions of the different parts of these engines are difficult to adjust, and that they are rarely found to be in correct proportion. Good models for the construction of double-cylinder engines are wanting, while excellent designs exist for the construction of engines on the plan of Watt. From this arises the diversity that we see in the results of the former, and the great uniformity that we have observed in the results of the latter.

Note I.—Among the attempts made to develop the motive power of heat by means of atmospheric air, we should mention those of MM. Niepce, which were made in France several years ago, by means of an apparatus called by the inventors a pyréolophore. The apparatus was made thus: There was a cylinder furnished with a piston, into which the atmospheric air was introduced at ordinary density. A very combustible material, reduced to a condition of extreme tenuity, was thrown into it, remained a moment in suspension in the air, and then flame was applied. The inflammation produced very nearly the same effect as if the elastic, fluid had been a mixture of air and combustible gas, of air and carburetted hydrogen gas, for example. There was a sort of explosion, and a sudden dilatation of the elastic fluid—a dilatation that was utilized by making it act upon the piston. The latter may have a motion of any amplitude whatever, and the motive power is thus realized. The air is next renewed, and the operation repeated.

This machine, very ingenious and interesting, especially on account of the novelty of its principle, fails in an essential point. The material used as a combustible (it was the dust of Lycopodium, used to produce flame in our theatres) was so expensive, that all the advantage was lost through that cause; and unfortunately it was difficult to employ a combustible of moderate price, since a very finely powdered substance was required which would burn quickly, spread rapidly, and leave little or no ash.

Instead of working as did MM. Niepce, it would seem to us preferable to compress the air by means of pumps, to make it traverse a perfectly closed furnace into which the combustible had been introduced in small portions by a mechanism easy of conception, to make it develop its action in a cylinder with a piston, or in any other variable space; finally, to throw it out again into the atmosphere, or even to make it pass under a steam-boiler in order to utilize the temperature remaining.

The principal difficulties that we should meet in this mode of operation would be to enclose the furnace in a sufficiently strong envelope, to keep the combustion meanwhile in the requisite state, to maintain the different parts of the apparatus at a moderate temperature, and to prevent rapid abrasion of the cylinder and of the piston. These difficulties do not appear to be insurmountable.

There have been made, it is said, recently in England, successful attempts to develop motive power through the action of heat on atmospheric air. We are entirely ignorant in what these attempts have consisted—if indeed they have really been made.

Note J.—The result given here was furnished by an engine whose large cylinder was 45 inches in diameter and 7 feet stroke. It is used in one of the mines of Cornwall called Wheal Abraham. This result should be considered as somewhat exceptional, for it was only temporary, continuing but a single month. Thirty millions of lbs. raised one English foot per bushel of coal of 88 lbs. is generally regarded as an excellent result for steam-engines. It is sometimes attained by engines of the Watt type, but very rarely surpassed. This latter product amounts, in French measures, to 104,000 kilograms raised one metre per kilogram of coal consumed.

According to what is generally understood by one horse-power, in estimating the duty of steam-engines, an engine of ten horse-power should raise per second 10 × 75 kilograms, or 750 kilograms, to a height of one metre, or more, per hour; 750 × 3600 = 2,700,000 kilograms to one metre. If we suppose that each kilogram of coal raised to this height 104,000 kilograms, it will be necessary, in order to ascertain how much coal is burnt in one hour by our ten-horse-power engine, to divide 2,700,000 by 104,000, which gives ${\tfrac {2700}{104}}$ = 26 kilograms. Now it is seldom that a ten-horse-power engine consumes less than 26 kilograms of coal per hour.