Every one knows that heat can produce motion. That it possesses vast motive power no one can doubt, in these days when the steam-engine is everywhere so well known.
To heat also are due the vast movements which take place on the earth. It causes the agitations of the atmosphere, the ascension of clouds, the fall of rain and of meteors, the currents of water which channel the surface of the globe, and of which man has thus far employed but a small portion. Even earthquakes and volcanic eruptions are the result of heat.
From this immense reservoir we may draw the moving force necessary for our purposes. Nature, in providing us with combustibles on all sides, has given us the power to produce, at all times and in all places, heat and the impelling power which is the result of it. To develop this power, to appropriate it to our uses, is the object of heat-engines.
The study of these engines is of the greatest interest, their importance is enormous, their use is continually increasing, and they seem destined to produce a great revolution in the civilized world.
Already the steam-engine works our mines, impels our ships, excavates our ports and our rivers, forges iron, fashions wood, grinds grains, spins and weaves our cloths, transports the heaviest burdens, etc. It appears that it must some day serve as a universal motor, and be substituted for animal power, waterfalls, and air currents.
Over the first of these motors it has the advantage of economy, over the two others the inestimable advantage that it can be used at all times and places without interruption.
If, some day, the steam-engine shall be so perfected that it can be set up and supplied with fuel at small cost, it will combine all desirable qualities, and will afford to the industrial arts a range the extent of which can scarcely be predicted. It is not merely that a powerful and convenient motor that can be procured and carried anywhere is substituted for the motors already in use, but that it causes rapid extension in the arts in which it is applied, and can even create entirely new arts.
The most signal service that the steam-engine has rendered to England is undoubtedly the revival of the working of the coal-mines, which had declined, and threatened to cease entirely, in consequence of the continually increasing difficulty of drainage, and of raising the coal.[5] We should rank second the benefit to iron manufacture, both by the abundant supply of coal substituted for wood just when the latter had begun to grow scarce, and by the powerful machines of all kinds, the use of which the introduction of the steam-engine has permitted or facilitated.
Iron and heat are, as we know, the supporters, the bases, of the mechanic arts. It is doubtful if there be in England a single industrial establishment of which the existence does not depend on the use of these agents, and which does not freely employ them. To take away to-day from England her steam-engines would be to take away at the same time her coal and iron. It would be to dry up all her sources of wealth, to ruin all on which her prosperity depends, in short, to annihilate that colossal power. The destruction of her navy, which she considers her strongest defence, would perhaps be less fatal.
The safe and rapid navigation by steamships may be regarded as an entirely new art due to the steam-engine. Already this art has permitted the establishment of prompt and regular communications across the arms of the sea, and on the great rivers of the old and new continents. It has made it possible to traverse savage regions where before we could scarcely penetrate. It has enabled us to carry the fruits of civilization over portions of the globe where they would else have been wanting for years. Steam navigation brings nearer together the most distant nations. It tends to unite the nations of the earth as inhabitants of one country. In fact, to lessen the time, the fatigues, the uncertainties, and the dangers of travel—is not this the same as greatly to shorten distances?[6]
The discovery of the steam-engine owed its birth, like most human inventions, to rude attempts which have been attributed to different persons, while the real author is not certainly known. It is, however, less in the first attempts that the principal discovery consists, than in the successive improvements which have brought steam-engines to the condition in which we find them to-day. There is almost as great a distance between the first apparatus in which the expansive force of steam was displayed and the existing machine, as between the first raft that man ever made and the modern vessel.
If the honor of a discovery belongs to the nation in which it has acquired its growth and all its developments, this honor cannot be here refused to England. Savery, Newcomen, Smeaton, the famous Watt, Woolf, Trevithick, and some other English engineers, are the veritable creators of the steam-engine. It has acquired at their hands all its successive degrees of improvement. Finally, it is natural that an invention should have its birth and especially be developed, be perfected, in that place where its want is most strongly felt.
Notwithstanding the work of all kinds done by steam-engines, notwithstanding the satisfactory condition to which they have been brought to-day, their theory is very little understood, and the attempts to improve them are still directed almost by chance.
The question has often been raised whether the motive power of heat[7] is unbounded, whether the possible improvements in steam-engines have an assignable limit,—a limit which the nature of things will not allow to be passed by any means whatever; or whether, on the contrary, these improvements may be carried on indefinitely. We have long sought, and are seeking to-day, to ascertain whether there are in existence agents preferable to the vapor of water for developing the motive power of heat; whether atmospheric air, for example, would not present in this respect great advantages. We propose now to submit these questions to a deliberate examination.
The phenomenon of the production of motion by heat has not been considered from a sufficiently general point of view. We have considered it only in machines the nature and mode of action of which have not allowed us to take in the whole extent of application of which it is susceptible. In such machines the phenomenon is, in a way, incomplete. It becomes difficult to recognize its principles and study its laws.
In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent. It is necessary to establish principles applicable not only to steam-engines[8] but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated.
Machines which do not receive their motion from heat, those which have for a motor the force of men or of animals, a waterfall, an air-current, etc., can be studied even to their smallest details by the mechanical theory. All cases are foreseen, all imaginable movements are referred to these general principles, firmly established, and applicable under all circumstances. This is the character of a complete theory. A similar theory is evidently needed for heat-engines. We shall have it only when the laws of Physics shall be extended enough, generalized enough, to make known beforehand all the effects of heat acting in a determined manner on any body.
We will suppose in what follows at least a superficial knowledge of the different parts which compose an ordinary steam-engine; and we consider it unnecessary to explain what are the furnace, boiler, steam-cylinder, piston, condenser, etc.
The production of motion in steam-engines is always accompanied by a circumstance on which we should fix our attention. This circumstance is the re-establishing of equilibrium in the caloric; that is, its passage from a body in which the temperature is more or less elevated, to another in which it is lower. What happens in fact in a steam-engine actually in motion? The caloric developed in the furnace by the effect of the combustion traverses the walls of the boiler, produces steam, and in some way incorporates itself with it. The latter carrying it away, takes it first into the cylinder, where it performs some function, and from thence into the condenser, where it is liquefied by contact with the cold water which it encounters there. Then, as a final result, the cold water of the condenser takes possession of the caloric developed by the combustion. It is heated by the intervention of the steam as if it had been placed directly over the furnace. The steam is here only a means of transporting the caloric. It fills the same office as in the heating of baths by steam, except that in this case its motion is rendered useful.
We easily recognize in the operations that we have just described the re-establishment of equilibrium in the caloric, its passage from a more or less heated body to a cooler one. The first of these bodies, in this case, is the heated air of the furnace; the second is the condensing water. The re-establishment of equilibrium of the caloric takes place between them, if not completely, at least partially, for on the one hand the heated air, after having performed its function, having passed round the boiler, goes out through the chimney with a temperature much below that which it had acquired as the effect of combustion; and on the other hand, the water of the condenser, after having liquefied the steam, leaves the machine with a temperature higher than that with which it entered.
The production of motive power is then due in steam-engines not to an actual consumption of caloric, but _to its transportation from a warm body to a cold body_, that is, to its re-establishment of equilibrium—an equilibrium considered as destroyed by any cause whatever, by chemical action such as combustion, or by any other. We shall see shortly that this principle is applicable to any machine set in motion by heat.
According to this principle, the production of heat alone is not sufficient to give birth to the impelling power: it is necessary that there should also be cold; without it, the heat would be useless. And in fact, if we should find about us only bodies as hot as our furnaces, how can we condense steam? What should we do with it if once produced? We should not presume that we might discharge it into the atmosphere, as is done in some engines;[9] the atmosphere would not receive it. It does receive it under the actual condition of things, only because it fulfils the office of a vast condenser, because it is at a lower temperature; otherwise it would soon become fully charged, or rather would be already saturated.[10]
Wherever there exists a difference of temperature, wherever it has been possible for the equilibrium of the caloric to be re-established, it is possible to have also the production of impelling power. Steam is a means of realizing this power, but it is not the only one. All substances in nature can be employed for this purpose, all are susceptible of changes of volume, of successive contractions and dilatations, through the alternation of heat and cold. All are capable of overcoming in their changes of volume certain resistances, and of thus developing the impelling power. A solid body—a metallic bar for example—alternately heated and cooled increases and diminishes in length, and can move bodies fastened to its ends. A liquid alternately heated and cooled increases and diminishes in volume, and can overcome obstacles of greater or less size, opposed to its dilatation. An aeriform fluid is susceptible of considerable change of volume by variations of temperature. If it is enclosed in an expansible space, such as a cylinder provided with a piston, it will produce movements of great extent. Vapors of all substances capable of passing into a gaseous condition, as of alcohol, of mercury, of sulphur, etc., may fulfil the same office as vapor of water. The latter, alternately heated and cooled, would produce motive power in the shape of permanent gases, that is, without ever returning to a liquid state. Most of these substances have been proposed, many even have been tried, although up to this time perhaps without remarkable success.
We have shown that in steam-engines the motive power is due to a re-establishment of equilibrium in the caloric; this takes place not only for steam-engines, but also for every heat-engine—that is, for every machine of which caloric is the motor. Heat can evidently be a cause of motion only by virtue of the changes of volume or of form which it produces in bodies.
These changes are not caused by uniform temperature, but rather by alternations of heat and cold. Now to heat any substance whatever requires a body warmer than the one to be heated; to cool it requires a cooler body. We supply caloric to the first of these bodies that we may transmit it to the second by means of the intermediary substance. This is to re-establish, or at least to endeavor to re-establish, the equilibrium of the caloric.
It is natural to ask here this curious and important question: Is the motive power of heat invariable in quantity, or does it vary with the agent employed to realize it as the intermediary substance, selected as the subject of action of the heat?
It is clear that this question can be asked only in regard to a given quantity of caloric,[11] the difference of the temperatures also being given. We take, for example, one body _A_ kept at a temperature of 100° and another body _B_ kept at a temperature of 0°, and ask what quantity of motive power can be produced by the passage of a given portion of caloric (for example, as much as is necessary to melt a kilogram of ice) from the first of these bodies to the second. We inquire whether this quantity of motive power is necessarily limited, whether it varies with the substance employed to realize it, whether the vapor of water offers in this respect more or less advantage than the vapor of alcohol, of mercury, a permanent gas, or any other substance. We will try to answer these questions, availing ourselves of ideas already established.
We have already remarked upon this self-evident fact, or fact which at least appears evident as soon as we reflect on the changes of volume occasioned by heat: _wherever there exists a difference of temperature, motive power can be produced_. Reciprocally, wherever we can consume this power, it is possible to produce a difference of temperature, it is possible to occasion destruction of equilibrium in the caloric. Are not percussion and the friction of bodies actually means of raising their temperature, of making it reach spontaneously a higher degree than that of the surrounding bodies, and consequently of producing a destruction of equilibrium in the caloric, where equilibrium previously existed? It is a fact proved by experience, that the temperature of gaseous fluids is raised by compression and lowered by rarefaction. This is a sure method of changing the temperature of bodies, and destroying the equilibrium of the caloric as many times as may be desired with the same substance. The vapor of water employed in an inverse manner to that in which it is used in steam-engines can also be regarded as a means of destroying the equilibrium of the caloric. To be convinced of this we need but to observe closely the manner in which motive power is developed by the action of heat on vapor of water. Imagine two bodies _A_ and _B_, kept each at a constant temperature, that of _A_ being higher than that of _B_. These two bodies, to which we can give or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We will call the first the furnace and the second the refrigerator.
If we wish to produce motive power by carrying a certain quantity of heat from the body _A_ to the body _B_ we shall proceed as follows:
(1) To borrow caloric from the body _A_ to make steam with it—that is, to make this body fulfil the function of a furnace, or rather of the metal composing the boiler in ordinary engines—we here assume that the steam is produced at the same temperature as the body _A_.
(2) The steam having been received in a space capable of expansion, such as a cylinder furnished with a piston, to increase the volume of this space, and consequently also that of the steam. Thus rarefied, the temperature will fall spontaneously, as occurs with all elastic fluids; admit that the rarefaction may be continued to the point where the temperature becomes precisely that of the body _B_.
(3) To condense the steam by putting it in contact with the body _B_, and at the same time exerting on it a constant pressure until it is entirely liquefied. The body _B_ fills here the place of the injection-water in ordinary engines, with this difference, that it condenses the vapor without mingling with it, and without changing its own temperature.[12]
The operations which we have just described might have been performed in an inverse direction and order. There is nothing to prevent forming vapor with the caloric of the body _B_, and at the temperature of that body, compressing it in such a way as to make it acquire the temperature of the body _A_, finally condensing it by contact with this latter body, and continuing the compression to complete liquefaction.
By our first operations there would have been at the same time production of motive power and transfer of caloric from the body _A_ to the body _B_. By the inverse operations there is at the same time expenditure of motive power and return of caloric from the body _B_ to the body _A_. But if we have acted in each case on the same quantity of vapor, if there is produced no loss either of motive power or caloric, the quantity of motive power produced in the first place will be equal to that which would have been expended in the second, and the quantity of caloric passed in the first case from the body _A_ to the body _B_ would be equal to the quantity which passes back again in the second from the body _B_ to the body _A_; so that an indefinite number of alternative operations of this sort could be carried on without in the end having either produced motive power or transferred caloric from one body to the other.
Now if there existed any means of using heat preferable to those which we have employed, that is, if it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body _B_ return to the body _A_ from the refrigerator to the furnace, to restore the initial conditions, and thus to be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without consumption either of caloric or of any other agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible.[13] We should then conclude that _the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever_. We will soon give a second more rigorous demonstration of this theory. This should be considered only as an approximation. (See page 59.)
We have a right to ask, in regard to the proposition just enunciated, the following questions: What is the sense of the word _maximum_ here? By what sign can it be known that this maximum is attained? By what sign can it be known whether the steam is employed to greatest possible advantage in the production of motive power?
Since every re-establishment of equilibrium in the caloric may be the cause of the production of motive power, every re-establishment of equilibrium which shall be accomplished without production of this power should be considered as an actual loss. Now, very little reflection would show that all change of temperature which is not due to a change of volume of the bodies can be only a useless re-establishment of equilibrium in the caloric.[14] The necessary condition of the maximum is, then, _that in the bodies employed to realize the motive power of heat there should not occur any change of temperature which may not be due to a change of volume_. Reciprocally, every time that this condition is fulfilled the maximum will be attained. This principle should never be lost sight of in the construction of heat-engines; it is its fundamental basis. If it cannot be strictly observed, it should at least be departed from as little as possible.
Every change of temperature which is not due to a change of volume or to chemical action (an action that we provisionally suppose not to occur here) is necessarily due to the direct passage of the caloric from a more or less heated body to a colder body. This passage occurs mainly by the contact of bodies of different temperatures; hence such contact should be avoided as much as possible. It cannot probably be avoided entirely, but it should at least be so managed that the bodies brought in contact with each other differ as little as possible in temperature. When we just now supposed, in our demonstration, the caloric of the body _A_ employed to form steam, this steam was considered as generated at the temperature of the body _A_; thus the contact took place only between bodies of equal temperatures; the change of temperature occurring afterwards in the steam was due to dilatation, consequently to a change of volume. Finally, condensation took place also without contact of bodies of different temperatures. It occurred while exerting a constant pressure on the steam brought in contact with the body _B_ of the same temperature as itself. The conditions for a maximum are thus found to be fulfilled. In reality the operation cannot proceed exactly as we have assumed. To determine the passage of caloric from one body to another, it is necessary that there should be an excess of temperature in the first, but this excess may be supposed as slight as we please. We can regard it as insensible in theory, without thereby destroying the exactness of the arguments.
A more substantial objection may be made to our demonstration, thus: When we borrow caloric from the body _A_ to produce steam, and when this steam is afterwards condensed by its contact with the body _B_, the water used to form it, and which we considered at first as being of the temperature of the body _A_, is found at the close of the operation at the temperature of the body _B_. It has become cool. If we wish to begin again an operation similar to the first, if we wish to develop a new quantity of motive power with the same instrument, with the same steam, it is necessary first to re-establish the original condition—to restore the water to the original temperature. This can undoubtedly be done by at once putting it again in contact with the body _A_; but there is then contact between bodies of different temperatures, and loss of motive power.[15] It would be impossible to execute the inverse operation, that is, to return to the body _A_ the caloric employed to raise the temperature of the liquid.
This difficulty may be removed by supposing the difference of temperature between the body _A_ and the body _B_ indefinitely small. The quantity of heat necessary to raise the liquid to its former temperature will be also indefinitely small and unimportant relatively to that which is necessary to produce steam—a quantity always limited.
The proposition found elsewhere demonstrated for the case in which the difference between the temperatures of the two bodies is indefinitely small, may be easily extended to the general case. In fact, if it operated to produce motive power by the passage of caloric from the body _A_ to the body _Z_, the temperature of this latter body being very different from that of the former, we should imagine a series of bodies _B_, _C_, _D_ ... of temperatures intermediate between those of the bodies _A_, _Z_, and selected so that the differences from _A_ to _B_, from _B_ to _C_, etc., may all be indefinitely small. The caloric coming from _A_ would not arrive at _Z_ till after it had passed through the bodies _B_, _C_, _D_, etc., and after having developed in each of these stages maximum motive power. The inverse operations would here be entirely possible, and the reasoning of page 52 would be strictly applicable.
According to established principles at the present time, we can compare with sufficient accuracy the motive power of heat to that of a waterfall. Each has a maximum that we cannot exceed, whatever may be, on the one hand, the machine which is acted upon by the water, and whatever, on the other hand, the substance acted upon by the heat. The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the _height of its fall_,[16] that is to say, the difference of temperature of the bodies between which the exchange of caloric is made. In the waterfall the motive power is exactly proportional to the difference of level between the higher and lower reservoirs. In the fall of caloric the motive power undoubtedly increases with the difference of temperature between the warm and the cold bodies; but we do not know whether it is proportional to this difference. We do not know, for example, whether the fall of caloric from 100 to 50 degrees furnishes more or less motive power than the fall of this same caloric from 50 to zero. It is a question which we propose to examine hereafter.
We shall give here a second demonstration of the fundamental proposition enunciated on page 56, and present this proposition under a more general form than the one already given.
When a gaseous fluid is rapidly compressed its temperature rises. It falls, on the contrary, when it is rapidly dilated. This is one of the facts best demonstrated by experiment. We will take it for the basis of our demonstration.[17]
If, when the temperature of a gas has been raised by compression, we wish to reduce it to its former temperature without subjecting its volume to new changes, some of its caloric must be removed. This caloric might have been removed in proportion as pressure was applied, so that the temperature of the gas would remain constant. Similarly, if the gas is rarefied we can avoid lowering the temperature by supplying it with a certain quantity of caloric. Let us call the caloric employed at such times, when no change of temperature occurs, _caloric due to change of volume_. This denomination does not indicate that the caloric appertains to the volume: it does not appertain to it any more than to pressure, and might as well be called _caloric due to the change of pressure_. We do not know what laws it follows relative to the variations of volume: it is possible that its quantity changes either with the nature of the gas, its density, or its temperature. Experiment has taught us nothing on this subject. It has only shown us that this caloric is developed in greater or less quantity by the compression of the elastic fluids.
[Illustration:
FIG. 1. ]
This preliminary idea being established, let us imagine an elastic fluid, atmospheric air for example, shut up in a cylindrical vessel, _abcd_ (Fig. 1), provided with a movable diaphragm or piston, _cd_. Let there be also two bodies, _A_ and _B_, kept each at a constant temperature, that of _A_ being higher than that of _B_. Let us picture to ourselves now the series of operations which are to be described:
(1) Contact of the body _A_ with the air enclosed in the space _abcd_ or with the wall of this space—a wall that we will suppose to transmit the caloric readily. The air becomes by such contact of the same temperature as the body _A_; _cd_ is the actual position of the piston.
(2) The piston gradually rises and takes the position _ef_. The body _A_ is all the time in contact with the air, which is thus kept at a constant temperature during the rarefaction. The body _A_ furnishes the caloric necessary to keep the temperature constant.
(3) The body _A_ is removed, and the air is then no longer in contact with any body capable of furnishing it with caloric. The piston meanwhile continues to move, and passes from the position _ef_ to the position _gh_. The air is rarefied without receiving caloric, and its temperature falls. Let us imagine that it falls thus till it becomes equal to that of the body _B_; at this instant the piston stops, remaining at the position _gh_.
(4) The air is placed in contact with the body _B_; it is compressed by the return of the piston as it is moved from the position _gh_ to the position _cd_. This air remains, however, at a constant temperature because of its contact with the body _B_, to which it yields its caloric.
(5) The body _B_ is removed, and the compression of the air is continued, which being then isolated, its temperature rises. The compression is continued till the air acquires the temperature of the body _A_. The piston passes during this time from the position _cd_ to the position _ik_.
(6) The air is again placed in contact with the body _A_. The piston returns from the position _ik_ to the position _ef_; the temperature remains unchanged.
(7) The step described under number 3 is renewed, then successively the steps 4, 5, 6, 3, 4, 5, 6, 3, 4, 5; and so on.
In these various operations the piston is subject to an effort of greater or less magnitude, exerted by the air enclosed in the cylinder; the elastic force of this air varies as much by reason of the changes in volume as of changes of temperature. But it should be remarked that with equal volumes, that is, for the similar positions of the piston, the temperature is higher during the movements of dilatation than during the movements of compression. During the former the elastic force of the air is found to be greater, and consequently the quantity of motive power produced by the movements of dilatation is more considerable than that consumed to produce the movements of compression. Thus we should obtain an excess of motive power—an excess which we could employ for any purpose whatever. The air, then, has served as a heat-engine; we have, in fact, employed it in the most advantageous manner possible, for no useless re-establishment of equilibrium has been effected in the caloric.
All the above-described operations may be executed in an inverse sense and order. Let us imagine that, after the sixth period, that is to say the piston having arrived at the position _ef_, we cause it to return to the position _ik_, and that at the same time we keep the air in contact with the body _A_. The caloric furnished by this body during the sixth period would return to its source, that is, to the body _A_, and the conditions would then become precisely the same as they were at the end of the fifth period. If now we take away the body _A_, and if we cause the piston to move from _ef_ to _cd_, the temperature of the air will diminish as many degrees as it increased during the fifth period, and will become that of the body _B_. We may evidently continue a series of operations the inverse of those already described. It is only necessary under the same circumstances to execute for each period a movement of dilatation instead of a movement of compression, and reciprocally.
The result of these first operations has been the production of a certain quantity of motive power and the removal of caloric from the body _A_ to the body _B_. The result of the inverse operations is the consumption of the motive power produced and the return of the caloric from the body _B_ to the body _A_; so that these two series of operations annul each other, after a fashion, one neutralizing the other.
The impossibility of making the caloric produce a greater quantity of motive power than that which we obtained from it by our first series of operations, is now easily proved. It is demonstrated by reasoning very similar to that employed at page 56; the reasoning will here be even more exact. The air which we have used to develop the motive power is restored at the end of each cycle of operations exactly to the state in which it was at first found, while, as we have already remarked, this would not be precisely the case with the vapor of water.[18]
We have chosen atmospheric air as the instrument which should develop the motive power of heat, but it is evident that the reasoning would have been the same for all other gaseous substances, and even for all other bodies susceptible of change of temperature through successive contractions and dilatations, which comprehends all natural substances, or at least all those which are adapted to realize the motive power of heat. Thus we are led to establish this general proposition:
_The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of the caloric._
We must understand here that each of the methods of developing motive power attains the perfection of which it is susceptible. This condition is found to be fulfilled if, as we remarked above, there is produced in the body no other change of temperature than that due to change of volume, or, what is the same thing in other words, if there is no contact between bodies of sensibly different temperatures.
Different methods of realizing motive power may be taken, as in the employment of different substances, or in the use of the same substance in two different states—for example, of a gas at two different densities.
This leads us naturally to those interesting researches on the aeriform fluids—researches which lead us also to new results in regard to the motive power of heat, and give us the means of verifying, in some particular cases, the fundamental proposition above stated.[19]
We readily see that our demonstration would have been simplified by supposing the temperatures of the bodies _A_ and _B_ to differ very little. Then the movements of the piston being slight during the periods 3 and 5, these periods might have been suppressed without influencing sensibly the production of motive power. A very little change of volume should suffice in fact to produce a very slight change of temperature, and this slight change of volume may be neglected in presence of that of the periods 4 and 6, of which the extent is unlimited.
If we suppress periods 3 and 5, in the series of operations above described, it is reduced to the following:
(1) Contact of the gas confined in _abcd_ (Fig. 2) with the body _A_, passage of the piston from _cd_ to _ef_.
[Illustration:
FIG. 2. FIG. 3. ]
(2) Removal of the body _A_, contact of the gas confined in _abef_ with the body _B_, return of the piston from _ef_ to _cd_.
(3) Removal of the body _B_, contact of the gas with the body _A_, passage of the piston from _cd_ to _ef_, that is, repetition of the first period, and so on.
The motive power resulting from the _ensemble_ of operations 1 and 2 will evidently be the difference between that which is produced by the expansion of the gas while it is at the temperature of the body _A_, and that which is consumed to compress this gas while it is at the temperature of the body _B_.
Let us suppose that operations 1 and 2 be performed on two gases of different chemical natures but under the same pressure—under atmospheric pressure, for example. These two gases will behave exactly alike under the same circumstances, that is, their expansive forces, originally equal, will remain always equal, whatever may be the variations of volume and of temperature, provided these variations are the same in both. This results obviously from the laws of Mariotte and MM. Gay-Lussac and Dalton—laws common to all elastic fluids, and in virtue of which the same relations exist for all these fluids between the volume, the expansive force, and the temperature.
Since two different gases at the same temperature and under the same pressure should behave alike under the same circumstances, if we subjected them both to the operations above described, they should give rise to equal quantities of motive power.
Now this implies, according to the fundamental proposition that we have established, the employment of two equal quantities of caloric; that is, it implies that the quantity of caloric transferred from the body _A_ to the body _B_ is the same, whichever gas is used.
The quantity of caloric transferred from the body _A_ to the body _B_ is evidently that which is absorbed by the gas in its expansion of volume, or that which this gas relinquishes during compression. We are led, then, to establish the following proposition:
_When a gas passes without change of temperature from one definite volume and pressure to another volume and another pressure equally definite, the quantity of caloric absorbed or relinquished is always the same, whatever may be the nature of the gas chosen as the subject of the experiment._
Take, for example, 1 litre of air at the temperature of 100° and under the pressure of one atmosphere. If we double the volume of this air and wish to maintain it at the temperature of 100°, a certain quantity of heat must be supplied to it. Now this quantity will be precisely the same if, instead of operating on the air, we operate upon carbonic-acid gas, upon nitrogen, upon hydrogen, upon vapor of water or of alcohol, that is, if we double the volume of 1 litre of these gases taken at the temperature of 100° and under atmospheric pressure.
It will be the same thing in the inverse sense if, instead of doubling the volume of gas, we reduce it one half by compression. The quantity of heat that the elastic fluids set free or absorb in their changes of volume has never been measured by any direct experiment, and doubtless such an experiment would be very difficult, but there exists a datum which is very nearly its equivalent. This has been furnished by the theory of sound. It deserves much confidence because of the exactness of the conditions which have led to its establishment. It consists in this:
Atmospheric air should rise one degree Centigrade when by sudden compression it experiences a reduction of volume of ¹⁄₁₁₆.[20]
Experiments on the velocity of sound having been made in air under the pressure of 760 millimetres of mercury and at the temperature of 6°, it is only to these two circumstances that our datum has reference. We will, however, for greater facility, refer it to the temperature 0°, which is nearly the same.
Air compressed ¹⁄₁₁₆, and thus heated one degree, differs from air heated directly one degree only in its density. The primitive volume being supposed to be _V_, the compression of ¹⁄₁₁₆ reduces it to _V_ − ¹⁄₁₁₆ _V_.
Direct heating under constant pressure should, according to the rule of M. Gay-Lussac, increase the volume of air ¹⁄₂₆₇ above what it would be at 0°: so the air is, on the one hand, reduced to the volume _V_ − ¹⁄₁₁₆ _V_; on the other, it is increased to _V_ + ¹⁄₂₆₇ _V_.
The difference between the quantities of heat which the air possesses in both cases is evidently the quantity employed to raise it directly one degree; so then the quantity of heat that the air would absorb in passing from the volume _V_ − ¹⁄₁₁₆ _V_ to the volume _V_ + ¹⁄₂₆₇ _V_ is equal to that which is required to raise it one degree.
Let us suppose now that, instead of heating one degree the air subjected to a constant pressure and able to dilate freely, we inclose it within an invariable space, and that in this condition we cause it to rise one degree in temperature. The air thus heated one degree will differ from the air compressed ¹⁄₁₁₆ only by its ¹⁄₁₁₆ greater volume. So then the quantity of heat that the air would set free by a reduction of volume of ¹⁄₁₁₆ is equal to that which would be required to raise it one degree Centigrade under constant volume. As the differences between the volumes _V_ − ¹⁄₁₁₆ _V_, _V_, and _V_ + ¹⁄₂₆₇ _V_ are small relatively to the volumes themselves, we may regard the quantities of heat absorbed by the air in passing from the first of these volumes to the second, and from the first to the third, as sensibly proportional to the changes of volume. We are then led to the establishment of the following relation:
The quantity of heat necessary to raise one degree air under constant pressure is to the quantity of heat necessary to raise one degree the same air under constant volume, in the ratio of the numbers
¹⁄₁₁₆ + ¹⁄₂₆₇ to ¹⁄₁₁₆;
or, multiplying both by 116 × 267, in the ratio of the numbers 267 + 116 to 267.
This, then, is the ratio which exists between the capacity of air for heat under constant pressure and its capacity under constant volume. If the first of these two capacities is expressed by unity, the other will be expressed by the number (267)/(267 + 116), or very nearly 0.700; their difference, 1 − 0.700 or 0.300, will evidently express the quantity of heat which will produce the increase of volume in the air when it is heated one degree under constant pressure.
According to the law of MM. Gay-Lussac and Dalton, this increase of volume would be the same for all other gases; according to the theory demonstrated on page 87, the heat absorbed by these equal increases of volume is the same for all the elastic fluids, which leads to the establishment of the following proposition:
_The difference between specific heat under constant pressure and specific heat under constant volume is the same for all gases._
It should be remarked here that all the gases are considered as taken under the same pressure, atmospheric pressure for example, and that the specific heats are also measured with reference to the volumes.
It is a very easy matter now for us to prepare a table of the specific heat of gases under constant volume, from the knowledge of their specific heats under constant pressure. Here is the table:
TABLE OF THE SPECIFIC HEAT OF GASES. ───────────────────────┬───────────────────────┬─────────────────────── NAMES OF GASES. │ Specific Heat under │Specific Heat at Const. │ Const. Press. │ Vol. ───────────────────────┼───────────────────────┼─────────────────────── Atmospheric Air, │ 1.000 │ 0.700 Hydrogen Gas, │ 0.903 │ 0.603 Carbonic Acid, │ 1.258 │ 0.958 Oxygen, │ 0.976 │ 0.676 Nitrogen, │ 1.000 │ 0.700 Protoxide of Nitrogen, │ 1.350 │ 1.050 Olefiant Gas, │ 1.553 │ 1.253 Oxide of Carbon, │ 1.034 │ 0.734 ───────────────────────┴───────────────────────┴───────────────────────
The first column is the result of the direct experiments of MM. Delaroche and Bérard on the specific heat of the gas under atmospheric pressure, and the second column is composed of the numbers of the first diminished by 0.300.
The numbers of the first column and those of the second are here referred to the same unit, to the specific heat of atmospheric air under constant pressure.
The difference between each number of the first column and the corresponding number of the second being constant, the relation between these numbers should be variable. Thus the relation between the specific heat of gases under constant pressure and the specific heat at constant volume, varies in different gases.
We have seen that air when it is subjected to a sudden compression of ¹⁄₁₁₆ of its volume rises one degree in temperature. The other gases through a similar compression should also rise in temperature. They should rise, but not equally, in inverse ratio with their specific heat at constant volume. In fact, the reduction of volume being by hypothesis always the same, the quantity of heat due to this reduction should likewise be always the same, and consequently should produce an elevation of temperature dependent only on the specific heat acquired by the gas after its compression, and evidently in inverse ratio with this specific heat. Thus we can easily form the table of the elevations of temperature of the different gases for a compression of ¹⁄₁₁₆.
TABLE OF THE ELEVATION OF TEMPERATURE<BR>OF _Gases through the Effect of Compression_. ──────────────────────┬──────────────────────────────────────────────── NAMES OF GASES. │ Elevation of Temperature for a Reduction of │ Volume of ¹⁄₁₁₆. ──────────────────────┼──────────────────────────────────────────────── │ ° Atmospheric Air, │ 1.000 Hydrogen Gas, │ 1.160 Carbonic Acid, │ 0.730 Oxygen, │ 1.035 Nitrogen, │ 1.000 Protoxide of Nitrogen,│ 0.667 Olefiant Gas, │ 0.558 Carbonic Oxide, │ 0.955 ──────────────────────┴────────────────────────────────────────────────
A second compression of ¹⁄₁₁₆ (of the altered volume), as we shall presently see, would also raise the temperature of these gases nearly as much as the first; but it would not be the same with a third, a fourth, a hundredth such compression. The capacity of gases for heat changes with their volume. It is not unlikely that it changes also with the temperature.
We shall now deduce from the general proposition stated on page 68 a second theory, which will serve as a corollary to that just demonstrated.
Let us suppose that the gas enclosed in the cylindrical space _abcd_ (Fig. 2) be transported into the space _a′b′c′d′_ (Fig. 3) of equal height, but of different base and wider. This gas would increase in volume, would diminish in density and in elastic force, in the inverse ratio of the two volumes _abcd_, _a′b′c′d′_. As to the total pressure exerted in each piston _cd_, _c′d′_, it would be the same from all quarters, for the surface of these pistons is in direct ratio to the volumes.
Let us suppose that we perform on the gas inclosed in _a′b′c′d′_ the operations described on page 70, and which were taken as having been performed upon the gas inclosed in _abcd_; that is, let us suppose that we have given to the piston _c′d′_ motions equal to those of the piston _cd_, that we have made it occupy successively the positions _c′d′_ corresponding to _cd_, and _e′f′_ corresponding to _ef_, and that at the same time we have subjected the gas by means of the two bodies _A_ and _B_ to the same variations of temperature as when it was inclosed in _abcd_. The total effort exercised on the piston would be found to be, in the two cases, always the same at the corresponding instants. This results solely from the law of Mariotte.[21] In fact, the densities of the two gases maintaining always the same ratio for similar positions of the pistons, and the temperatures being always equal in both, the total pressures exercised on the pistons will always maintain the same ratio to each other. If this ratio is, at any instant whatever, unity, the pressures will always be equal.
As, furthermore, the movements of the two pistons have equal extent, the motive power produced by each will evidently be the same; whence we should conclude, according to the proposition on page 68, that the quantities of heat consumed by each are the same, that is, that there passes from the body _A_ to the body _B_ the same quantity of heat in both cases.
The heat abstracted from the body _A_ and communicated to the body _B_, is simply the heat absorbed during the rarefaction of the gas, and afterwards liberated by its compression. We are therefore led to establish the following theorem:
_When an elastic fluid passes without change of temperature from the volume U to the volume V, and when a similar ponderable quantity of the same gas passes at the same temperature from the volume U′ to the volume V′, if the ratio of U′ to V′ is found to be the same as the ratio of U to V, the quantities of heat absorbed or disengaged in the two cases will be equal._
This theorem might also be expressed as follows:
_When a gas varies in volume without change of temperature, the quantities of heat absorbed or liberated by this gas are in arithmetical progression, if the increments or the decrements of volume are found to be in geometrical progression._
When a litre of air maintained at a temperature of ten degrees is compressed, and when it is reduced to one half a litre, a certain quantity of heat is set free. This quantity will be found always the same if the volume is further reduced from a half litre to a quarter litre, from a quarter litre to an eighth, and so on.
If, instead of compressing the air, we carry it successively to two litres, four litres, eight litres, etc., it will be necessary to supply to it always equal quantities of heat in order to maintain a constant temperature.
This readily accounts for the high temperature attained by air when rapidly compressed. We know that this temperature inflames tinder and even makes air luminous. If, for a moment, we suppose the specific heat of air to be constant, in spite of the changes of volume and temperature, the temperature will increase in arithmetical progression for reduction of volume in geometrical progression.
Starting from this datum, and admitting that one degree of elevation in the temperature corresponds to a compression of ¹⁄₁₁₆, we shall readily come to the conclusion that air reduced to ¹⁄₁₄ of its primitive volume should rise in temperature about 300 degrees, which is sufficient to inflame tinder.[22]
The elevation of temperature ought, evidently, to be still more considerable if the capacity of the air for heat becomes less as its volume diminishes. Now this is probable, and it also seems to follow from the experiments of MM. Delaroche and Bérard on the specific heat of air taken at different densities. (See the Mémoire in the _Annales de Chimie_, t. lxxxv. pp. 72, 224.)
The two theorems explained on pp. 72 and 81 suffice for the comparison of the quantities of heat absorbed or set free in the changes of volume of elastic fluids, whatever may be the density and the chemical nature of these fluids, provided always that they be taken and maintained at a certain invariable temperature. But these theories furnish no means of comparing the quantities of heat liberated or absorbed by elastic fluids which change in volume at different temperatures. Thus we are ignorant what relation exists between the heat relinquished by a litre of air reduced one half, the temperature being kept at zero, and the heat relinquished by the same litre of air reduced one half, the temperature being kept at 100°. The knowledge of this relation is closely connected with that of the specific heat of gases at various temperatures, and to some other data that Physics as yet does not supply.
The second of our theorems offers us a means of determining according to what law the specific heat of gases varies with their density.
Let us suppose that the operations described on p. 70, instead of being performed with two bodies, _A_, _B_, of temperatures differing indefinitely small, were carried on with two bodies whose temperatures differ by a finite quantity—one degree, for example. In a complete circle of operations the body _A_ furnishes to the elastic fluid a certain quantity of heat, which may be divided into two portions: (1) That which is necessary to maintain the temperature of the fluid constant during dilatation; (2) that which is necessary to restore the temperature of the fluid from that of the body _B_ to that of the body _A_, when, after having brought back this fluid to its primitive volume, we place it again in contact with the body _A_. Let us call the first of these quantities _a_ and the second _b_. The total caloric furnished by the body A will be expressed by _a_ + _b_.
The caloric transmitted by the fluid to the body _B_ may also be divided into two parts: one, _b′_, due to the cooling of the gas by the body _B_; the other, _a′_, which the gas abandons as a result of its reduction of volume. The sum of these two quantities is _a′_ + _b′_; it should be equal to _a_ + _b_, for, after a complete cycle of operations, the gas is brought back exactly to its primitive state. It has been obliged to give up all the caloric which has first been furnished to it. We have then
_a_ + _b_ = _a′_ + _b′_;
or rather,
_a_ − _a′_ = _b′_ − _b_.
Now, according to the theorem given on page 81, the quantities _a_ and _a′_ are independent of the density of the gas, provided always that the ponderable quantity remains the same and that the variations of volume be proportional to the original volume. The difference _a_ − _a′_ should fulfil the same conditions, and consequently also the difference _b′_ − _b_, which is equal to it. But _b′_ is the caloric necessary to raise the gas enclosed in _abcd_ (Fig. 2) one degree; _b′_ is the caloric surrendered by the gas when, enclosed in _abef_, it is cooled one degree. These quantities may serve as a measure for specific heats. We are then led to the establishment of the following proposition:
_The change in the specific heat of a gas caused by change of volume depends entirely on the ratio between the original volume and the altered volume._ That is, the difference of the specific heats does not depend on the absolute magnitude of the volumes, but only on their ratio.
This proposition might also be differently expressed, thus:
_When a gas increases in volume in geometrical progression, its specific heat increases in arithmetical progression._
Thus, _a_ being the specific heat of air taken at a given density, and _a_ + _h_ the specific heat for a density one half less, it will be, for a density equal to one quarter, _a_ + 2_h_; for a density equal to one eighth, _a_ + 3_h_; and so on.
The specific heats are here taken with reference to weight. They are supposed to be taken at an invariable volume, but, as we shall see, they would follow the same law if they were taken under constant pressure.
To what cause is the difference between specific heats at constant volume and at constant pressure really due? To the caloric required to produce in the second case increase of volume. Now, according to the law of Mariotte, increase of volume of a gas should be, for a given change of temperature, a determined fraction of the original volume, a fraction independent of pressure. According to the theorem expressed on page 76, if the ratio between the primitive volume and the altered volume is given, that determines the heat necessary to produce increase of volume. It depends solely on this ratio and on the weight of the gas. We must then conclude that:
_The difference between specific heat at constant pressure and specific heat at constant volume is always the same, whatever may be the density of the gas, provided the weight remains the same._
These specific heats both increase accordingly as the density of the gas diminishes, but their difference does not vary.[23]
Since the difference between the two capacities for heat is constant, if one increases in arithmetical progression the other should follow a similar progression: thus one law is applicable to specific heats at constant pressure.
We have tacitly assumed the increase of specific heat with that of volume. This increase is indicated by the experiments of MM. Delaroche and Bérard: in fact these physicists have found 0.967 for the specific heat of air under the pressure of 1 metre of mercury (see Mémoire already cited), taking for the unit the specific heat of the same weight of air under the pressure of 0^m.760.
According to the law that specific heats follow with relation to pressures, it is only necessary to have observed them in two particular cases to deduce them in all possible cases: it is thus that, making use of the experimental result of MM. Delaroche and Bérard which has just been given, we have prepared the following table of the specific heat of air under different pressures:
SPECIFIC HEAT OF AIR. ────────────────────────┬──────────────────────────────────────────── Pressure in Atmospheres.│Specific Heat, that of Air under Atmospheric │ Pressure being 1. ────────────────────────┼──────────────────────────────────────────── ¹⁄₁₀₂₄ │ 1.840 ¹⁄₅₁₂ │ 1.756 ¹⁄₂₅₆ │ 1.672 ¹⁄₁₂₈ │ 1.588 ¹⁄₆₄ │ 1.504 ¹⁄₃₂ │ 1.420 ¹⁄₁₆ │ 1.336 ⅛ │ 1.252 ¼ │ 1.165 ½ │ 1.084 1 │ 1.000 2 │ 0.916 4 │ 0.832 8 │ 0.748 16 │ 0.664 32 │ 0.580 64 │ 0.496 128 │ 0.412 256 │ 0.328 512 │ 0.244 1024 │ 0.160 ────────────────────────┴────────────────────────────────────────────
The first column is, as we see, a geometrical progression, and the second an arithmetical progression.
We have carried out the table to the extremes of compression and rarefaction. It may be believed that air would be liquefied before acquiring a density 1024 times its normal density, that is, before becoming more dense than water. The specific heat would become zero and even negative on extending the table beyond the last term. We think, furthermore, that the figures of the second column here decrease too rapidly. The experiments which serve as a basis for our calculation have been made within too contracted limits for us to expect great exactness in the figures which we have obtained, especially in the outside numbers.
Since we know, on the one hand, the law according to which heat is disengaged in the compression of gases, and on the other, the law according to which specific heat varies with volume, it will be easy for us to calculate the increase of temperature of a gas that has been compressed without being allowed to lose heat. In fact, the compression may be considered as composed of two successive operations: (1) compression at a constant temperature; (2) restoration of the caloric emitted. The temperature will rise through the second operation in inverse ratio with the specific heat acquired by the gas after the reduction of volume,—specific heat that we are able to calculate by means of the law demonstrated above. The heat set free by compression, according to the theorem of page 81, ought to be represented by an expression of the form
_s_ = _A_ + _B_ log _v_,
_s_ being this heat, _v_ the volume of the gas after compression, _A_ and _B_ arbitrary constants dependent on the primitive volume of the gas, on its pressure, and on the units chosen.
The specific heat varying with the volume according to the law just demonstrated, should be represented by an expression of the form
_z_ = _A′_ + _B′_ log _v_,
_A′_ and _B′_ being the different arbitrary constants of _A_ and _B_.
The increase of temperature acquired by the gas, as the effect of compression, is proportional to the ratio (_s_)/(_z_) or to the relation (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_). It can be represented by this ratio itself; thus, calling it _t_, we shall have
_t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_).
If the original volume of the gas is 1, and the original temperature zero, we shall have at the same time _t_ = 0, log _v_ = 0, whence _A_ = 0; _t_ will then express not only the increase of temperature, but the temperature itself above the thermometric zero.
We need not consider the formula that we have just given as applicable to very great changes in the volume of gases. We have regarded the elevation of temperature as being in inverse ratio to the specific heat; which tacitly supposes the specific heat to be constant at all temperatures. Great changes of volume lead to great changes of temperature in the gas, and nothing proves the constancy of specific heat at different temperatures, especially at temperatures widely separated. This constancy is only an hypothesis admitted for gases by analogy, to a certain extent verified for solid bodies and liquids throughout a part of the thermometric scale, but of which the experiments of MM. Dulong and Petit have shown the inaccuracy when it is desirable to extend it to temperatures far above 100°.[24]
According to a law of MM. Clement and Desormes, a law established by direct experiment, the vapor of water, under whatever pressure it may be formed, contains always, at equal weights, the same quantity of heat; which leads to the assertion that steam, compressed or expanded mechanically without loss of heat, will always be found in a saturated state if it was so produced in the first place. The vapor of water so made may then be regarded as a permanent gas, and should observe all the laws of one. Consequently the formula
_t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_)
should be applicable to it, and be found to accord with the table of tensions derived from the direct experiments of M. Dalton.
We may be assured, in fact, that our formula, with a convenient determination of arbitrary constants, represents very closely the results of experiment. The slight irregularities which we find therein do not exceed what we might reasonably attribute to errors of observation.[25]
We will return, however, to our principal subject, from which we have wandered too far—the motive power of heat.
We have shown that the quantity of motive power developed by the transfer of caloric from one body to another depends essentially upon the temperature of the two bodies, but we have not shown the relation between these temperatures and the quantities of motive power produced. It would at first seem natural enough to suppose that for equal differences of temperature the quantities of motive power produced are equal; that is, for example, the passage of a given quantity of caloric from a body, _A_, maintained at 100°, to a body, _B_, maintained at 50°, should give rise to a quantity of motive power equal to that which would be developed by the transfer of the same caloric from a body, _B_, at 50°, to a body, _C_, at zero. Such a law would doubtless be very remarkable, but we do not see sufficient reason for admitting it _à priori_. We will investigate its reality by exact reasoning.
Let us imagine that the operations described on p. 70 be conducted successively on two quantities of atmospheric air equal in weight and volume, but taken at different temperatures. Let us suppose, further, the differences of temperature between the bodies _A_ and _B_ equal, so these bodies would have for example, in one of these cases, the temperatures 100° and 100° − _h_ (_h_ being indefinitely small), and in the other 1° and 1° − _h_. The quantity of motive power produced is, in each case, the difference between that which the gas supplies by its dilatation and that which must be expended to restore its primitive volume. Now this difference is the same in both cases, as any one can prove by simple reasoning, which it seems unnecessary to give here in detail; hence the motive power produced is the same.
Let us now compare the quantities of heat employed in the two cases. In the first, the quantity of heat employed is that which the body _A_ furnishes to the air to maintain it at the temperature of 100° during its expansion. In the second, it is the quantity of heat which this same body should furnish to it, to keep its temperature at one degree during an exactly similar change of volume. If these two quantities of heat were equal, there would evidently result the law that we have already assumed. But nothing proves that it is so, and we shall find that these quantities are not equal.
The air that we shall first consider as occupying the space _abcd_ (Fig. 2), and having 1 degree of temperature, can be made to occupy the space _abef_, and to acquire the temperature of 100 degrees by two different means:
(1) We may heat it without changing its volume, then expand it, keeping its temperature constant.
(2) We may begin by expanding it, maintaining the temperature constant, then heat it, when it has acquired its greater volume.
Let _a_ and _b_ be the quantities of heat employed successively in the first of the two operations, and let _b′_ and _a′_ be the quantities of heat employed successively in the second. As the final result of these two operations is the same, the quantities of heat employed in both should be equal. We have then
_a_ + _b_ = _a′_ + _b′_,
whence
_a′_ − _a_ = _b_ − _b′_.
_a′_ is the quantity of heat required to cause the gas to rise from 1° to 100° when it occupies the space _abef_.
_a_ is the quantity of heat required to cause the gas to rise from 1° to 100° when it occupies the space _abcd_.
The density of the air is less in the first than in the second case, and according to the experiments of MM. Delaroche and Bérard, already cited on page 87, its capacity for heat should be a little greater.
The quantity _a′_ being found to be greater than the quantity _a_, _b_ should be greater than _b′_. Consequently, generalizing the proposition, we should say:
_The quantity of heat due to the change of volume of a gas is greater as the temperature is higher._
Thus, for example, more caloric is necessary to maintain at 100° the temperature of a certain quantity of air the volume of which is doubled, than to maintain at 1° the temperature of this same air during a dilatation exactly equal.
These unequal quantities of heat would produce, however, as we have seen, equal quantities of motive power for equal fall of caloric taken at different heights on the thermometric scale; whence we draw the following conclusion:
_The fall of caloric produces more motive power at inferior than at superior temperatures._
Thus a given quantity of heat will develop more motive power in passing from a body kept at 1 degree to another maintained at zero, than if these two bodies were at the temperature of 101° and 100°.
The difference, however, should be very slight. It would be nothing if the capacity of the air for heat remained constant, in spite of changes of density. According to the experiments of MM. Delaroche and Bérard, this capacity varies little—so little even, that the differences noticed might strictly have been attributed to errors of observation or to some circumstances of which we have failed to take account.
We are not prepared to determine precisely, with no more experimental data than we now possess, the law according to which the motive power of heat varies at different points on the thermometric scale. This law is intimately connected with that of the variations of the specific heat of gases at different temperatures—a law which experiment has not yet made known to us with sufficient exactness.[26]
We will endeavor now to estimate exactly the motive power of heat, and in order to verify our fundamental proposition, in order to determine whether the agent used to realize the motive power is really unimportant relatively to the quantity of this power, we will select several of them successively: atmospheric air, vapor of water, vapor of alcohol.
Let us suppose that we take first atmospheric air. The operation will proceed according to the method indicated on page 70. We will make the following hypotheses: The air is taken under atmospheric pressure. The temperature of the body _A_ is ¹⁄₁₀₀₀ of a degree above zero, that of the body _B_ is zero. The difference is, as we see, very slight—a necessary condition here.
The increase of volume given to the air in our operation will be ¹⁄₁₁₆ + ¹⁄₂₆₇ of the primitive volume; this is a very slight increase, absolutely speaking, but great relatively to the difference of temperature between the bodies _A_ and _B_.
The motive power developed by the whole of the two operations described (page 70) will be very nearly proportional to the increase of volume and to the difference between the two pressures exercised by the air, when it is found at the temperatures 0°.001 and zero.
This difference is, according to the law of M. Gay-Lussac, ¹⁄₂₆₇₀₀₀ of the elastic force of the gas, or very nearly ¹⁄₂₆₇₀₀₀ of the atmospheric pressure.
The atmospheric pressure balances at 10.40 metres head of water; ¹⁄₂₆₇₀₀₀ of this pressure equals ¹⁄₂₆₇₀₀₀ × 10^m.40 of head of water.
As to the increase of volume, it is, by supposition, ¹⁄₁₁₆ + ¹⁄₂₆₇ of the original volume, that is, of the volume occupied by one kilogram of air at zero, a volume equal to 0^{mc}.77, allowing for the specific weight of the air. So then the product,
(¹⁄₁₁₆ + ¹⁄₂₆₇) × 0.77 × ¹⁄₂₆₇₀₀₀ × 10.40,
will express the motive power developed. This power is estimated here in cubic metres of water raised one metre.
If we carry out the indicated multiplications, we find the value of the product to be 0.000000372.
Let us endeavor now to estimate the quantity of heat employed to give this result; that is, the quantity of heat passed from the body _A_ to the body _B_.
The body _A_ furnishes:
(1) The heat required to carry the temperature of one kilogram of air from zero to 0°.001;
(2) The quantity necessary to maintain at this temperature the temperature of the air when it experiences a dilatation of
¹⁄₁₁₆ + ¹⁄₂₆₇.
The first of these quantities of heat being very small in comparison with the second, we may disregard it. The second is, according to the reasoning on page 74, equal to that which would be necessary to increase one degree the temperature of one kilogram of air subjected to atmospheric pressure.
According to the experiments of MM. Delaroche and Bérard on the specific heat of gases, that of air is, for equal weights, 0.267 that of water. If, then, we take for the unit of heat the quantity necessary to raise 1 kilogram of water 1 degree, that which will be required to raise 1 kilogram of air 1 degree would have for its value 0.267. Thus the quantity of heat furnished by the body _A_ is
0.267 units.
This is the heat capable of producing 0.000000372 units of motive power by its fall from 0°.001 to zero.
For a fall a thousand times greater, for a fall of one degree, the motive power will be very nearly a thousand times the former, or
0.000372.
If, now, instead of 0.267 units of heat we employ 1000 units, the motive power produced will be expressed by the proportion
(0.267)/(0.000372) = (1000)/(x), whence x = (372)/(267) = 1.395.
Thus 1000 units of heat passing from a body maintained at the temperature of 1 degree to another body maintained at zero would produce, in acting upon the air,
1.395 units of motive power.
We will now compare this result with that furnished by the action of heat on the vapor of water.
[Illustration:
FIG. 4. ]
Let us suppose one kilogram of liquid water enclosed in the cylindrical vessel _abcd_ (Fig. 4), between the bottom _ab_ and the piston _cd_. Let us suppose, also, the two bodies _A_, _B_ maintained each at a constant temperature, that of _A_ being a very little above that of _B_. Let us imagine now the following operations:
(1) Contact of the water with the body _A_, movement of the piston from the position _cd_ to the position _ef_, formation of steam at the temperature of the body _A_ to fill the vacuum produced by the extension of volume. We will suppose the space _abef_ large enough to contain all the water in a state of vapor.
(2) Removal of the body _A_, contact of the vapor with the body _B_, precipitation of a part of this vapor, diminution of its elastic force, return of the piston from _ef_ to _ab_, liquefaction of the rest of the vapor through the effect of the pressure combined with the contact of the body _B_.
(3) Removal of the body _B_, fresh contact of the water with the body _A_, return of the water to the temperature of this body, renewal of the former period, and so on.
The quantity of motive power developed in a complete cycle of operations is measured by the product of the volume of the vapor multiplied by the difference between the tensions that it possesses at the temperature of the body _A_ and at that of the body _B_. As to the heat employed, that is to say, transported from the body _A_ to the body _B_, it is evidently that which was necessary to turn the water into vapor, disregarding always the small quantity required to restore the temperature of the liquid water from that of _B_ to that of _A_.
Suppose the temperature of the body _A_ 100 degrees, and that of the body _B_ 99 degrees: the difference of the tensions will be, according to the table of M. Dalton, 26 millimetres of mercury or 0^m.36 head of water.
The volume of the vapor is 1700 times that of the water. If we operate on one kilogram, that will be 1700 litres, or 1^{mc}.700.
Thus the value of the motive power developed is the product
1.700 × 0.36 = 0.611 units,
of the kind of which we have previously made use.
The quantity of heat employed is the quantity required to turn into vapor water already heated to 100°. This quantity is found by experiment. We have found it equal to 550°, or, to speak more exactly, to 550 of our units of heat.
Thus 0.611 units of motive power result from the employment of 550 units of heat. The quantity of motive power resulting from 1000 units of heat will be given by the proportion
⁵⁵⁰⁄₀.611 = 1000/_x_, whence _x_ = ⁶¹¹⁄₅₅₀ = 1.112.
Thus 1000 units of heat transported from one body kept at 100 degrees to another kept at 99 degrees will produce, acting upon vapor of water, 1.112 units of motive power.
The number 1.112 differs by about ¼ from the number 1.395 previously found for the value of the motive power developed by 1000 units of heat acting upon the air; but it should be observed that in this case the temperatures of the bodies _A_ and _B_ were 1 degree and zero, while here they are 100 degrees and 99 degrees. The difference is much the same; but it is not found at the same height in the thermometric scale. To make an exact comparison, it would have been necessary to estimate the motive power developed by the steam formed at 1 degree and condensed at zero. It would also have been necessary to know the quantity of heat contained in the steam formed at one degree.
The law of MM. Clement and Desormes referred to on page 92 gives us this datum. The constituent heat of vapor of water being always the same at any temperature at which vaporization takes place, if 550 degrees of heat are required to vaporize water already brought up to 100 degrees, 550 + 100 or 650 will be required to vaporize the same weight of water taken at zero.
Making use of this datum and reasoning exactly as we did for water at 100 degrees, we find, as is easily seen,
1.290
for the motive power developed by 1000 units of heat acting upon the vapor of water between one degree and zero. This number approximates more closely than the first to
1.395.
It differs from it only ¹⁄₁₃, an error which does not exceed probable limits, considering the great number of data of different sorts of which we have been obliged to make use in order to arrive at this approximation. Thus is our fundamental law verified in a special case.[27]
We will examine another case in which vapor of alcohol is acted upon by heat. The reasoning is precisely the same as for the vapor of water. The data alone are changed. Pure alcohol boils under ordinary pressure at 78°.7 Centigrade. One kilogram absorbs, according to MM. Delaroche and Bérard, 207 units of heat in undergoing transformation into vapor at this same temperature, 78°.7.
The tension of the vapor of alcohol at one degree below the boiling-point is found to be diminished ¹⁄₂₅. It is ¹⁄₂₅ less than the atmospheric pressure; at least, this is the result of the experiment of M. Bétancour reported in the second part of _l’Architecture hydraulique_ of M. Prony, pp. 180, 195.[28]
If we use these data, we find that, in acting upon one kilogram of alcohol at the temperatures of 78°.7 and 77°.7, the motive power developed will be 0.251 units.
This results from the employment of 207 units of heat. For 1000 units the proportion must be
(207)/(0.254) = (1000)/(_x_), whence _x_ = 1.230.
This number is a little more than the 1.112 resulting from the use of the vapor of water at the temperatures 100° and 99°; but if we suppose the vapor of water used at the temperatures 78° and 77°, we find, according to the law of MM. Clement and Desorme, 1.212 for the motive power due to 1000 units of heat. This latter number approaches, as we see, very nearly to 1.230. There is a difference of only ¹⁄₅₀.
We should have liked to be able to make other approximations of this sort—to be able to calculate, for example, the motive power developed by the action of heat on solids and liquids, by the congelation of water, and so on; but Physics as yet refuses us the necessary data.[29]
The fundamental law that we propose to confirm seems to us to require, however, in order to be placed beyond doubt, new verifications. It is based upon the theory of heat as it is understood to-day, and it should be said that this foundation does not appear to be of unquestionable solidity. New experiments alone can decide the question. Meanwhile we can apply the theoretical ideas expressed above, regarding them as exact, to the examination of the different methods proposed up to date, for the realization of the motive power of heat.
It has sometimes been proposed to develop motive power by the action of heat on solid bodies. The mode of procedure which naturally first occurs to the mind is to fasten immovably a solid body—a metallic bar, for example—by one of its extremities; to attach the other extremity to a movable part of the machine; then, by successive heating and cooling, to cause the length of the bar to vary, and so to produce motion. Let us try to decide whether this method of developing motive power can be advantageous. We have shown that the condition of the most effective employment of heat in the production of motion is, that all changes of temperature occurring in the bodies should be due to changes of volume. The nearer we come to fulfilling this condition the more fully will the heat be utilized. Now, working in the manner just described, we are very far from fulfilling this condition: change of temperature is not due here to change of volume; all the changes are due to contact of bodies differently heated—to the contact of the metallic bar, either with the body charged with furnishing heat to it, or with the body charged with carrying it off.
The only means of fulfilling the prescribed condition would be to act upon the solid body exactly as we did on the air in the operations described on page 92. But for this we must be able to produce, by a single change of volume of the solid body, considerable changes of temperature, that is, if we should want to utilize considerable falls of caloric. Now this appears impracticable. In short, many considerations lead to the conclusion that the changes produced in the temperature of solid or liquid bodies through the effect of compression and rarefaction would be but slight.
(1) We often observe in machines (particularly in steam-engines) solid pieces which endure considerable strain in one way or another, and although these efforts may be sometimes as great as the nature of the substances employed permits, the variations of temperature are scarcely perceptible.
(2) In the action of striking medals, in that of the rolling-mill, of the draw-plate, the metals undergo the greatest compression to which we can submit them, employing the hardest and strongest tools. Nevertheless the elevation of temperature is not great. If it were, the pieces of steel used in these operations would soon lose their temper.
(3) We know that it would be necessary to exert on solids and liquids a very great strain in order to produce in them a reduction of volume comparable to that which they experience in cooling (cooling from 100° to zero, for example). Now the cooling requires a greater abstraction of caloric than would simple reduction of volume. If this reduction were produced by mechanical means, the heat set free would not then be able to make the temperature of the body vary as many degrees as the cooling makes it vary. It would, however, necessitate the employment of a force undoubtedly very considerable.
Since solid bodies are susceptible of little change of temperature through changes of volume, and since the condition of the most effective employment of heat for the development of motive power is precisely that all change of temperature should be due to a change of volume, solid bodies appear but ill fitted to realize this power.
The same remarks apply to liquids. The same reasons may be given for rejecting them.[30]
We are not speaking now of practical difficulties. They will be numberless. The motion produced by the dilatation and compression of solid or liquid bodies would only be very slight. In order to give them sufficient amplitude we should be forced to make use of complicated mechanisms. It would be necessary to employ materials of the greatest strength to transmit enormous pressure; finally, the successive operations would be executed very slowly compared to those of the ordinary steam-engine, so that apparatus of large dimensions and heavy cost would produce but very ordinary results.
The elastic fluids, gases or vapors, are the means really adapted to the development of the motive power of heat. They combine all the conditions necessary to fulfil this office. They are easy to compress; they can be almost infinitely expanded; variations of volume occasion in them great changes of temperature; and, lastly, they are very mobile, easy to heat and to cool, easy to transport from one place to another, which enables them to produce rapidly the desired effects. We can easily conceive a multitude of machines fitted to develop the motive power of heat through the use of elastic fluids; but in whatever way we look at it, we should not lose sight of the following principles:
(1) The temperature of the fluid should be made as high as possible, in order to obtain a great fall of caloric, and consequently a large production of motive power.
(2) For the same reason the cooling should be carried as far as possible.
(3) It should be so arranged that the passage of the elastic fluid from the highest to the lowest temperature should be due to increase of volume; that is, it should be so arranged that the cooling of the gas should occur spontaneously as the effect of rarefaction. The limits of the temperature to which it is possible to bring the fluid primarily, are simply the limits of the temperature obtainable by combustion; they are very high.
The limits of cooling are found in the temperature of the coldest body of which we can easily and freely make use; this body is usually the water of the locality.
As to the third condition, it involves difficulties in the realization of the motive power of heat when the attempt is made to take advantage of great differences of temperature, to utilize great falls of heat. In short, it is necessary then that the gas, by reason of its rarefaction, should pass from a very high temperature to a very low one, which requires a great change of volume and of density, which requires also that the gas be first taken under a very heavy pressure, or that it acquire by its dilatation an enormous volume—conditions both difficult to fulfil. The first necessitates the employment of very strong vessels to contain the gas at a very high temperature and under very heavy pressure. The second necessitates the use of vessels of large dimensions. These are, in a word, the principal obstacles which prevent the utilization in steam-engines of a great part of the motive power of the heat. We are obliged to limit ourselves to the use of a slight fall of caloric, while the combustion of the coal furnishes the means of procuring a very great one.
It is seldom that in steam-engines the elastic fluid is produced under a higher pressure than six atmospheres—a pressure corresponding to about 160° Centigrade, and it is seldom that condensation takes place at a temperature much under 40°. The fall of caloric from 160° to 40° is 120°, while by combustion we can procure a fall of 1000° to 2000°.
In order to comprehend this more clearly, let us recall what we have termed the fall of caloric. This is the passage of the heat from one body, _A_, having an elevated temperature, to another, _B_, where it is lower. We say that the fall of the caloric is 100° or 1000° when the difference of temperature between the bodies _A_ and _B_ is 100° or 1000°.
In a steam-engine which works under a pressure of six atmospheres the temperature of the boiler is 160°. This is the body _A_. It is kept, by contact with the furnace, at the constant temperature of 160°, and continually furnishes the heat necessary for the formation of steam. The condenser is the body _B_. By means of a current of cold water it is kept at a nearly constant temperature of 40°. It absorbs continually the caloric brought from the body _A_ by the steam. The difference of temperature between these two bodies is 160° − 40°, or 120°. Hence we say that the fall of caloric is here 120°.
Coal being capable of producing, by its combustion, a temperature higher than 1000°, and the cold water, which is generally used in our climate, being at about 10°, we can easily procure a fall of caloric of 1000°, and of this only 120° are utilized by steam-engines. Even these 120° are not wholly utilized. There is always considerable loss due to useless re-establishments of equilibrium in the caloric.
[Illustration:
Fig. 5. ]
It is easy to see the advantages possessed by high-pressure machines over those of lower pressure. _This superiority lies essentially in the power of utilizing a greater fall of caloric._ The steam produced under a higher pressure is found also at a higher temperature, and as, further, the temperature of condensation remains always about the same, it is evident that the fall of caloric is more considerable. But to obtain from high-pressure engines really advantageous results, it is necessary that the fall of caloric should be most profitably utilized. It is not enough that the steam be produced at a high temperature: it is also necessary that by the expansion of its volume its temperature should become sufficiently low. A good steam-engine, therefore, should not only employ steam under heavy pressure, _but under successive and very variable pressures, differing greatly from one another, and progressively decreasing_.[31]
In order to understand in some sort _à posteriori_ the advantages of high-pressure engines, let us suppose steam to be formed under atmospheric pressure and introduced into the cylindrical vessel _abcd_ (Fig. 5), under the piston _cd_, which at first touches the bottom _ab_. The steam, after having moved the piston from _ab_ to _cd_, will continue finally to produce its results in a manner with which we will not concern ourselves.
Let us suppose that the piston having moved to _cd_ is forced downward to _ef_, without the steam being allowed to escape, or any portion of its caloric to be lost. It will be driven back into the space _abef_, and will increase at the same time in density, elastic force, and temperature. If the steam, instead of being produced under atmospheric pressure, had been produced just when it was being forced back into _abef_, and so that after its introduction into the cylinder it had made the piston move from _ab_ to _ef_, and had moved it simply by its extension of volume, from _ef_ to _cd_, the motive power produced would have been more considerable than in the first case. In fact, the movement of the piston, while equal in extent, would have taken place under the action of a greater pressure, though variable, and though progressively decreasing.
The steam, however, would have required for its formation exactly the same quantity of caloric, only the caloric would have been employed at a higher temperature.
It is considerations of this nature which have led to the making of double-cylinder engines—engines invented by Mr. Hornblower, improved by Mr. Woolf, and which, as regards economy of the combustible, are considered the best. They consist of a small cylinder, which at each pulsation is filled more or less (often entirely) with steam, and of a second cylinder having usually a capacity quadruple that of the first, and which receives no steam except that which has already operated in the first cylinder. Thus the steam when it ceases to act has at least quadrupled in volume. From the second cylinder it is carried directly into the condenser, but it is conceivable that it might be carried into a third cylinder quadruple the second, and in which its volume would have become sixteen times the original volume. The principal obstacle to the use of a third cylinder of this sort is the capacity which it would be necessary to give it, and the large dimensions which the openings for the passage of the steam must have. We will say no more on this subject, as we do not propose here to enter into the details of construction of steam-engines. These details call for a work devoted specially to them, and which does not yet exist, at least in France.[32]
If the expansion of the steam is mainly limited by the dimensions of the vessels in which the dilatation must take place, the degree of condensation at which it is possible to use it at first is limited only by the resistance of the vessels in which it is produced, that is, of the boilers.
In this respect we have by no means attained the best possible results. The arrangement of the boilers generally in use is entirely faulty, although the tension of the steam rarely exceeds from four to six atmospheres. They often burst and cause severe accidents. It will undoubtedly be possible to avoid such accidents, and meantime to raise the steam to much greater pressures than is usually done.
Besides the high-pressure double-cylinder engines of which we have spoken, there are also high-pressure engines of one cylinder. The greater part of these latter have been constructed by two ingenious English engineers, Messrs. Trevithick and Vivian. They employ the steam under a very high pressure, sometimes eight to ten atmospheres, but they have no condenser. The steam, after it has been introduced into the cylinder, undergoes therein a certain increase of volume, but preserves always a pressure higher than atmospheric. When it has fulfilled its office it is thrown out into the atmosphere. It is evident that this mode of working is fully equivalent, in respect to the motive power produced, to condensing the steam at 100°, and that a portion of the useful effect is lost. But the engines working thus dispense with condenser and air-pump. They are less costly than the others, less complicated, occupy less space, and can be used in places where there is not sufficient water for condensation. In such places they are of inestimable advantage, since no others could take their place. These engines are principally employed in England to move coal-wagons on railroads laid either in the interior of mines or outside of them.
We have, further, only a few remarks to make upon the use of permanent gases and other vapors than that of water in the development of the motive power of heat.
Various attempts have been made to produce motive power by the action of heat on atmospheric air. This gas presents, as compared with vapor of water, both advantages and disadvantages, which we will proceed to examine.
(1) It presents, as compared with vapor of water, a notable advantage in that, having for equal volume a much less capacity for heat, it would cool more rapidly by an equal increase of volume. (This fact is proved by what has already been stated.) Now we have seen how important it is to produce by change of volume the greatest possible changes of temperature.
(2) Vapors of water can be formed only through the intervention of a boiler, while atmospheric air could be heated directly by combustion carried on within its own mass. Considerable loss could thus be prevented, not only in the quantity of heat, but also in its temperature. This advantage belongs exclusively to atmospheric air. Other gases do not possess it. They would be even more difficult to heat than vapor of water.
(3) In order to give to air great increase of volume, and by that expansion to produce a great change of temperature, it must first be taken under a sufficiently high pressure; then it must be compressed with a pump or by some other means before heating it. This operation would require a special apparatus, an apparatus not found in steam-engines. In the latter, water is in a liquid state when injected into the boiler, and to introduce it requires but a small pump.
(4) The condensing of the vapor by contact with the refrigerant body is much more prompt and much easier than is the cooling of air. There might, of course, be the expedient of throwing the latter out into the atmosphere, and there would be also the advantage of avoiding the use of a refrigerant, which is not always available, but it would be requisite that the increase of the volume of the air should not reduce its pressure below that of the atmosphere.
(5) One of the gravest inconveniences of steam is that it cannot be used at high temperatures without necessitating the use of vessels of extraordinary strength. It is not so with air for which there exists no necessary relation between the elastic force and the temperature. Air, then, would seem more suitable than steam to realize the motive power of falls of caloric from high temperatures. Perhaps in low temperatures steam may be more convenient. We might conceive even the possibility of making the same heat act successively upon air and vapor of water. It would be only necessary that the air should have, after its use, an elevated temperature, and instead of throwing it out immediately into the atmosphere, to make it envelop a steam-boiler, as if it issued directly from a furnace.
The use of atmospheric air for the development of the motive power of heat presents in practice very great, but perhaps not insurmountable, difficulties. If we should succeed in overcoming them, it would doubtless offer a notable advantage over vapor of water.[33]
As to the other permanent gases, they should be absolutely rejected. They have all the inconveniences of atmospheric air, with none of its advantages. The same can be said of other vapors than that of water, as compared with the latter.
If we could find an abundant liquid body which would vaporize at a higher temperature than water, of which the vapor would have, for the same volume, a less specific heat, which would not attack the metals employed in the construction of machines, it would undoubtedly merit the preference. But nature provides no such body.
The use of the vapor of alcohol has been proposed. Machines have even been constructed for the purpose of using it, by avoiding the mixture of its vapor with the water of condensation, that is, by applying the cold body externally instead of introducing it into the machine. It has been thought that a remarkable advantage might be secured by using the vapor of alcohol in that it possesses a stronger tension than the vapor of water at the same temperature. We can see in this only a fresh obstacle to be overcome. The principal defect of the vapor of water is its excessive tension at an elevated temperature; now this defect exists still more strongly in the vapor of alcohol. As to the relative advantage in a greater production of motive power,—an advantage attributed to it,—we know by the principles above demonstrated that it is imaginary.
It is thus upon the use of atmospheric air and vapor of water that subsequent attempts to perfect heat-engines should be based. It is to utilize by means of these agents the greatest possible falls of caloric that all efforts should be directed.
Finally, we will show how far we are from having realized, by any means at present known, all the motive power of combustibles.
One kilogram of carbon burnt in the calorimeter furnishes a quantity of heat capable of raising one degree Centigrade about 7000 kilograms of water, that is, it furnishes 7000 units of heat according to the definition of these units given on page 100.
The greatest fall of caloric attainable is measured by the difference between the temperature produced by combustion and that of the refrigerant bodies. It is difficult to perceive any other limits to the temperature of combustion than those in which the combination between oxygen and the combustible may take place. Let us assume, however, that 1000° may be this limit, and we shall certainly be below the truth. As to the temperature of the refrigerant, let us suppose it 0°. We estimated approximately (page 104) the quantity of motive power that 1000 units of heat develop between 100° and 99°. We found it to be 1.112 units of power, each equal to 1 metre of water raised to a height of 1 metre.
If the motive power were proportional to the fall of caloric, if it were the same for each thermometric degree, nothing would be easier than to estimate it from 1000° to 0°. Its value would be
1.112 × 1000 = 1112.
But as this law is only approximate, and as possibly it deviates much from the truth at high temperatures, we can only make a very rough estimate. We will suppose the number 1112 reduced one half, that is, to 560.
Since a kilogram of carbon produces 7000 units of heat, and since the number 560 is relatively 1000 units, it must be multiplied by 7, which gives
7 × 560 = 3920.
This is the motive power of 1 kilogram of carbon.
In order to compare this theoretical result with that of experiment, let us ascertain how much motive power a kilogram of carbon actually develops in the best-known steam-engines.
The engines which, up to this time, have shown the best results are the large double-cylinder engines used in the drainage of the tin and copper mines of Cornwall. The best results that have been obtained with them are as follows:
65 millions of lbs. of water have been raised one English foot by the bushel of coal burned (the bushel weighing 88 lbs.). This is equivalent to raising, by a kilogram of coal, 195 cubic metres of water to a height of 1 metre, producing thereby 195 units of motive power per kilogram of coal burned.
195 units are only the twentieth of 3920, the theoretical maximum; consequently ¹⁄₂₀ only of the motive power of the combustible has been utilized.
We have, nevertheless, selected our example from among the best steam-engines known.
Most engines are greatly inferior to these. The old engine of Chaillot, for example, raised twenty cubic metres of water thirty-three metres, for thirty kilograms of coal consumed, which amounts to twenty-two units of motive power per kilogram,—a result nine times less than that given above, and one hundred and eighty times less than the theoretical maximum.
We should not expect ever to utilize in practice all the motive power of combustibles. The attempts made to attain this result would be far more hurtful than useful if they caused other important considerations to be neglected. The economy of the combustible is only one of the conditions to be fulfilled in heat-engines. In many cases it is only secondary. It should often give precedence to safety, to strength, to the durability of the engine, to the small space which it must occupy, to small cost of installation, etc. To know how to appreciate in each case, at their true value, the considerations of convenience and economy which may present themselves; to know how to discern the more important of those which are only accessories; to balance them properly against each other, in order to attain the best results by the simplest means: such should be the leading characteristics of the man called to direct, to co-ordinate among themselves the labors of his comrades, to make them co-operate towards one useful end, of whatsoever sort it may be.
[Illustration:
(_To face p. 127._) ]
IV.[34] CARNOT’S THEORY OF THE MOTIVE POWER OF HEAT.[35] WITH NUMERICAL RESULTS DEDUCED FROM REGNAULT’S EXPERIMENTS ON STEAM.[36]
BY SIR WILLIAM THOMSON [LORD KELVIN].
1. The presence of heat may be recognized in every natural object; and there is scarcely an operation in nature which is not more or less affected by its all-pervading influence. An evolution and subsequent absorption of heat generally give rise to a variety of effects; among which may be enumerated, chemical combinations or decompositions; the fusion of solid substances; the vaporization of solids or liquids; alterations in the dimensions of bodies, or in the statical pressure by which their dimensions may be modified; mechanical resistance overcome; electrical currents generated. In many of the actual phenomena of nature several or all of these effects are produced together; and their complication will, if we attempt to trace the agency of heat in producing any individual effect, give rise to much perplexity. It will, therefore, be desirable, in laying the foundation of a physical theory of any of the effects of heat, to discover or to imagine phenomena free from all such complication, and depending on a definite thermal agency; in which the relation between the cause and effect, traced through the medium of certain simple operations, may be clearly appreciated. Thus it is that Carnot, in accordance with the strictest principles of philosophy, enters upon the investigation of the theory of the motive power of heat.
2. The sole effect to be contemplated in investigating the motive power of heat is _resistance overcome_, or, as it is frequently called, “_work performed_,” or “_mechanical effect_.” The questions to be resolved by a complete theory of the subject are the following:
(1) What is the precise nature of the thermal agency by means of which _mechanical effect_ is to be produced, without effects of any other kind?
(2) How may the amount of this thermal agency necessary for performing a given quantity of work be estimated?
3. In the following paper I shall commence by giving a short abstract of the reasoning by which Carnot is led to an answer to the first of these questions; I shall then explain the investigation by which, in accordance with his theory, the experimental elements necessary for answering the second question are indicated; and, in conclusion, I shall state the _data_ supplied by Regnault’s recent observations on steam, and apply them to obtain, as approximately as the present state of experimental science enables us to do, a complete solution of the question.
I. On the nature of Thermal agency, considered as a motive power.
4. There are [at present known] two, and only two, distinct ways in which mechanical effect can be obtained from heat. One of these is by means of the alterations of volume, which bodies may experience through the action of heat; the other is through the medium of electric agency. Seebeck’s discovery of thermo-electric currents enables us at present to conceive of an electro-magnetic engine supplied from a thermal origin, being used as a motive power; but this discovery was not made until 1821, and the subject of thermo-electricity can only have been generally known in a few isolated facts, with reference to the electrical effects of heat upon certain crystals, at the time when Carnot wrote. He makes no allusion to it, but confines himself to the method for rendering thermal agency available as a source of mechanical effect, by means of the expansions and contractions of bodies.
5. A body expanding or contracting under the action of force may, in general, either produce mechanical effect by overcoming resistance, or receive mechanical effect by yielding to the action of force. The amount of mechanical effect thus developed will depend not only on the calorific agency concerned, but also on the alteration in the physical condition of the body. Hence, after allowing the volume and temperature of the body to change, we must restore it to its original temperature and volume; and then we may estimate the aggregate amount of mechanical effect developed as due solely to the thermal origin.
6. Now the ordinarily-received, and almost universally-acknowledged, principles with reference to “quantities of caloric” and “latent heat” lead us to conceive that, at the end of a cycle of operations, when a body is left in precisely its primitive physical condition, if it has absorbed any heat during one part of the operations, it must have given out again exactly the same amount during the remainder of the cycle. The truth of this principle is considered as axiomatic by Carnot, who admits it as the foundation of his theory; and expresses himself in the following terms regarding it, in a note on one of the passages of his treatise:[37]
“In our demonstrations we tacitly assume that after a body has experienced a certain number of transformations, if it be brought identically to its primitive physical state as to density, temperature, and molecular constitution, it must contain the same quantity of heat as that which it initially possessed; or, in other words, we suppose that the quantities of heat lost by the body under one set of operations are precisely compensated by those which are absorbed in the others. This fact has never been doubted; it has at first been admitted without reflection, and afterwards verified, in many cases, by calorimetrical experiments. To deny it would be to overturn the whole theory of heat, in which it is the fundamental principle. It must be admitted, however, that the chief foundations on which the theory of heat rests, would require a most attentive examination. Several experimental facts appear nearly inexplicable in the actual state of this theory.”
7. Since the time when Carnot thus expressed himself, the necessity of a most careful examination of the entire experimental basis of the theory of heat has become more and more urgent. Especially all those assumptions depending on the idea that heat is a _substance_, invariable in quantity; not convertible into any other element, and incapable of being _generated_ by any physical agency; in fact the acknowledged principles of latent heat,—would require to be tested by a most searching investigation before they ought to be admitted, as they usually have been, by almost every one who has been engaged on the subject, whether in combining the results of experimental research, or in general theoretical investigations.
8. The extremely important discoveries recently made by Mr. Joule of Manchester, that heat is evolved in every part of a closed electric conductor, moving in the neighborhood of a magnet,[38] and that heat is _generated_ by the friction of fluids in motion, seem to overturn the opinion commonly held that heat cannot be _generated_, but only produced from a source, where it has previously existed either in a sensible or in a latent condition.
In the present state of science, however, no operation is known by which heat can be absorbed into a body without either elevating its temperature or becoming latent, and producing some alteration in its physical condition; and the fundamental axiom adopted by Carnot may be considered as still the most probable basis for an investigation of the motive power of heat; although this, and with it every other branch of the theory of heat, may ultimately require to be reconstructed upon another foundation, when our experimental data are more complete. On this understanding, and to avoid a repetition of doubts, I shall refer to Carnot’s fundamental principle, in all that follows, as if its truth were thoroughly established.
9. We are now led to the conclusion that the origin of motive power, developed by the alternate expansions and contractions of a body, must be found in the agency of heat entering the body and leaving it; since there cannot, at the end of a complete cycle, when the body is restored to its primitive physical condition, have been any absolute absorption of heat, and consequently no conversion of heat, or caloric, into mechanical effect; and it remains for us to trace the precise nature of the circumstances under which heat must enter the body, and afterwards leave it, so that mechanical effect may be produced. As an example, we may consider that machine for obtaining motive power from heat with which we are most familiar—the steam-engine.
10. Here, we observe, that heat enters the machine from the furnace, through the sides of the boiler, and that heat is continually abstracted by the water employed for keeping the condenser cool. According to Carnot’s fundamental principle, the quantity of heat thus discharged, during a complete revolution (or double stroke) of the engine, must be precisely equal to that which enters the water of the boiler;[39] provided the total mass of water and steam be invariable, and be restored to its primitive physical condition (which will be the case rigorously, if the condenser be kept cool by the external application of cold water instead of by injection, as is more usual in practice), and if the condensed water be restored to the boiler at the end of each complete revolution. Thus we perceive that a certain quantity of heat is _let down_ from a hot body, the metal of the boiler, to another body at a lower temperature, the metal of the condenser; and that there results from this transference of heat a certain development of mechanical effect.
11. If we examine any other case in which mechanical effect is obtained from a thermal origin, by means of the alternate expansions and contractions of any substance whatever, instead of the water of a steam-engine, we find that a similar transference of heat is effected, and we may therefore answer the first question proposed, in the following manner:
_The thermal agency by which mechanical effect may be obtained is the transference of heat from one body to another at a lower temperature._
11. On the measurement of Thermal Agency, considered with reference to its equivalent of mechanical effect.
12. A _perfect_ thermodynamic engine of any kind is a machine by means of which the greatest possible amount of mechanical effect can be obtained from a given thermal agency; and, therefore, if in any manner we can construct or imagine a perfect engine which may be applied for the transference of a given quantity of heat from a body at any given temperature to another body at a lower given temperature, and if we can evaluate the mechanical effect thus obtained, we shall be able to answer the question at present under consideration, and so to complete the theory of the motive power of heat. But whatever kind of engine we may consider with this view, it will be necessary for us to prove that it is a perfect engine; since the transference of the heat from one body to the other may be wholly, or partially, effected by conduction through a solid,[40] without the development of mechanical effect; and, consequently, engines may be constructed in which the whole or any portion of the thermal agency is wasted. Hence it is of primary importance to discover the criterion of a perfect engine. This has been done by Carnot, who proves the following proposition:
13. _A perfect thermodynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency, if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced._[41]
14. This proposition will be made clearer by the applications of it which are given later (§ 29), in the cases of the air-engine and the steam-engine, than it could be by any general explanation; and it will also appear, from the nature of the operations described in those cases, and the principles of Carnot’s reasoning, that a perfect engine may be constructed with any substance of an indestructible texture as the alternately expanding and contracting medium. Thus we might conceive thermodynamic engines founded upon the expansions and contractions of a perfectly elastic solid, or of a liquid; or upon the alterations of volume experienced by substances in passing from the liquid to the solid state,[42] each of which being perfect, would produce the same amount of mechanical effect from a given thermal agency; but there are two cases which Carnot has selected as most worthy of minute attention, because of their peculiar appropriateness for illustrating the general principles of his theory, no less than on account of their very great practical importance: the steam-engine, in which the substance employed as the transferring medium is water, alternately in the liquid state and in the state of vapor; and the air-engine, in which the transference is effected by means of the alternate expansions and contractions of a medium always in the gaseous state. The details of an actually practicable engine of either kind are not contemplated by Carnot in his general theoretical reasonings, but he confines himself to the ideal construction, in the simplest possible way in each case, of an engine in which the economy is perfect. He thus determines the degree of perfectibility which cannot be surpassed; and by describing a conceivable method of attaining to this perfection by an air-engine or a steam-engine, he points out the proper objects to be kept in view in the practical construction and working of such machines. I now proceed to give an outline of these investigations.
CARNOT’S THEORY OF THE STEAM-ENGINE.
15. Let _CDF_{2}E_{2}_ be a cylinder, of which the curved surface is perfectly impermeable to heat, with a piston also impermeable to heat, fitted in it; while the fixed bottom _CD_, itself with no capacity for heat, is possessed of perfect conducting power. Let _K_ be an impermeable stand, such that when the cylinder is placed upon it the contents below the piston can neither gain nor lose heat. Let _A_ and _B_ be two bodies permanently retained at constant temperatures, _S°_ and _T°_, respectively, of which the former is higher than the latter. Let the cylinder, placed on the impermeable stand, _K_, be partially filled with water, at the temperature _S_, of the body _A_, and (there being no air below it) let the piston be placed in a position _EF_, near the surface of the water. The pressure of the vapor above the water will tend to push up the piston, and must be resisted by a force applied to the piston,[43] till the commencement of the operations, which are conducted in the following manner:
[Illustration]
(1) The cylinder being placed on the body _A_, so that the water and vapor may be retained at the temperature _S_, _let the piston rise any convenient height EE_{1}, to a position E_{1}F_{1}, performing work by the pressure of the vapor below it during its ascent_.
[During this operation a certain quantity, _H_, of heat, the amount of latent heat in the fresh vapor which is formed, is abstracted from the body _A_.]
(2) The cylinder being removed, and placed on the impermeable stand _K, let the piston rise gradually, till, when it reaches a position E_{2}F_{2}, the temperature of the water and vapor is T, the same as that of the body B_.
[During this operation the fresh vapor continually formed requires heat to become latent; and, therefore, as the contents of the cylinder are protected from any accession of heat, their temperature sinks.]
(3) The cylinder being removed from _K_, and placed on _B, let the piston be pushed down, till, when it reaches the position E_{3}F_{3}, the quantity of heat evolved and abstracted by B amounts to that which, during the first operation, was taken from A_.
[Note of Nov. 5, 1881. The specification of this operation, with a view to the return to the primitive condition, intended as the conclusion to the four operations, is the only item in which Carnot’s temporary and provisional assumption of the materiality of heat has effect. To exclude this hypothesis, Prof. James Thomson has suggested the following corrected specification for the third operation: _Let the piston be pushed down, till it reaches a position E_{3}F_{3}, determined so as to fulfil the condition, that at the end of the fourth operation the primitive temperature S shall be reached_:[44]]
[During this operation the temperature of the contents of the cylinder is retained constantly at _T°_, and all the latent heat of the vapor which is condensed into water at the same temperature is given out to _B_.]
(4) The cylinder being removed from _B_, and placed on the impermeable stand, _let the piston be pushed down from E_{3}F_{3} to its original position EF_.
[During this operation, the impermeable stand preventing any loss of heat, the temperature of the water and air must rise continually, till (since the quantity of heat evolved during the third operation was precisely equal to that which was previously absorbed) at the conclusion it reaches its primitive value, _S_, in virtue of Carnot’s fundamental axiom.]
[Note of Nov. 5, 1881. With Prof. James Thomson’s correction of operation (3), the words in virtue of “Carnot’s Fundamental Axiom” must be replaced by “the condition fulfilled by operation (3),” in the description of the results of operation (4).]
16. At the conclusion of this cycle of operations[45] the total thermal agency has been the _letting down_ of _H_ units of heat from the body _A_, at the temperature _S_, to _B_, at the lower temperature _T_; and the aggregate of the mechanical effect has been a certain amount of _work produced_, since during the ascent of the piston in the first and second operations, the temperature of the water and vapor, and therefore the pressure of the vapor on the piston, was on the whole higher than during the descent, in the third and fourth operations. It remains for us actually to evaluate this aggregate amount of work performed; and for this purpose the following graphical method of representing the mechanical effect developed in the several operations, taken from Mons. Clapeyron’s paper, is extremely convenient.
17. Let _OX_ and _OY_ be two lines at right angles to one another. Along _OX_ measure off distances _ON_{1}_, _N_{1}N_{2}_, _N_{2}N_{3}_, _N_{3}O_, respectively proportional to the spaces described by the piston during the four successive operations described above; and, with reference to these four operations respectively, let the following constructions be made:
(1) Along _OY_ measure a length _OA_, to represent the pressure of the saturated vapor at the temperature _S_; and draw _AA_{1}_ parallel to _OX_, and let it meet an ordinate through _N_{1}_, in _A_{1}_.
(2) Draw a curve _A_{1}PA_ such that, if _ON_ represent, at any instant during the second operation, the distance of the piston from its primitive position, _NP_ shall represent the pressure of the vapor at the same instant.
(3) Through _A__{2} draw _A_{2}A_{3}_ parallel to _OX_, and let it meet an ordinate through _N_{3}_ in _A_{3}_.
(4) Draw the curve _A_{3}A_ such that the abscissa and ordinate of any point in it may represent respectively the distances of the piston from its primitive position, and the pressure of the vapor, at each instant during the fourth operation. The last point of this curve must, according to Carnot’s fundamental principle, coincide with _A_, since the piston is, at the end of the cycle of operations, again in its primitive position, and the pressure of the vapor is the same as it was at the beginning.
[Illustration]
18. Let us now suppose that the lengths, _ON_{1}_, _N_{1}N_{2}_, _N_{2}N_{3}_, and _N_{3}O_, _represent numerically_ the volumes of the spaces moved through by the piston during the successive operations. It follows that the mechanical effect obtained during the first operation will be _numerically represented_ by the area _AA_{1}N_{1}O_; that is, the number of superficial units in this area will be equal to the number of “foot-pounds” of work performed by the ascending piston during the first operation. The work performed by the piston during the second operation will be similarly represented by the area _A_{1}A_{2}N_{2}N_{1}_. Again, during the third operation a certain amount of work is spent on the piston, which will be represented by the area _A_{2}A_{3}N_{3}N_{2}_; and lastly, during the fourth operation, work is spent in pushing the piston to an amount represented by the area _A_{3}AON_{3}_.
19. Hence the mechanical effect (represented by the area _OAA_{1}A_{2}N_{2}_) which was obtained during the first and second operations, exceeds the work (represented by _N_{2}A_{2}A_{3}AO_) spent during the third and fourth, by an amount represented by the area of the quadrilateral figure _AA_{1}A_{2}A_{3}_; and, consequently, it only remains for us to evaluate this area, that we may determine the total mechanical effect gained in a complete cycle of operations. Now, from experimental data, at present nearly complete, as will be explained below, we may determine the length of the line _AA_{1}_ for the given temperature _S_, and a given absorption _H_, of heat, during the first operation; and the length of _A_{2}A_{3}_ for the given lower temperature _T_, and the evolution of the same quantity of heat during the fourth operation: and the curves _A_{1}PA_{2}_, _A_{3}P′A_ may be drawn as graphical representations of actual observations. The figure being thus constructed, its area may be measured, and we are, therefore, in possession of a graphical method of determining the amount of mechanical effect to be obtained from any given thermal agency. As, however, it is merely the area of the figure which it is required to determine, it will not be necessary to be able to describe each of the curves _A_{1}PA_{2}_, _A_{3}P′A_, but it will be sufficient to know the difference of the abscissas corresponding to any equal ordinates in the two; and the following analytical method of completing the problem is the most convenient for leading to the actual numerical results.
20. Draw any line _PP′_ parallel to _OX_, meeting the curvilinear sides of the quadrilateral in _P_ and _P′_. Let ξ denote the length of this line, and _p_ its distance from _OX_. The area of the figure, according to the integral calculus, will be denoted by the expression
∫_{_p_{3}_} ^{_p_{1}_} ξ_dp_,
where _p_{1}_ and _p_{3}_ (the limits of integration indicated according to Fourier’s notation) denote the lines _OA_ and _N_{3}A_{3}_, which represent respectively the pressures during the first and third operations. Now, by referring to the construction described above, we see that ξ is the difference of the volumes below the piston at corresponding instants of the second and fourth operations, or instants at which the saturated steam and the water in the cylinder have the same pressure _p_, and consequently the same temperature, which we may denote by _t_. Again, throughout the second operation the entire contents of the cylinder possess a greater amount of heat by _H_ units than during the fourth; and, therefore, at any instant of the second operation there is as much more steam as contains _H_ units of latent heat than at the corresponding instant of the fourth operation. Hence if _k_ denote the latent heat in a unit of saturated steam at the temperature _t_, the volume of the steam at the two corresponding instants must differ by (_H_)/(_k_). Now, if σ denote the ratio of the density of the steam to that of the water, the volume (_H_)/(_k_) of steam will be formed from the volume σ (_H_)/(_k_) of water; and consequently we have, for the difference of volumes of the entire contents at the corresponding instants,
ξ = (1 - σ)(_H_)/(_k_).
Hence the expression for the area of the quadrilateral figure becomes
∫^{_p_{1}_}_{_p_{3}_}(1 - σ)(_H_)/(_k_)_dp_.
Now, σ, _k_, and _p_, being quantities which depend upon the temperature, may be considered as functions of _t_; and it will be convenient to modify the integral so as to make _t_ the independent variable. The limits will be from _t_ = _T_ to _t_ = _S_, and, if we denote by _M_ the value of the integral, we have the expression
_M_ = _H_ ∫_{_T_}^{_S_}(1 - σ)((_dp_/_dt_)/_k_)_dt_. (1)
for the total amount of mechanical effect gained by the operations described above.
21. If the interval of temperatures be extremely small,—so small that (1 − σ)(_dp_)/(_dt_/_k_) will not sensibly vary for values of _t_ between _T_ and _S_,—the preceding expression becomes simply
_Μ_ = (1 - σ)(_dp_)/(_dt_)/(_k_). _Η_(_S_ - _Τ_). (2)
This might, of course, have been obtained at once by supposing the breadth of the quadrilateral figure _AA_{1}A_{2}A_ to be extremely small compared with its length, and then taking for its area, as an approximate value, the product of the breadth into the line _AA_{1}_, or the line _A_{3}A_{2}_, or any line of intermediate magnitude.
The expression (2) is rigorously correct for any interval _S_ − _T_, if the mean value of (1 − σ)((_dp_/_dt_)/_k_) for that interval be employed as the coefficient of _H_(_S_ − _T_).
CARNOT’S THEORY OF THE AIR-ENGINE.
22. In the ideal air-engine imagined by Carnot four operations performed upon a mass of air or gas enclosed in a closed vessel of variable volume constitute a complete cycle, at the end of which the medium is left in its primitive physical condition; the construction being the same as that which was described above for the steam-engine, a body _A_, permanently retained at the temperature _S_, and _B_ at the temperature _T_; an impermeable stand _K_; and a cylinder and piston, which in this case contains a mass of air at the temperature _S_, instead of water in the liquid state, at the beginning and end of a cycle of operations. The four successive operations are conducted in the following manner:
(1) The cylinder is laid on the body _A_, so that the air in it is kept at the temperature _S_; and the piston is allowed to rise, performing work.
(2) The cylinder is placed on the impermeable stand _K_, so that its contents can neither gain nor lose heat, and the piston is allowed to rise farther, still performing work, till the temperature of the air sinks to _T_.
(3) The cylinder is placed on _B_, so that the air is retained at the temperature _T_, and the piston is pushed down till the air gives out to the body _B_ as much heat as it had taken in from _A_, during the first operation.
[Note of Nov. 5, 1881. To eliminate the assumption of the materiality of heat, make Professor James Thomson’s correction here also; as above in § 15; or take Maxwell’s rearrangement of the cycle described in the foot-note to § 15, p. 144.]
(4) The cylinder is placed on _K_, so that no more heat can be taken in or given out, and the piston is pushed down to its primitive position.
23. _At the end of the fourth operation the temperature must have reached its primitive value S, in virtue of_ CARNOT’S _axiom_.
24. Here, again, as in the former case, we observe that work is performed by the piston during the first two operations; and during the third and fourth work is spent upon it, but to a less amount, since the pressure is on the whole less during the third and fourth operations than during the first and second, on account of the temperature being lower. Thus, at the end of a complete cycle of operations, mechanical effect has been obtained; and the thermal agency from which it is drawn is the taking of a certain quantity of heat from _A_, and _letting it down_, through the medium of the engine, to the body _B_ at a lower temperature.
25. To estimate the actual amount of effect thus obtained, it will be convenient to consider the alterations of volume of the mass of air in the several operations as extremely small. We may afterwards pass by the integral calculus, or, practically, by summation to determine the mechanical effect whatever be the amplitudes of the different motions of the piston.
26. Let _dq_ be the quantity of heat absorbed during the first operation, which is evolved again during the third; and let _dv_ be the corresponding augmentation of volume which takes place while the temperature remains constant, as it does during the first operation.[46] The diminution of volume in the third operation must be also equal to _dv_, or only differ from it by an infinitely small quantity of the second order. During the second operation we may suppose the volume to be increased by an infinitely small quantity φ; which will occasion a diminution of pressure and a diminution of temperature, denoted respectively by ω and τ. During the fourth operation there will be a diminution of volume and an increase of pressure and temperature, which can only differ, by infinitely small quantities of the second order, from the changes in the other direction, which took place in the second operation, and they also may, therefore, be denoted by φ, ω, and τ, respectively. The alteration of pressure during the first and third operations may at once be determined by means of Mariotte’s law, since in them the temperature remains constant. Thus, if, at the commencement of the cycle, the volume and pressure be _v_ and _p_, they will have become _v_ + _dv_ and _pv_/(_v_ + _dv_) at the end of the first operation. Hence the diminution of pressure during the first operation is _p_ − _pv_/(_v_ + _dv_) or _pdv_/(_v_ + _dv_) and therefore, if we neglect infinitely small quantities of the second order, we have _pdv_/_v_ for the diminution of pressure during the first operation; which to the same degree of approximation, will be equal to the increase of pressure during the third. If _t_ + τ and _t_ be taken to denote the superior and inferior limits of temperature, we shall thus have for the volume, the temperature, and the pressure at the commencements of the four successive operations, and at the end of the cycle, the following values respectively:
(1) _v_, _t_ + τ, _p_; (2) _v_ + _dv_, _t_ + τ, _p_(1 − (_dv_)/(_v_)); (3) _v_ + _dv_ + φ, _t_, _p_(1 − (_dv_)/(_v_)) − ω; (4) _v_ + φ, _t_, _p_ − ω; (5) _v_, _t_ + τ, _p_.
Taking the mean of the pressures at the beginning and end of each operation, we find
(1) _p_(1 − ½(_dv_)/(_v_)),
(2) _p_(1 − (_dv_)/(_v_)) − ½ω,
(3) _p_(1 − ½(_dv_)/(_v_))) − ω,
(4) _p_ − ½ω,
which, as we are neglecting infinitely small quantities of the second order, will be the expressions for the mean pressures during the four successive operations. Now, the mechanical effect gained or spent, during any of the operations, will be found by multiplying the mean pressure by the increase or diminution of volume which takes place; and we thus find
(1) _p_(1 − ½(_dv_)/(_v_))_dv_,
(2) {_p_(1 − (_dv_)/(_v_)) − ½ω}φ,
(3) {_p_(1 − ½(_dv_)/(_v_)) − ω}_dv_,
(4) (_p_ − ½ω)φ.
for the amounts gained during the first and second, and spent during the third and fourth operations; and hence, by addition and subtraction, we find
ω_dv_ − _p_φ(_dv_)/(_v_), or (_v_ω − _p_φ)(_dv_)/(_v_),
for the aggregate amount of mechanical effect gained during the cycle of operations. It only remains for us to express this result in terms of _dq_ and τ, on which the given thermal agency depends. For this purpose we remark that φ and ω are alterations of volume and pressure which take place along with a change of temperature τ, and hence, by the laws of compressibility and expansion, we may establish a relation[47] between them in the following manner:
Let _p_{0}_ be the pressure of the mass of air when reduced to the temperature zero, and confined in a volume _v_{0}_; then, whatever be _v_{0}_, the product _p_{0}v_{0}_ will, by the law of compressibility, remain constant; and, if the temperature be elevated from 0 to _t_ + τ, and the gas be allowed to expand freely without any change of pressure, its volume will be increased in the ratio of 1 to 1 + _E_(_t_ + τ), where _E_ is very nearly equal to .00366 (the Centigrade scale of the air-thermometer being referred to), whatever be the gas employed, according to the researches of Regnault and of Magnus on the expansion of gases by heat. If, now, the volume be altered arbitrarily with the temperature continually at _t_ + τ, the product of the pressure and volume will remain constant; and therefore we have
_pv_ = _p_{0}v_{0}_{1 + _E_(_t_ + τ)}.
Similarly,
(_p_ − ω)(_v_ + φ) = _p_{0}v_{0}_{1 + _Et_}.
Hence, by subtraction, we have
_v_ω − _p_φ + ωφ = _p_{0}v_{0}E_τ,
or, neglecting the product ωφ,
_v_ω − _p_φ = _p_{0}v_{0}E_τ.
Hence the preceding expression for mechanical effect, gained in the cycle of operations, becomes
_p_{0}v_{0}_. _E_τ . _dv_/_v_.
Or, as we may otherwise express it,
(_Ep_{0}v_{0}_)/(_vdq_/_dv_). _dq_. τ.
Hence, if we denote by _M_ the mechanical effect due to _H_ units of heat descending through the same interval τ, which might be obtained by repeating the cycle of operations described above, (_H_)/(_dq_) times, we have
_M_ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (3)
27. If the _amplitudes_ of the operations had been finite, so as to give rise to an absorption of _H_ units of heat during the first operation, and a lowering of temperature from _S_ to _T_ during the second, the amount of work obtained would have been found to be expressed by means of a double definite integral thus:[48]
_M_ = ∫_{0}^{_H_} _dq_ ∫_{_T_}^{_S_} _dt_. (_Ep_{0}v_{0}_)/(_vdq_/_dv_), ⎫ or ⎬. (4) _M_ = _Ep_{0}v_{0}_ ∫_{0}^{_H_} ∫_{_T_}^{_S_} (1)/(_v_) (_dv_)/(_dq_). _dtdq_; ⎭
this second form being sometimes more convenient.
28. The preceding investigations, being founded on the approximate laws of compressibility and expansion (known as the law of Mariotte and Boyle, and the law of Dalton and Gay-Lussac), would require some slight modifications to adapt them to cases in which the gaseous medium employed is such as to present sensible deviations from those laws. Regnault’s very accurate experiments show that the deviations are insensible, or very nearly so, for the ordinary gases at ordinary pressures; although they may be considerable for a medium, such as sulphurous acid, or carbonic acid under high pressure, which approaches the physical condition of a vapor at saturation; and therefore, in general, and especially in practical applications to real air-engines, it will be unnecessary to make any modification in the expressions. In cases where it may be necessary, there is no difficulty in making the modifications, when the requisite data are supplied by experiment.
29.[49] Either the steam-engine or the air-engine, according to the arrangements described above, gives all the mechanical effect that can possibly be obtained from the thermal agency employed. For it is clear that in either case the operations may be performed in the reverse order, with every thermal and mechanical effect reversed. Thus, in the steam-engine, we may commence by placing the cylinder on the impermeable stand, allow the piston to rise, performing work, to the position _E_{3}F_{3}_; we may then place it on the body _B_, and allow it to rise, performing work, till it reaches _E_{2}F_{2}_ after that the cylinder may be placed again on the impermeable stand, and the piston may be pushed down to _E_{1}F_{1}_; and, lastly, the cylinder being removed to the body _A_, the piston may be pushed down to its primitive position. In this inverse cycle of operations a certain amount of work has been spent, precisely equal, as we readily see, to the amount of mechanical effect gained in the direct cycle described above; and heat has been abstracted from _B_, and deposited in the body _A_, at a higher temperature, to an amount precisely equal to that which in the direct style was _let down_ from _A_ to _B_. Hence it is impossible to have an engine which will derive more mechanical effect from the same thermal agency than is obtained by the arrangement described above; since, if there could be such an engine, it might be employed to perform, as a part of its whole work, the inverse cycle of operations, upon an engine of the kind we have considered, and thus to continually restore the heat from _B_ to _A_, which has descended from _A_ to _B_ for working itself; so that we should have a complex engine, giving a residual amount of mechanical effect without any thermal agency, or alteration of materials, which is an impossibility in nature. The same reasoning is applicable to the air-engine; and we conclude, generally, that any two engines, constructed on the principles laid down above, whether steam-engines with different liquids, an air-engine and a steam-engine, or two air-engines with different gases, must derive the same amount of mechanical effect from the same thermal agency.
30. Hence, by comparing the amounts of mechanical effect obtained by the steam-engine and the air-engine from the letting down of the _H_ units of heat from _A_ at the temperature (_t_ + τ) to _B_ at _t_, according to the expressions (2) and (3), we have
_M_ = (1 − σ)(_dp_)/(_kdt_). _H_τ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (5)
If we denote the coefficient of _Η_τ in these equal expressions by μ, which maybe called “Carnot’s coefficient,” we have
μ = (1 − σ)(_dp_)/(_kdt_) = (_Ep_{0}v_{0}_)/(_vdq_/_dv_), (6)
and we deduce the following very remarkable conclusions:
(1) For the saturated vapors of all different liquids, at the same temperature, the value of (1 − σ)(_dp_/_kdt_) must be the same.
(2) For any different gaseous masses, at the same temperature, the value of _Ep_{0}v_{0}_/(_vdq_/_dv_) must be the same.
(3) The values of these expressions for saturated vapors and for gases, at the same temperature, must be the same.
31. No conclusion can be drawn _a priori_ regarding the values of this coefficient μ for different temperatures, which can only be determined, or compared, by experiment. The results of a great variety of experiments, in different branches of physical science (Pneumatics and Acoustics), cited by Carnot and by Clapeyron, indicate that the values of μ for low temperatures exceed the values for higher temperatures; a result amply verified by the continuous series of experiments performed by Regnault on the saturated vapor of water for all temperatures from 0° to 230°, which, as we shall see later, give values for μ gradually diminishing from the inferior limit to the superior limit of temperature. When, by observation, μ has been determined as a function of the temperature, the amount of mechanical effect, _M_, deducible from _H_ units of heat descending from a body at the temperature _S_ to a body at the temperature _T_, may be calculated from the expression
_M_ = _H_ ∫_{_S_}^{_T_} μ_dt_, (7)
which is, in fact, what either of the equations (1) for the steam-engine, or (4) for the air-engine, becomes, when the notation μ, for Carnot’s multiplier, is introduced.
The values of this integral may be practically obtained, in the most convenient manner, by first determining, from observation, the mean values of μ for the successive degrees of the thermometric scale, and then adding the values for all the degrees within the limits of the extreme temperatures _S_ and _T_.[50]
32. The complete theoretical investigation of the motive power of heat is thus reduced to the experimental determination of the coefficient μ; and may be considered as perfect, when, by any series of experimental researches whatever, we can find a value of μ for every temperature within practical limits. The special character of the experimental researches, whether with reference to gases or with reference to vapors, necessary and sufficient for this object, is defined and restricted in the most precise manner, by the expressions (6) for μ, given above.
33. The object of Regnault’s great work, referred to in the title of this paper, is the experimental determination of the various physical elements of the steam-engine; and when it is complete, it will furnish all the _data_ necessary for the calculation of μ. The valuable researches already published in a first part of that work make known the latent heat of a given weight, and the pressure, of saturated steam for all temperatures between 0° and 230° Cent. of the air-thermometer. Besides these data, however, the density of saturated vapor must be known, in order that _k_, the latent heat of a unit of volume, may be calculated from Regnault’s determination of the latent heat of a given weight.[51] Between the limits of 0° and 100°, it is probable, from various experiments which have been made, that the density of vapor follows very closely the simple laws which are so accurately verified by the ordinary gases;[52] and thus it may be calculated from Regnault’s table giving the pressure at any temperature within those limits. Nothing as yet is known with accuracy as to the density of saturated steam between 100 and 230°, and we must be contented at present to estimate it by calculation from Regnault’s table of pressures; although, when accurate experimental researches on the subject shall have been made, considerable deviations from the laws of Boyle and Dalton, on which this calculation is founded, may be discovered.
34. Such are the experimental data on which the mean values of μ for the successive degrees of the air-thermometer, from 0 to 230°, at present laid before the Royal Society, is founded. The unit of length adopted is the English foot; the unit of weight, the pound; the unit of work, a “foot-pound;” and the unit of heat that quantity which, when added to a pound of water at 0°, will produce an elevation of 1° in temperature. The mean value of μ for any degree is found to a sufficient degree of approximation by taking, in place of σ, _dp_/_dt_ and _k_; in the expression
(1 − σ). (_dp_)/(_kdt_);
the mean values of those elements; or, what is equivalent to the corresponding accuracy of approximation, by taking, in place of σ and _k_ respectively, the mean of the values of those elements for the limits of temperature, and in place of _dp_/_dt_, the difference of the values of _p_, at the same limits.
35. In Regnault’s work (at the end of the eighth memoir), a table of the pressures of saturated steam for the successive temperatures 0°, 1°, 2°, ... 230°, expressed in millimetres of mercury, is given. On account of the units adopted in this paper, these pressures must be estimated in pounds on the square foot, which we may do by multiplying each number of millimetres by 2.7896, the weight in pounds of a sheet of mercury, one millimetre thick, and a square foot in area.
36. The value of _k_, the latent heat of a cubic foot, for any temperature _t_, is found from λ, the latent heat of a pound of saturated steam, by the equation
_k_ = (_p_)/(760). (1 + .00366 × 100)/(1 + .00366 × _t_). × .036869[53] . λ,
where _p_ denotes the pressure in millimetres, and λ the latent heat of a pound of saturated steam; the values of λ being calculated by the empirical formula[54]
λ = (606.5 + 0.305_t_) − (_t_ + .00002_t_^2 + 0.0000003_t_^3),
given by Regnault as representing, between the extreme limits of his observations, the latent heat of a unit weight of saturated steam.
