CARNOT: ON THE MOTIVE POWER OF HEAT 16. A practical and theoretical question of the greatest im portance was first answered by Sadi Carnot about this time in his Reflexions sur la puissance motrice du feu (1824). How much motive power (defined by Carnot as weight lifted through a cer tain height) can be obtained from heat alone by means of an engine repeating a regular succession or "cycle" of operations continuously? Is the efficiency limited, and, if so, how is it limited? Are other agents preferable to steam for developing motive power from heat? In discussing this problem, we cannot do better than follow Carnot's reasoning which, in its main fea tures, could hardly be improved at the present day. Carnot points out that in order to obtain an answer to this question, it is neces sary to consider the essential conditions of the process, apart from the mechanism of the engine and the working substance or agent employed. Work cannot be said to be produced from heat alone unless nothing but heat is supplied, and the working sub stance and all parts of the engine are at the end of the process in precisely the same state as at the beginning.
For instance a mass of compressed air, if allowed to expand in a cylinder at constant temperature, will do work, and will at the same time absorb a quantity of heat which, as we now know, is the thermal equivalent of the work done. But this work cannot be said to have been produced solely from the heat absorbed in the process, because the air at the end of the process is in a changed condition, and could not be restored to its original state at the same temperature without having work done upon it pre cisely equal to that obtained by its expansion. The process could not be repeated indefinitely without a continual supply of com pressed air. The source of the work in this case is work pre viously done in compressing the air, and no part of the work is really generated at the expense of heat alone, unless the compres sion is effected at a lower temperature than the expansion.
Carnot's Axiom. Carnot here, and throughout his reasoning, makes a fundamental assumption, which he states as follows: "When a body has undergone any changes and after a certain number of transformations is brought back identically to its orig inal state, considered relatively to density, temperature and mode of aggregation, it must contain the same quantity of heat as it con tained originally." Clausius (Pogg. Ann. 79, p. 369) and others have misinterpreted this assumption, and have taken it to mean that the quantity of heat required to produce any given change of state is independent of the manner in which the change is effected, which Carnot does not here assume.
Heat, according to Carnot, in the type of engine we are con sidering, can evidently be a cause of motive power only by virtue of changes of volume or form produced by alternate heating and cooling. This involves the existence of hot and cold bodies to act as boiler and condenser, or source and sink of heat, respec tively. Wherever there exists a difference of temperature, it is possible to have the production of motive power from heat; and conversely, production of motive power, from heat alone, is im possible without difference of temperature. In other words the production of motive power from heat is not merely a question of the consumption of heat, but always requires transference of heat from hot to cold. What then are the conditions which en able the difference of temperature to be most advantageously employed in the production of motive power, and how much motive power can be obtained with a given difference of tempera ture from a given quantity of heat? Carnot's Rule for Maximum Etect.—In order to realize the maximum effect, it is necessary that, in the process employed, there should not be any direct interchange of heat between bodies at different temperatures. Direct transference of heat by conduc tion or radiation between bodies at different temperatures is equivalent to wasting a difference of temperature which might have been utilized to produce motive power. The working sub stance must throughout every stage of the process be in equilib rium with itself (i.e., at uniform temperature and pressure) and also with external bodies, such as the boiler and condenser, at such times as it is put in communication with them. In the actual engine there is always some interchange of heat between the steam and the cylinder, and some loss of heat to external bodies. There may also be some difference of temperature between the boiler steam and the cylinder on admission, or between the waste steam and the condenser at release. These differences represent losses of efficiency which may be reduced indefinitely, at least in imagination, by suitable means, and designers had even at that date been very successful in reducing them. All such losses are supposed to be absent in deducing the ideal limit of efficiency, beyond which it would be impossible to go.
17. Carnot's Description of His Ideal Cycle.—Carnot first gives a rough illustration of an incomplete cycle, using steam much in the same way as it is employed in an ordinary steam engine. After expansion down to condenser pressure the steam is completely condensed to water, and is then returned as cold water to the hot boiler. He points out that the last step does not con form exactly to the condition he laid down, because although the water is restored to its initial state, there is direct passage of heat from a hot body to a cold body in the last process. He points out that this difficulty might be over come by supposing the difference of temperature small, and by employing a series of engines, each working through a small range, to cover a finite interval of temperature. Having established the general notions of a perfect cycle, he proceeds to give a more exact illustra tion, employing a gas as the working sub stance. He takes as the basis of his demonstration the well-established experi mental fact that a gas is heated by rapid compression and cooled by rapid ex pansion, and that if compressed or ex panded slowly in contact with conducting bodies, the gas will give out heat in com pression or absorb heat in expansion while its temperature remains constant. He then goes on to say : "This preliminary notion being settled, let us imagine an elastic fluid, atmospheric air for example, enclosed in a cylinder abcd (fig. 5) fitted with a movable diaphragm or piston cd. Let there also be two bodies A, B, each maintained at a constant tempera ture, that of A being more elevated than that of B. Let us now suppose the following series of operations to be performed: "I. Contact of the body A with the air contained in the space abcd, or with the bottom of the cylinder, which we will suppose to transmit heat easily. The air is now at the temperature of the body A, and cd is the actual position of the piston.
"3. The body A is removed, and the air no longer being in con tact with any body capable of giving it heat, the piston continues nevertheless to rise, and passes from the position ef to gli. The air expands without receiving heat and its temperature falls. Let us imagine that it falls until it is just equal to that of the body B. At this moment the piston is stopped and occupies the position gh.
"4. The air is placed in contact with the body B ; it is com pressed by the return of the piston, which is brought from the position gh to the position cd. The air remains meanwhile at a constant temperature, because of its contact with the body B to which it gives up its heat.
"5. The body B is removed, and the compression of the air is continued. The air being now isolated, rises in temperature. The compression is continued until the air has acquired the tempera ture of the body A. The piston passes meanwhile from the posi tion cd to the position ik.
"6. The air is replaced in contact with the body A, and the piston returns from the position ik to the position ef, the tern perature remaining invariable.
"7. The period described under (3) is repeated, then succes sively the periods (4), (5), ; (3), (4), (5), ; (3), (4), (5), (6) ; and so on.
"During these operations the air enclosed in the cylinder exerts an effort more or less great on the piston. The pressure of the air varies both on account of changes of volume and on account of changes of temperature ; but it should be observed that for equal volumes, that is to say, for like positions of the piston, the tem perature is higher during the dilatation than during the com pression. Since the pressure is greater during the expansion, the quantity of motive power produced by the dilatation is greater than that consumed by the compression. We shall thus obtain a balance of motive power, which may be employed for any pur pose. The air has served as working substance in a heat-engine; it has also been employed in the most advantageous manner pos sible, since no useless re-establishment of the equilibrium of heat has been allowed to occur.
"All the operations above described may be executed in the reverse order and direction. Let us imagine that after the sixth period, that is to say, when the piston has reached the position ef, we make it return to the position ik, and that at the same time we keep the air in contact with the hot body A; the heat furnished by this body during the sixth period will return to its source, that is, to the body A, and everything will be as it was at the end of the fifth period. If now we remove the body A, and if we make the piston move from ik to cd, the temperature of the air will de crease by just as many degrees as it increased during the fifth period, and will become that of the body B. We can evidently continue in this way a series of operations the exact reverse of those which were previously described ; it suffices to place oneself in the same circumstances and to execute for each period a move ment of expansion in place of a movement of compression, and vice versa.
"The result of the first series of operations was the production of a certain quantity of motive power, and the transport of heat from the body A to the body B ; the result of the reverse opera tions is the consumption of the motive power produced in the first case, and the return of heat from the body B to the body A, in such sort that these two series of operations annul and neu tralize each other.
"The impossibility of producing by the agency of heat alone a quantity of motive power greater than that which we have ob tained in our first series of operations is now easy to prove. It is demonstrated by reasoning exactly similar to that which we have already given. The reasoning will have in this case a greater de gree of exactitude ; the air of which we made use to develop the motive power is brought back at the end of each cycle of opera tions precisely to its initial state, whereas this was not quite exactly the case for the vapour of water, as we have already remarked." IS. Proof of Carnot's Principle.—Carnot considered the proof too obvious to be worth repeating, but, unfortunately, his previous demonstration, referring to an incomplete cycle, is not so exactly worded that exception cannot be taken to it. We will therefore repeat his proof in a slightly more definite and exact form. Suppose that a reversible engine R, working in the cycle above described, takes a quantity of heat Q from the source in each cycle, and performs a quantity of useful work If it were possible for any other engine S, working with the same two bodies A and B as source and refrigerator, to perform a greater amount of useful work IV, per cycle for the same quantity of heat Q taken from the source, it would suffice to take a portion W, of this motive power (since IV, is by hypothesis greater than IV, ) to drive the engine R backwards, and return a quantity of heat Q to the source in each cycle. The process might be repeated indefi nitely, and we should obtain at each repetition a balance of useful work W,–IV„ without taking any heat from the source, which is contrary to experience. Whether the quantity of heat taken from the condenser by R is equal to that given to the condenser by S is immaterial. The hot body A might be a comparatively small boiler, since no heat is taken from it. The cold body B might be the ocean, or the whole earth. We might thus obtain without any consumption of fuel a practically unlimited supply of motive power. Which is absurd.
Carnot's Statement of his Principle.—If the foregoing reasoning be admitted, we must conclude with Carnot that the motive power obtainable from heat is independent of the agents employed to realize it. The efficiency is fixed solely by the temperatures of the bodies between which, in the last resort, the transfer of heat is effected. "We must understand here that each of the methods of developing motive power attains the perfection of which it is susceptible. This condition is fulfilled if, according to our rule, there is produced in the body no change of temperature that is not due to change of volume, or in other words, if there is no direct interchange of heat between bodies of sensibly different temperatures." It is characteristic of a state of frictionless mechanical equilib rium that an indefinitely small difference of pressure suffices to upset the equilibrium and reverse the motion. Similarly in thermal equilibrium between bodies at the same temperature, an indef initely small difference of temperature suffices to reverse the trans fer of heat. Carnot's rule is therefore the criterion of the reversibil ity of a cycle of operations as regards transfer of heat. It is as sumed that the ideal engine is mechanically reversible, that there is not, for instance, any communication between reservoirs of gas or vapour at sensibly different pressures, and that there is no waste of power in friction. If there is equilibrium both mechanical and thermal at every stage of the cycle, the ideal engine will be per fectly reversible. That is to say, all its operations will be exactly reversed as regards transfer of heat and work, when the operations are performed in the reverse order and direction. On this under standing Carnot's principle may be put in a different way, which is often adopted, but is really only the same thing put in different words : The efficiency of a perfectly reversible engine is the maxi mum possible, and is a function solely of the limits of temperature between which it works. This result depends essentially on the existence of a state of thermal equilibrium defined by equality of temperature, and independent, in the majority of cases, of the state of a body in other respects. In order to apply the principle to the calculation and prediction of results, it is sufficient to deter mine the manner in which the efficiency depends on the tempera ture for one particular case, since the efficiency must be the same for all reversible engines.
19. Carnot endeavoured to test his results by calculating the amount of work obtainable from an engine on his cycle, using steam as a working substance, and comparing the result with an en gine performing a similar cycle, using air as the working substance. He found, using the experimental data available at that time, that the work to be obtained from i,000 gram-calories of heat was roughly independent of the working substance, being about I•13 kilogrammetre per kilo-calorie per I ° fall at ioo° C, i.e., when the limits of temperature of the cycle are I OI ° C and Ioo° C. He was able to show that the efficiency per degree fall probably diminished with rise of temperature, but the experimental data at that time were too inconsistent to suggest the true relation. He took as the analytical expression of his principle that the efficiency W/Q of a perfect engine taking in heat Q at a temperature t° C, and re jecting heat at the temperature o° C, must be some function Ft of the temperature t, which would be the same for all sub stances. The efficiency per degree fall at a temperature t he rep resented by F't, the derived function of Ft. The function F't would be the same for all substances at the same temperature, but would have different values at different temperatures. In terms of this function, which is generally known as Carnot's func tion, the results obtained in the previous section might be ex pressed as follows : "The increase of volume of a mixture of liquid and vapour per unit-mass vaporized at any temperature, multiplied by the in crease of vapour-pressure per degree, is equal to the product of the function F't by the latent heat of vaporization.
"The difference of the specific heats, or the latent heat of ex pansion for any substance, multiplied by the function F't, is equal to the product of the expansion per degree at constant pres sure by the increase of pressure per degree at constant volume." Since the last two coefficients are the same for all gases if equal volumes are taken, Carnot concluded that : "The difference of the specific heats at constant pressure and volume is the same for equal volumes of all gases at the same temperature and pressure." Taking the expression W =RT log for the whole work done by a gas obeying the gaseous laws pv=RT in expanding at a tem perature T from a volume i (unity) to a volume r, or for a ratio of expansion r, and putting W' = R log for the work done in a cycle of range i°, Carnot obtained the expression for the heat absorbed by a gas in isothermal expansion.
Q = R log (2) He gives several important deductions which follow from this formula, which is the analytical expression of the experimental result already quoted as having been discovered subsequently by Dulong. Employing the above expression for the latent heat of expansion, Carnot deduced a general expression for the specific heat of a gas at constant volume on the basis of the caloric the ory. He showed that if the specific heat was independent of the temperature (the hypothesis already adopted by Laplace and Poisson) the function F'(t) must be of the form F'(t) =R/C (t+to) (3) where C and are unknown constants. A similar result follows from his expression for the difference of the specific heats. If this is assumed to be constant and equal to C, the expression for F'(t) becomes R/CT, which is the same as the above if Assuming the specific heat to be also independent of the volume, he shows that the function F'(t) should be constant. But this assumption is inconsistent with the caloric theory of latent heat of expansion, which requires the specific heat to be a function of the volume. It appears in fact impossible to reconcile Carnot's principle with the caloric theory on any simple assumptions. As Carnot remarks : "The main principles on which the theory of heat rests require most careful examination. Many experimental facts appear almost inexplicable in the present state of this theory." Carnot's work was subsequently put in a more complete analyt ical form by B. P. E. Clapeyron (bourn. de l'dc, polytechn., 1832), who also made use of Watt's indicator diagram for the first time in discussing physical problems. Clapeyron gave the general ex pressions for the latent heat of a vapour, and for the latent heat of isothermal expansion of any substance, in terms of Carnot's function, employing the notation of the calculus. The expressions he gave are the same in form as those in use at the present time. He also gave the general expression for Carnot's function, and endeavoured to find its variation with temperature; but having no better data he succeeded no better than Carnot. Unfortu nately, in describing Carnot's cycle, he assumed the caloric the ory of heat, and made some unnecessary mistakes, which Carnot (who, we now know, was a believer in the mechanical theory) had been very careful to avoid. Clapeyron directs one to compress the gas at the lower temperature in contact with the body B until the heat disengaged is equal to that which has been absorbed at the higher temperature. He assumes that the gas at this point con tains the same quantity of heat as it contained in its original state at the higher temperature, and that, when the body B is removed, the gas will be restored to its original temperature, when com pressed to its initial volume. This mistake is still attributed to Carnot, and regarded as a fatal objection to his reasoning by many writers.
Prof. W. Thomson (Lord Kelvin) stated (Phil. Mag., 1852) that "Carnot's original demonstration utterly fails," and he in troduced the "corrections" attributed to James Thomson and Clerk Maxwell respectively. In reality Carnot's original demon stration requires no correction.