Part 16

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; 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, 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.

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, 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, till the commencement of the operations, which are conducted in the following manner:

(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_.

(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.

(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.

(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.

16. At the conclusion of this cycle of operations 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.

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}} ξdp_,

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