Part 32

F′t = (N)/(T) = (N)/(Ct + C{1}_).

Multiplying both by dt and integrating, we have

Ft = (N)/(C) log (Ct + C{1}) + C{2};

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

Ft = A log (1 + (t)/(B)). (6)

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

In equation (5), the first member represents, as we have remarked, the specific heat of the air occupying the volume v. Experiment having shown that this heat varies little in spite of the quite considerable changes of volume, it is necessary that the coefficient T′ of log v should be a very small quantity. If we consider it nothing, and, after having multiplied by dt the equation

T′ = 0,

we take the integral of it, we find

T = C, constant quantity;

but

T = N/F′t,

whence

F′t = N/T = N/C = A;

whence we deduce finally, by a second integration,

Ft = At + B.

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

Ft = At;

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

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

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

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

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

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

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

Portions of the descent from the top of the Decreasing pressure of the cylinder. steam, the entire pressure being 1. Steam arriving 0.05 freely from the 1.000 Total pressure. boiler. 0.10 „ 1.000 „ 0.15 „ 1.000 „ 0.20 „ 1.000 „ Quarter 0.25 „ 1.000 „ The steam being cut off and the 0.30 descent taking 0.830 place only by expansion. 0.35 „ 0.714 0.40 „ 0.625 0.45 „ 0.555 Half 0.50 „ 0.500 Half original pressure. 0.55 „ 0.454 0.60 „ 0.417 0.65 „ 0.385 0.70 „ 0.375 0.75 „ 0.333 One third. 0.80 „ 0.312 0.85 „ 0.294 0.90 „ 0.277 0.95 „ 0.262 Bottom of cylinder 1.00 „ 0.025 Quarter. Total, 11.583

Mean pressure (11.583)/(20) = 0.579.

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