_x_ = 80°;
which shows that air compressed one half rises 80°.
All this is subject to the hypothesis that the specific heat of air
does not change, although the volume diminishes. But if, for the
reasons hereafter given (pp. 86, 89), we regard the specific heat of
air compressed one half as reduced in the relation of 700 to 616, the
number 80° must be multiplied by ⁷⁰⁰⁄₆₁₆, which raises it to 90°.
Footnote 23:
MM. Gay-Lussac and Welter have found by direct experiments, cited in
the _Mécanique Céleste_ and in the _Annales de Chimie et de Physique_,
July, 1822, p. 267, that the ratio between the specific heat at
constant pressure and the specific heat at constant volume varies very
little with the density of the gas. According to what we have just
seen, the difference should remain constant, and not the ratio. As,
further, the specific heat of gases for a given weight varies very
little with the density, it is evident that the ratio itself
experiences but slight changes.
The ratio between the specific heat of atmospheric air at constant
pressure and at constant volume is, according to MM. Gay-Lussac and
Welter, 1.3748, a number almost constant for all pressures, and even
for all temperatures. We have come, through other considerations, to
the number (267 + 116)/(267) = 1.44, which differs from the former
(1)/(20), and we have used this number to prepare a table of the
specific heats of gases at constant volume. So we need not regard this
table as very exact, any more than the table given on p. 89. These
tables are mainly intended to demonstrate the laws governing specific
heats of aeriform fluids.
Footnote 24:
Note C, Appendix B.
Footnote 25:
Note D, Appendix B.
Footnote 26:
Note E, Appendix B.
Footnote 27:
We find (_Annales de Chimie et de Physique_, July, 1818, p. 294) in a
memoir of M. Petit an estimate of the motive power of heat applied to
air and to vapor of water. This estimate leads us to attribute a great
advantage to atmospheric air, but it is derived by a method of
considering the action of heat which is quite imperfect.
Footnote 28:
Note F, Appendix B.
Footnote 29:
Those that we need are the expansive force acquired by solids and
liquids by a given increase of temperature, and the quantity of heat
absorbed or relinquished in the changes of volume of these bodies.
Footnote 30:
The recent experiments of M. Oerstedt on the compressibility of water
have shown that, for a pressure of five atmospheres, the temperature
of this liquid exhibits no appreciable change. (_See Annales de Chimie
et de Physique_, Feb. 1823, p. 192.)
Footnote 31:
Note G, Appendix B.
Footnote 32:
We find in the work called _De la Richesse Minérale_, by M. Heron de
Villefosse, vol. iii. p. 50 and following, a good description of the
steam-engines actually in use in mining. In England the steam-engine
has been very fully discussed in the _Encyclopedia Britannica_. Some
of the data here employed are drawn from the latter work.
Footnote 33:
Note I, Appendix B.
Footnote 34:
From _Transactions of the Edinburgh Royal Society_, xiv. 1849;
_Annales de Chimie_, xxxv. 1852.
Footnote 35:
Published in 1824, in a work entitled “_Réflexions sur la Puissance
Motrice du Feu, et sur les Machines Propres à Developer cette
Puissance. Par S. Carnot._” [Note of Nov. 5, 1881. The original work
has now been republished, with a biographical notice, Paris, 1878.]
Footnote 36:
An account of the first part of a series of researches undertaken by
Mons. Regnault, by order of the late French Government, for
ascertaining the various physical data of importance in the theory of
the steam-engine, has been recently published (under the title
“_Relation des Expériences_,” etc.) in the _Mémoires de l’Institut_,
of which it constitutes the twenty-first volume (1847). The second
part of these researches has not yet been published. [Note of Nov. 5,
1881. The continuation of these researches has now been published;
thus we have for the whole series, vol. i. in 1847; vol. ii. in 1862;
and vol. iii. in 1870.]
Footnote 37:
Carnot, p. 67.
Footnote 38:
The _evolution_ of heat in a fixed conductor, through which a
galvanic current is sent from any source whatever, has long been
known to the scientific world; but it was pointed out by Mr. Joule
that we cannot infer from any previously-published experimental
researches, the actual _generation_ of heat when the current
originates in electro-magnetic induction; since the question occurs,
_is the heat which is evolved in one part of the closed conductor
merely transferred from those parts which are subject to the
inducing influence?_ Mr. Joule, after a most careful experimental
investigation with reference to this question, finds that it must be
answered in the negative. (See a paper “On the Calorific Effects of
Magneto-Electricity, and on the Mechanical Value of Heat; by J. P.
Joule, Esq.” Read before the British Association at Cork in 1843,
and subsequently communicated by the Author to the _Philosophical
Magazine_, vol. xxiii., pp. 263, 347, 435.)
Before we can finally conclude that heat is absolutely generated in
such operations, it would be necessary to prove that the inducing
magnet does not become lower in temperature, and thus compensate for
the heat evolved in the conductor. I am not aware that any examination
with reference to the truth of this conjecture has been instituted;
but, in the case where the inducing body is a pure electro-magnet
(without any iron), the experiments actually performed by Mr. Joule
render the conclusion probable that the heat evolved in the wire of
the electro-magnet is not affected by the inductive action, otherwise
than through the reflected influence which increases the strength of
its own current.
Footnote 39:
So generally is Carnot’s principle tacitly admitted as an axiom, that
its application in this case has never, so far as I am aware, been
questioned by practical engineers. (1849).
Footnote 40:
When “thermal agency” is thus spent in conducting heat through a
solid, what becomes of the mechanical effect which it might produce?
Nothing can be lost in the operations of nature—no energy can be
destroyed. What effect, then, is produced in place of the mechanical
effect which is lost? A perfect theory of heat imperatively demands an
answer to this question; yet no answer can be given in the present
state of science. A few years ago, a similar confession must have been
made with reference to the mechanical effect lost in a fluid set in
motion in the interior of a rigid closed vessel, and allowed to come
to rest by its own internal friction; but in this case the foundation
of a solution of the difficulty has been actually found in Mr. Joule’s
discovery of the generation of heat, by the internal friction of a
fluid in motion. Encouraged by this example, we may hope that the very
perplexing question in the theory of heat, by which we are at present
arrested, will before long be cleared up. [Note of Sept., 1881. The
Theory of the Dissipation of Energy completely answers this question
and removes the difficulty.]
It might appear that the difficulty would be entirely avoided by
abandoning Carnot’s fundamental axiom; a view which is strongly urged
by Mr. Joule (at the conclusion of his paper “On the Changes of
Temperature produced by the Rarefaction and Condensation of Air.”
_Phil. Mag._, May 1845, vol. xxvi.) If we do so, however, we meet with
innumerable other difficulties—insuperable without farther
experimental investigation, and an entire reconstruction of the theory
of heat from its foundation. It is in reality to experiment that we
must look—either for a verification of Carnot’s axiom, and an
explanation of the difficulty we have been considering; or for an
entirely new basis of the Theory of Heat.
Footnote 41:
For a demonstration, see § 29.
Footnote 42:
A case minutely examined in another paper, to be laid before the
Society at the present meeting. “Theoretical Considerations on the
Effect of Pressure in Lowering the Freezing-point of Water,” by Prof.
James Thomson.
Footnote 43:
In all that follows, the pressure of the atmosphere on the upper side
of the piston will be included in the applied forces, which, in the
successive operations described, are sometimes overcome by the upward
motion, and sometimes yielded to in the motion downwards. It will be
unnecessary, in reckoning at the end of a cycle of operations, to take
into account the work thus spent upon the atmosphere, and the
restitution which has been made, since these precisely compensate for
one another.
Footnote 44:
[Note of Nov. 5, 1881. Maxwell has simplified the correction by
beginning the cycle with Carnot’s second operation, and completing it
through his third, fourth, and first operations, with his third
operation nearly as follows:
_let the piston be pushed down to any position E_{3}F_{3}_;
then Carnot’s fourth operation altered to the following:
_let the piston be pushed down from E_{3}F_{3} until the temperature
reaches its primitive value S_;
and lastly, Carnot’s first operation altered to the following:
_let the piston rise to its primitive position_.]
Footnote 45:
In Carnot’s work some perplexity is introduced with reference to the
temperature of the water, which, in the operations he describes, is
not brought back exactly to what it was at the commencement; but the
difficulty which arises is explained by the author. No such difficulty
occurs with reference to the cycle of operation described in the text,
for which I am indebted to Mons. Clapeyron.
Footnote 46:
Thus, _dq_/_dv_ will be the partial differential coefficient, with
respect to _v_, of that function of _v_ and _t_ which expresses the
quantity of heat that must be added to a mass of air when in a
“standard” state (such as at the temperature zero, and under the
atmospheric pressure), to bring it to the temperature _t_ and the
volume _v_. That there is such a function, of two independent
variables _v_ and _t_, is merely an analytical expression of Carnot’s
fundamental axiom, as applied to a mass of air. The general principle
may be analytically stated in the following terms:—If _Mdv_ denote the
accession of heat received by a mass of any kind, not possessing a
destructible texture, when the volume is increased by _dv_, the
temperature being kept constant, and if _Ndt_ denote the amount of
heat which must be supplied to raise the temperature by _dt_, without
any alteration of volume; then _Mdv_ + _Ndt_ must be the differential
of a function of _v_ and _t_. [Note of Nov. 5, 1881. In the corrected
theory it is (_M_ − _Jp_)_dv_ + _Ndt_, that is a complete
differential, not _Mdv_ + _Ndt_. See _Dynamical Theory of Heat_ (Art.
XLVIII., below), § 20.]
Footnote 47:
We might also investigate another relation, to express the fact that
there is no accession or removal of heat during either the second or
the fourth operation; but it will be seen that this will not affect
the result in the text, although it would enable us to determine both
φ and ω in terms of τ.
Footnote 48:
This result might have been obtained by applying the usual notation of
the integral calculus to express the area of the curvilinear
quadrilateral, which, according to Clapeyron’s graphical construction,
would be found to represent the entire mechanical effect gained in the
cycle of operations of the air-engine. It is not necessary, however,
to enter into the details of this investigation, as the formula (3),
and the consequences derived from it, include the whole theory of the
air-engine, in the best practical form; and the investigation of it
which I have given in the text will probably give as clear a view of
the reasoning on which it is founded as could be obtained by the
graphical method, which in this case is not so valuable as it is from
its simplicity in the case of the steam-engine.
Footnote 49:
This paragraph is the demonstration, referred to above, of the
proposition stated in § 13, as it is readily seen that it is
applicable to any conceivable kind of thermodynamic engine.
Footnote 50:
The results of these investigations are exhibited in Tables I and II.
Footnote 51:
It is, comparatively speaking, of little consequence to know
accurately the value of σ, for the factor (1 − σ) of the expression
for μ, since it is so small (being less than ¹⁄₁₇₀₀ for all
temperatures between 0° and 100°) that, unless all the data are known
with more accuracy than we can count upon at present, we might neglect
it altogether, and take _dp_/_kdt_ simply, as the expression for μ,
without committing any error of important magnitude.
Footnote 52:
This is well established, within the ordinary atmospheric limits, in
Regnault’s Études Météorologiques, in the _Annales de Chimie_, vol.
xv., 1846.
Footnote 53:
It appears that the vol. of 1 kilog. must be 1.69076 according to the
data here assumed.
The density of saturated steam at 100° is taken as ¹⁄₁₆₉₃.5 of that of
water at its maximum. Rankine takes it as ¹⁄₁₆₉₆.
Footnote 54:
The part of this expression in the first vinculum (see Regnault, end
of ninth memoir) is what is known as “the total heat” of a pound of
steam, or the amount of heat necessary to convert a pound of water at
0° into a pound of saturated steam at _t°_; which, according to
“Watt’s law” thus approximately verified, would be constant. The
second part, which would consist of the single term _t_, if the
specific heat of water were constant for all temperatures, is the
number of thermic units necessary to raise the temperature of a pound
of water from 0° to _t°_, and expresses empirically the results of
Regnault’s experiments on the specific heat of water (see end of the
tenth memoir), described in the work already referred to.
Footnote 55:
In strictness, the 230th is the last degree for which the experimental
data are complete; but the data for the 231st may readily be assumed
in a sufficiently satisfactory manner.
Footnote 56:
The numbers here tabulated may also be regarded as _the actual values
of μ for t_ = ½, _t_ = 1½, _t_ = 2½, _t_ = 3½, etc.
Footnote 57:
For at the end of the fourth operation the whole mass is liquid, and
at the temperature _S_. Now, this state might be arrived at by first
compressing the vapor into water at the temperature _t_, and then
raising the temperature of the liquid to _S_; and however this state
may be arrived at, there cannot, on the whole, be any heat added to or
subtracted from the contents of the cylinder, since, during the fourth
operation, there is neither gain nor loss of heat. This reasoning is,
of course, founded on Carnot’s fundamental principle, which is tacitly
assumed in the commonly-received ideas connected with “Watt’s law,”
the “latent heat of steam,” and “the total heat of steam.”
Footnote 58:
Thus, from Carnot’s calculations, we find, in the case of alcohol
4.035, and in the case of water 3.648, instead of 3.963 and 3.658,
which are Clapeyron’s results in the same cases.
Footnote 59:
A still closer agreement must be expected when more accurate
experimental data are afforded with reference to the other media.
Mons. Regnault informs me that he is engaged in completing some
researches, from which we may expect, possibly before the end of the
present year, to be furnished with all the data for five or six
different liquids which we possess at present for water. It is
therefore to be hoped that, before long, a most important test of the
validity of Carnot’s theory will be afforded.
Footnote 60:
The _Napierian_ logarithm of _V_/_V′_ is here understood.
Footnote 61:
Carnot varies the statement of his theorem, and illustrates it in a
passage, pp. 81, 82, of which the following is translation:
“_When a gas varies in volume without any change of temperature, the
quantities of heat absorbed or evolved by this gas are in arithmetical
progression, if the augmentation or diminutions of volume are in
geometrical progression._
“When we compress a litre of air maintained at the temperature 10°,
and reduce it to half a litre, it disengages a certain quantity of
heat. If, again, the volume be reduced from half a litre to a quarter
of a litre, from a quarter to an eighth, and so on the quantities of
heat successively evolved will be the same.
“If, in place of compressing the air, we allow it to expand to two
litres, four litres, eight litres, etc., it will be necessary to
supply equal quantities of heat to maintain the temperature always at
the same degree.”
Footnote 62:
The best figure (1896) is _J_ = 778 ft.-lbs. = 1 B.T.U., or _J_ =
426.8 kgm. = 1 calorie, and probably with great accuracy.
Footnote 63:
Or the capacity of a unit of volume for heat.
Footnote 64:
Carnot suggests a combination of the two principles, with air as the
medium for receiving the heat at a very high temperature from the
furnace; and a second medium, alternately in the state of saturated
vapor and liquid water, to receive the heat, discharged at an
intermediate temperature from the air, and transmit it to the coldest
part of the apparatus. It is possible that a complex arrangement of
this kind might be invented which would enable us to take the heat at
a higher temperature, and discharge it at a lower temperature than
would be practicable in any simple air-engine or simple steam-engine.
If so, it would no doubt be equally possible, and perhaps more
convenient, to employ steam alone, but to use it at a very high
temperature not in contact with water in the hottest part of the
apparatus, instead of, as in the steam-engine, always in a saturated
state.
Footnote 65:
It is probably this invention to which Carnot alludes in the following
passage: “Il a été fait, dit-on, tout récemment en Angleterre des
essais heureux sur le développement de la puissance motrice par
l’action de la chaleur sur l’air atmosphérique. Nous ignorons
entièrement ne quoi ces essais ont consisté, si toutefois ils sont
réels.”
Footnote 66:
From this point of view, we see very clearly how imperfect is the
steam-engine, even after all Watt’s improvements. For to “push the
principle of expansion to the utmost,” we must allow the steam, before
leaving the cylinder, to expand until its pressure is the same as that
of the vapor in the condenser. According to “Watt’s law,” its
temperature would then be the same as (actually a little above, as
Regnault has shown) that of the condenser, and hence the steam-engine
worked in this most advantageous way has in reality the very fault
that Watt found in Newcomen’s engine. This defect is partially
remedied by Hornblower’s system of using a separate expansion
cylinder, an arrangement the advantages of which did not escape
Carnot’s notice, although they have not been recognized extensively
among practical engineers, until within the last few years.
Footnote 67:
I am indebted to the kindness of Professor Gordon of Glasgow for the
information regarding the various cases given in the text.
Footnote 68:
In different Cornish engines, the pressure in the boiler is from 2½ to
5 atmospheres; and, therefore, as we find from Regnault’s table of the
pressure of saturated steam, the temperature of the water in the
boiler must, in all of them, lie between 128° and 152°. For the better
class of engines, the average temperature of the water in the boiler
may be estimated at 140°, the corresponding pressure of steam being 3½
atmospheres.
Footnote 69:
This number agrees very closely with the number corresponding to the
fall from 100° to 0°, given in Table II. Hence, the fall from 140° to
30° of the scale of the air-thermometer is equivalent, with reference
to motive power, to the fall from 100° to 0°.
Footnote 70:
It being assumed that the temperatures of the boiler and condenser are
the same as those of the Cornish engines. If, however, the pressure be
lower, two atmospheres, for instance, the numbers would stand thus:
The temperature in the boiler would be only 121. Consequently, for
each pound of steam evaporated, only 614 units of heat would be
required; and therefore the work performed for each unit of heat
transmitted would be 160.3 foot-pounds, which is _more_ than according
to the estimate in the text. On the other hand, the range of
temperatures, or the fall utilized, is only from 131 to 30, instead of
from 140 to 30°, and consequently (Table II.), the theoretical duty
for each unit of heat is only 371 foot-pounds. Hence, if the engine,
to work according to the specification, requires a pressure of only 15
lbs. on the square inch (i.e., a total steam-pressure of two
atmospheres), its performance is (160.3)/(371) or 43.2 per cent of its
theoretical duty.
Footnote 71:
If, in this case again, the pressure required in the boiler to make
the engine work according to the contract were only 15 lbs. on the
square inch, we should have a different estimate of the economy, for
which see Table B, at the end of this paper.
Footnote 72:
These engines are provided with separate expansion cylinders, which
have been recently added to them by Mr. M‘Naught of Glasgow.
Footnote 73:
[Note added March 15, 1881. Total work for thermal unit, 1390 (Joule),
377.1 corrected by the dynamical theory, March 15, 1851.
377.1 = .2713 × 1390,
253 = .1820 × 1390 = (1)/(5.49) × 1390.]
Footnote 74:
Pressure 15 lbs. on the square inch.
------------------------------------------------------------------------