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]