which, he observed, agrees in form with that found by Rankine.
21. _The Absolute Scale of Temperature._--Since Carnot's function is the
same for all substances at the same temperature, and is a function of
the temperature only, it supplies a means of measuring temperature
independently of the properties of any particular substance. This
proposal was first made by Lord Kelvin (_Phil. Mag._, 1848), who
suggested that the degree of temperature should be chosen so that the
efficiency of a perfect engine at any point of the scale should be the
same, or that Carnot's function F't should be constant. This would give
the simplest expression for the efficiency on the caloric theory, but
the scale so obtained, when the values of Carnot's function were
calculated from Regnault's observations on steam, was found to differ
considerably from the scale of the mercury or air-thermometer. At a
later date, when it became clear that the value of Carnot's function was
very nearly proportional to the reciprocal of the temperature T measured
from the absolute zero of the gas thermometer, he proposed a simpler
method (_Phil. Trans._, 1854), namely, to define absolute temperature
[theta] as proportional to the reciprocal of Carnot's function. On this
definition of absolute temperature, the expression ([theta]1 -
[theta]0)/[theta]1 for the efficiency of a Carnot cycle with limits
[theta]1 and [theta]0 would be exact, and it became a most important
problem to determine how far the temperature T by gas thermometer
differed from the absolute temperature [theta]. With this object he
devised a very delicate method, known as the "porous plug experiment"
(see THERMODYNAMICS) of testing the deviation of the gas thermometer
from the absolute scale. The experiments were carried out in conjunction
with Joule, and finally resulted in showing (_Phil. Trans._, 1862, "On
the Thermal Effects of Fluids in Motion") that the deviations of the air
thermometer from the absolute scale as above defined are almost
negligible, and that in the case of the gas hydrogen the deviations are
so small that a thermometer containing this gas may be taken for all
practical purposes as agreeing exactly with the absolute scale at all
ordinary temperatures. For this reason the hydrogen thermometer has
since been generally adopted as the standard.
22. _Availability of Heat of Combustion._--Taking the value 1.13
kilogrammetres per kilo-calorie for 1 deg. C. fall of temperature at 100
deg. C., Carnot attempted to estimate the possible performance of a
steam-engine receiving heat at 160 deg. C. and rejecting it at 40 deg.
C. Assuming the performance to be simply proportional to the temperature
fall, the work done for 120 deg. fall would be 134 kilogrammetres per
kilo-calorie. To make an accurate calculation required a knowledge of
the variation of the function F't with temperature. Taking the accurate
formula of S 20, the work obtainable is 118 kilogrammetres per
kilo-calorie, which is 28% of 426, the mechanical equivalent of the
kilo-calorie in kilogrammetres. Carnot pointed out that the fall of 120
deg. C. utilized in the steam-engine was only a small fraction of the
whole temperature fall obtainable by combustion, and made an estimate of
the total power available if the whole fall could be utilized, allowing
for the probable diminution of the function F't with rise of
temperature. His estimate was 3.9 million kilogrammetres per kilogramme
of coal. This was certainly an over-estimate, but was surprisingly
close, considering the scanty data at his disposal.
In reality the fraction of the heat of combustion available, even in an
ideal engine and apart from practical limitations, is much less than
might be inferred from the efficiency formula of the Carnot cycle. In
applying this formula to estimate the availability of the heat it is
usual to take the temperature obtainable by the combustion of the fuel
as the upper limit of temperature in the formula. For carbon burnt _in
air_ at constant pressure without any loss of heat, the products of
combustion might be raised 2300 deg. C. in temperature, assuming that
the specific heats of the products were constant and that there was no
dissociation. If all the heat could be supplied to the working fluid at
this temperature, that of the condenser being 40 deg. C., the possible
efficiency by the formula of S 20 would be 89%. But the combustion
obviously cannot maintain so high a temperature if heat is being
continuously abstracted by a boiler. Suppose that [theta]' is the
maximum temperature of combustion as above estimated, [theta]" the
temperature of the boiler, and [theta]^0 that of the condenser. Of the
whole heat supplied by combustion represented by the rise of temperature
[theta]' - [theta]^0, the fraction ([theta]' - [theta]")/([theta]'
-[theta]^0) is the maximum that could be supplied to the boiler, the
fraction ([theta]" - [theta]^0)/([theta]' - [theta]^0) being carried
away with the waste gases. Of the heat supplied to the boiler, the
fraction ([theta]' - [theta]^0)/[theta]" might theoretically be
converted into work. The problem in the case of an engine using a
separate working fluid, like a steam-engine, is to find what must be the
temperature [theta]" of the boiler in order to obtain the largest
possible fraction of the heat of combustion in the form of work. It is
easy to show that [theta]" must be the geometric mean of [theta]' and
[theta]^0, or [theta]" = [root]([theta]'[theta]^0). Taking [theta]'
-[theta]^0 = 2300 deg. C., and [theta]^0 = 313 deg. Abs. as before, we
find [theta]" = 903 deg. Abs. or 630 deg. C. The heat supplied to the
boiler is then 74.4% of the heat of combustion, and of this 65.3% is
converted into work, giving a maximum possible efficiency of 49% in
place of 89%. With the boiler at 160 deg. C., the possible efficiency,
calculated in a similar manner, would be 26.3%, which shows that the
possible increase of efficiency by increasing the temperature range is
not so great as is usually supposed. If the temperature of the boiler
were raised to 300 deg. C., corresponding to a pressure of 1260 lb. per
sq. in., which is occasionally surpassed in modern flash-boilers, the
possible efficiency would be 40%. The waste heat from the boiler,
supposed perfectly efficient, would be in this case 11%, of which less
than a quarter could be utilized in the form of work. Carnot foresaw
that in order to utilize a larger percentage of the heat of combustion
it would be necessary to employ a series of working fluids, the waste
heat from one boiler and condenser serving to supply the next in the
series. This has actually been effected in a few cases, e.g. steam and
SO2, when special circumstances exist to compensate for the extra
complication. Improvements in the steam-engine since Carnot's time have
been mainly in the direction of reducing waste due to condensation and
leakage by multiple expansion, superheating, &c. The gain by increased
temperature range has been comparatively small owing to limitations of
pressure, and the best modern steam-engines do not utilize more than 20%
of the heat of combustion. This is in reality a very respectable
fraction of the ideal limit of 40% above calculated on the assumption of
1260 lb. initial pressure, with a perfectly efficient boiler and
complete expansion, and with an ideal engine which does not waste
available motive power by complete condensation of the steam before it
is returned to the boiler.
23. _Advantages of Internal Combustion._--As Carnot pointed out, the
chief advantage of using atmospheric air as a working fluid in a
heat-engine lies in the possibility of imparting heat to it directly by
internal combustion. This avoids the limitation imposed by the use of a
separate boiler, which as we have seen reduces the possible efficiency
at least 50%. Even with internal combustion, however, the full range of
temperature is not available, because the heat cannot conveniently in
practice be communicated to the working fluid at constant temperature,
owing to the large range of expansion at constant temperature required
for the absorption of a sufficient quantity of heat. Air-engines of this
type, such as Stirling's or Ericsson's, taking in heat at constant
temperature, though theoretically the most perfect, are bulky and
mechanically inefficient. In practical engines the heat is generated by
the combustion of an explosive mixture at constant volume or at constant
pressure. The heat is not all communicated at the highest temperature,
but over a range of temperature from that of the mixture at the
beginning of combustion to the maximum temperature. The earliest
instance of this type of engine is the lycopodium engine of M. M.
Niepce, discussed by Carnot, in which a combustible mixture of air and
lycopodium powder at atmospheric pressure was ignited in a cylinder, and
did work on a piston. The early gas-engines of E. Lenoir (1860) and N.
Otto and E. Langen (1866), operated in a similar manner with
illuminating gas in place of lycopodium. Combustion in this case is
effected practically at constant volume, and the maximum efficiency
theoretically obtainable is 1 - log_e r/(r - 1), where r is the ratio of
the maximum temperature [theta]' to the initial temperature [theta]^0.
In order to obtain this efficiency it would be necessary to follow
Carnot's rule, and expand the gas after ignition without loss or gain of
heat from [theta]' down to [theta]^0, and then to compress it at
[theta]^0 to its initial volume. If the rise of temperature in
combustion were 2300 deg. C., and the initial temperature were 0 deg. C.
or 273 deg. Abs., the theoretical efficiency would be 73.3%, which is
much greater than that obtainable with a boiler. But in order to reach
this value, it would be necessary to expand the mixture to about 270
times its initial volume, which is obviously impracticable. Owing to
incomplete expansion and rapid cooling of the heated gases by the large
surface exposed, the actual efficiency of the Lenoir engine was less
than 5%, and of the Otto and Langen, with more rapid expansion, about
10%. Carnot foresaw that in order to render an engine of this type
practically efficient, it would be necessary to compress the mixture
before ignition. Compression is beneficial in three ways: (1) it permits
a greater range of expansion after ignition; (2) it raises the mean
effective pressure, and thus improves the mechanical efficiency and the
power in proportion to size and weight; (3) it reduces the loss of heat
during ignition by reducing the surface exposed to the hot gases. In the
modern gas or petrol motor, compression is employed as in Carnot's
cycle, but the efficiency attainable is limited not so much by
considerations of temperature as by limitations of volume. It is
impracticable before combustion at constant volume to compress a rich
mixture to much less than 1/5th of its initial volume, and, for
mechanical simplicity, the range of expansion is made equal to that of
compression. The cycle employed was patented in 1862 by Beau de Rochas
(d. 1892), but was first successfully carried out by Otto (1876). It
differs from the Carnot cycle in employing reception and rejection of
heat at constant volume instead of at constant temperature. This cycle
is not so efficient as the Carnot cycle for given limits of temperature,
but, _for the given limits of volume imposed_, it gives a much higher
efficiency than the Carnot cycle. The efficiency depends only on the
range of temperature in expansion and compression, and is given by the
formula ([theta]' - [theta]")/[theta]', where [theta]' is the maximum
temperature, and [theta]" the temperature at the end of expansion. The
formula is the same as that for the Carnot cycle with the same range of
temperature in expansion. The ratio [theta]'/[theta]" is r^([gamma] -
1), where r is the given ratio of expansion or compression, and [gamma]
is the ratio of the specific heats of the working fluid. Assuming the
working fluid to be a perfect gas with the same properties as air, we
should have [gamma] = 1.41. Taking r = 5, the formula gives 48% for the
maximum possible efficiency. The actual products of combustion vary with
the nature of the fuel employed, and have different properties from air,
but the efficiency is found to vary with compression in the same manner
as for air. For this reason a committee of the Institution of Civil
Engineers in 1905 recommended the adoption of the air-standard for
estimating the effects of varying the compression ratio, and defined the
relative efficiency of an internal combustion engine as the ratio of its
observed efficiency to that of a perfect air-engine with the same
compression.
24. _Effect of Dissociation, and Increase of Specific Heat._--One of the
most important effects of heat is the decomposition or dissociation of
compound molecules. Just as the molecules of a vapour combine with
evolution of heat to form the more complicated molecules of the liquid,
and as the liquid molecules require the addition of heat to effect their
separation into molecules of vapour; so in the case of molecules of
different kinds which combine with evolution of heat, the reversal of
the process can be effected either by the agency of heat, or indirectly
by supplying the requisite amount of energy by electrical or other
methods. Just as the latent heat of vaporization diminishes with rise of
temperature, and the pressure of the dissociated vapour molecules
increases, so in the case of compound molecules in general the heat of
combination diminishes with rise of temperature, and the pressure of the
products of dissociation increases. There is evidence that the compound
carbon dioxide, CO2, is partly dissociated into carbon monoxide and
oxygen at high temperatures, and that the proportion dissociated
increases with rise of temperature. There is a very close analogy
between these phenomena and the vaporization of a liquid. The laws which
govern dissociation are the same fundamental laws of thermodynamics, but
the relations involved are necessarily more complex on account of the
presence of different kinds of molecules, and present special
difficulties for accurate investigation in the case where dissociation
does not begin to be appreciable until a high temperature is reached. It
is easy, however, to see that the general effect of dissociation must be
to diminish the available temperature of combustion, and all experiments
go to show that in ordinary combustible mixtures the rise of temperature
actually attained is much less than that calculated as in S 22, on the
assumption that the whole heat of combustion is developed and
communicated to products of constant specific heat. The defect of
temperature observed can be represented by supposing that the specific
heat of the products of combustion increases with rise of temperature.
This is the case for CO2 even at ordinary temperatures, according to
Regnault, and probably also for air and steam at higher temperatures.
Increase of specific heat is a necessary accompaniment of dissociation,
and from some points of view may be regarded as merely another way of
stating the facts. It is the most convenient method to adopt in the case
of products of combustion consisting of a mixture of CO2 and steam with
a large excess of inert gases, because the relations of equilibrium of
dissociated molecules of so many different kinds would be too complex to
permit of any other method of expression. It appears from the researches
of Dugald Clerk, H. le Chatelier and others that the apparent specific
heat of the products of combustion in a gas-engine may be taken as
approximately .34 to .33 in place of .24 at working temperatures between
1000 deg. C. and 1700 deg. C., and that the ratio of the specific heats
is about 1.29 in place of 1.41. This limits the availability of the heat
of combustion by reducing the rise of temperature actually obtainable in
combustion at constant volume by 30 or 40%, and also by reducing the
range of temperature [theta]'/[theta]" for a given ratio of expansions r
from r^(.41) to r^(.29). The formula given in S 21 is no longer quite
exact, because the ratio of the specific heats of the mixture during
compression is not the same as that of the products of combustion during
expansion. But since the work done depends principally on the expansion
curve, the ratio of the range of temperature in expansion ([theta]'
-[theta]") to the maximum temperature [theta]' will still give a very
good approximation to the possible efficiency. Taking r = 5, as before,
for the compression ratio, the possible efficiency is reduced from 48%
to 38%, if [gamma] = 1.29 instead of 1.41. A large gas-engine of the
present day with r = 5 may actually realize as much as 34% indicated
efficiency, which is 90% of the maximum possible, showing how perfectly
all avoidable heat losses have been minimized.
It is often urged that the gas-engine is relatively less efficient than
the steam-engine, because, although it has a much higher absolute
efficiency, it does not utilize so large a fraction of its temperature
range, reckoning that of the steam-engine from the temperature of the
boiler to that of the condenser, and that of the gas-engine from the
maximum temperature of combustion to that of the air. This is not quite
fair, and has given rise to the mistaken notion that "there is an
immense margin for improvement in the gas-engine," which is not the case
if the practical limitations of volume are rightly considered. If
expansion could be carried out in accordance with Carnot's principle of
maximum efficiency, down to the lower limit of temperature [theta]0,
with rejection of heat at [theta]0 during compression to the original
volume V0, it would no doubt be possible to obtain an ideal efficiency
of nearly 80%. But this would be quite impracticable, as it would
require expansion to about 100 times v0, or 500 times the compression
volume. Some advantage no doubt might be obtained by carrying the
expansion beyond the original volume. This has been done, but is not
found to be worth the extra complication. A more practical method, which
has been applied by Diesel for liquid fuel, is to introduce the fuel at
the end of compression, and adjust the supply in such a manner as to
give combustion at nearly constant pressure. This makes it possible to
employ higher compression, with a corresponding increase in the ratio of
expansion and the theoretical efficiency. With a compression ratio of
14, an indicated efficiency of 40% has been obtained In this way, but
owing to additional complications the brake efficiency was only 31%,
which is hardly any improvement on the brake efficiency of 30% obtained
with the ordinary type of gas-engine. Although Carnot's principle makes
it possible to calculate in every case what the limiting possible
efficiency would be for any kind of cycle if all heat losses were
abolished, it is very necessary, in applying the principle to practical
cases, to take account of the possibility of avoiding the heat losses
which are supposed to be absent, and of other practical limitations in
the working of the actual engine. An immense amount of time and
ingenuity has been wasted in striving to realize impossible margins of
ideal efficiency, which a close study of the practical conditions would
have shown to be illusory. As Carnot remarks at the conclusion of his
essay: "Economy of fuel is only one of the conditions a heat-engine must
satisfy; in many cases it is only secondary, and must often give way to
considerations of safety, strength and wearing qualities of the machine,
of smallness of space occupied, or of expense in erecting. To know how
to appreciate justly in each case the considerations of convenience and
economy, to be able to distinguish the essential from the accessory, to
balance all fairly, and finally to arrive at the best result by the
simplest means, such must be the principal talent of the man called on
to direct and co-ordinate the work of his fellows for the attainment of
a useful object of any kind."