_v_ω − _p_φ + ωφ = _p_{0}v_{0}E_τ,
or, neglecting the product ωφ,
_v_ω − _p_φ = _p_{0}v_{0}E_τ.
Hence the preceding expression for mechanical effect, gained in the
cycle of operations, becomes
_p_{0}v_{0}_. _E_τ . _dv_/_v_.
Or, as we may otherwise express it,
(_Ep_{0}v_{0}_)/(_vdq_/_dv_). _dq_. τ.
Hence, if we denote by _M_ the mechanical effect due to _H_ units of
heat descending through the same interval τ, which might be obtained by
repeating the cycle of operations described above, (_H_)/(_dq_) times,
we have
_M_ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (3)
27. If the _amplitudes_ of the operations had been finite, so as to give
rise to an absorption of _H_ units of heat during the first operation,
and a lowering of temperature from _S_ to _T_ during the second, the
amount of work obtained would have been found to be expressed by means
of a double definite integral thus:[48]
_M_ = ∫_{0}^{_H_} _dq_ ∫_{_T_}^{_S_} _dt_. (_Ep_{0}v_{0}_)/(_vdq_/_dv_), ⎫
or ⎬. (4)
_M_ = _Ep_{0}v_{0}_ ∫_{0}^{_H_} ∫_{_T_}^{_S_} (1)/(_v_) (_dv_)/(_dq_). _dtdq_; ⎭
this second form being sometimes more convenient.
28. The preceding investigations, being founded on the approximate laws
of compressibility and expansion (known as the law of Mariotte and
Boyle, and the law of Dalton and Gay-Lussac), would require some slight
modifications to adapt them to cases in which the gaseous medium
employed is such as to present sensible deviations from those laws.
Regnault’s very accurate experiments show that the deviations are
insensible, or very nearly so, for the ordinary gases at ordinary
pressures; although they may be considerable for a medium, such as
sulphurous acid, or carbonic acid under high pressure, which approaches
the physical condition of a vapor at saturation; and therefore, in
general, and especially in practical applications to real air-engines,
it will be unnecessary to make any modification in the expressions. In
cases where it may be necessary, there is no difficulty in making the
modifications, when the requisite data are supplied by experiment.
29.[49] Either the steam-engine or the air-engine, according to the
arrangements described above, gives all the mechanical effect that can
possibly be obtained from the thermal agency employed. For it is clear
that in either case the operations may be performed in the reverse
order, with every thermal and mechanical effect reversed. Thus, in the
steam-engine, we may commence by placing the cylinder on the impermeable
stand, allow the piston to rise, performing work, to the position
_E_{3}F_{3}_; we may then place it on the body _B_, and allow it to
rise, performing work, till it reaches _E_{2}F_{2}_ after that the
cylinder may be placed again on the impermeable stand, and the piston
may be pushed down to _E_{1}F_{1}_; and, lastly, the cylinder being
removed to the body _A_, the piston may be pushed down to its primitive
position. In this inverse cycle of operations a certain amount of work
has been spent, precisely equal, as we readily see, to the amount of
mechanical effect gained in the direct cycle described above; and heat
has been abstracted from _B_, and deposited in the body _A_, at a higher
temperature, to an amount precisely equal to that which in the direct
style was _let down_ from _A_ to _B_. Hence it is impossible to have an
engine which will derive more mechanical effect from the same thermal
agency than is obtained by the arrangement described above; since, if
there could be such an engine, it might be employed to perform, as a
part of its whole work, the inverse cycle of operations, upon an engine
of the kind we have considered, and thus to continually restore the heat
from _B_ to _A_, which has descended from _A_ to _B_ for working itself;
so that we should have a complex engine, giving a residual amount of
mechanical effect without any thermal agency, or alteration of
materials, which is an impossibility in nature. The same reasoning is
applicable to the air-engine; and we conclude, generally, that any two
engines, constructed on the principles laid down above, whether
steam-engines with different liquids, an air-engine and a steam-engine,
or two air-engines with different gases, must derive the same amount of
mechanical effect from the same thermal agency.
30. Hence, by comparing the amounts of mechanical effect obtained by the
steam-engine and the air-engine from the letting down of the _H_ units
of heat from _A_ at the temperature (_t_ + τ) to _B_ at _t_, according
to the expressions (2) and (3), we have
_M_ = (1 − σ)(_dp_)/(_kdt_). _H_τ = (_Ep_{0}v_{0}_)/(_vdq_/_dv_). _H_τ. (5)
If we denote the coefficient of _Η_τ in these equal expressions by μ,
which maybe called “Carnot’s coefficient,” we have
μ = (1 − σ)(_dp_)/(_kdt_) = (_Ep_{0}v_{0}_)/(_vdq_/_dv_), (6)
and we deduce the following very remarkable conclusions:
(1) For the saturated vapors of all different liquids, at the same
temperature, the value of (1 − σ)(_dp_/_kdt_) must be the same.
(2) For any different gaseous masses, at the same temperature, the value
of _Ep_{0}v_{0}_/(_vdq_/_dv_) must be the same.
(3) The values of these expressions for saturated vapors and for gases,
at the same temperature, must be the same.
31. No conclusion can be drawn _a priori_ regarding the values of this
coefficient μ for different temperatures, which can only be determined,
or compared, by experiment. The results of a great variety of
experiments, in different branches of physical science (Pneumatics and
Acoustics), cited by Carnot and by Clapeyron, indicate that the values
of μ for low temperatures exceed the values for higher temperatures; a
result amply verified by the continuous series of experiments performed
by Regnault on the saturated vapor of water for all temperatures from 0°
to 230°, which, as we shall see later, give values for μ gradually
diminishing from the inferior limit to the superior limit of
temperature. When, by observation, μ has been determined as a function
of the temperature, the amount of mechanical effect, _M_, deducible from
_H_ units of heat descending from a body at the temperature _S_ to a
body at the temperature _T_, may be calculated from the expression
_M_ = _H_ ∫_{_S_}^{_T_} μ_dt_, (7)
which is, in fact, what either of the equations (1) for the
steam-engine, or (4) for the air-engine, becomes, when the notation μ,
for Carnot’s multiplier, is introduced.
The values of this integral may be practically obtained, in the most
convenient manner, by first determining, from observation, the mean
values of μ for the successive degrees of the thermometric scale, and
then adding the values for all the degrees within the limits of the
extreme temperatures _S_ and _T_.[50]
32. The complete theoretical investigation of the motive power of heat
is thus reduced to the experimental determination of the coefficient μ;
and may be considered as perfect, when, by any series of experimental
researches whatever, we can find a value of μ for every temperature
within practical limits. The special character of the experimental
researches, whether with reference to gases or with reference to vapors,
necessary and sufficient for this object, is defined and restricted in
the most precise manner, by the expressions (6) for μ, given above.
33. The object of Regnault’s great work, referred to in the title of
this paper, is the experimental determination of the various physical
elements of the steam-engine; and when it is complete, it will furnish
all the _data_ necessary for the calculation of μ. The valuable
researches already published in a first part of that work make known the
latent heat of a given weight, and the pressure, of saturated steam for
all temperatures between 0° and 230° Cent. of the air-thermometer.
Besides these data, however, the density of saturated vapor must be
known, in order that _k_, the latent heat of a unit of volume, may be
calculated from Regnault’s determination of the latent heat of a given
weight.[51] Between the limits of 0° and 100°, it is probable, from
various experiments which have been made, that the density of vapor
follows very closely the simple laws which are so accurately verified by
the ordinary gases;[52] and thus it may be calculated from Regnault’s
table giving the pressure at any temperature within those limits.
Nothing as yet is known with accuracy as to the density of saturated
steam between 100 and 230°, and we must be contented at present to
estimate it by calculation from Regnault’s table of pressures; although,
when accurate experimental researches on the subject shall have been
made, considerable deviations from the laws of Boyle and Dalton, on
which this calculation is founded, may be discovered.
34. Such are the experimental data on which the mean values of μ for the
successive degrees of the air-thermometer, from 0 to 230°, at present
laid before the Royal Society, is founded. The unit of length adopted is
the English foot; the unit of weight, the pound; the unit of work, a
“foot-pound;” and the unit of heat that quantity which, when added to a
pound of water at 0°, will produce an elevation of 1° in temperature.
The mean value of μ for any degree is found to a sufficient degree of
approximation by taking, in place of σ, _dp_/_dt_ and _k_; in the
expression
(1 − σ). (_dp_)/(_kdt_);
the mean values of those elements; or, what is equivalent to the
corresponding accuracy of approximation, by taking, in place of σ and
_k_ respectively, the mean of the values of those elements for the
limits of temperature, and in place of _dp_/_dt_, the difference of the
values of _p_, at the same limits.
35. In Regnault’s work (at the end of the eighth memoir), a table of the
pressures of saturated steam for the successive temperatures 0°, 1°,
2°, ... 230°, expressed in millimetres of mercury, is given. On account
of the units adopted in this paper, these pressures must be estimated in
pounds on the square foot, which we may do by multiplying each number of
millimetres by 2.7896, the weight in pounds of a sheet of mercury, one
millimetre thick, and a square foot in area.
36. The value of _k_, the latent heat of a cubic foot, for any
temperature _t_, is found from λ, the latent heat of a pound of
saturated steam, by the equation
_k_ = (_p_)/(760). (1 + .00366 × 100)/(1 + .00366 × _t_). × .036869[53] . λ,
where _p_ denotes the pressure in millimetres, and λ the latent heat of
a pound of saturated steam; the values of λ being calculated by the
empirical formula[54]
λ = (606.5 + 0.305_t_) − (_t_ + .00002_t_^2 + 0.0000003_t_^3),
given by Regnault as representing, between the extreme limits of his
observations, the latent heat of a unit weight of saturated steam.