BY DE VOLSON WOOD, PROFESSOR OF ENGINEERING IN STEVENS INSTITUTE OF
TECHNOLOGY.
It is clearly intimated by Mr. Hanssen, in his determination of the
mechanical equivalent of heat, published in the Scientific American
Supplement, No. 642, April 21, 1888, that his object is to determine
the _absolute_ value of this constant. With his data he finds it to be
771.89 foot pounds. But the determination by direct experiment gives a
larger value. Thus, the most reliable experiments--those of Joule and
Rowland--give values exceeding by several units that found by Hanssen.
A committee of the British Association, appointed for this purpose,
reported in 1876 that sixty of the most reliable of Joule's
experiments gave the mean value 774.1. The experiments were made with
water at a temperature of about 60° F., according to the mercurial
thermometer, and reduced to its value at the temperature of melting
ice, according to the formula given by Regnault for the variation of
the specific heat of water at varying temperature under the constant
pressure of one atmosphere. According to this formula the specific
heat of water increases with the temperature above the melting point
of ice, so that the equivalent would be somewhat less at 32° F. than
at 60° F. It will be found in Regnault's _Relation des Experiences_
that he experimented on water at high temperatures, but more recently
Professor Rowland has found that the specific heat of water is
_greater_ at 40° F. than at 60° F., thus reversing between these
limits the law given by Regnault; the increase, as given by the most
probable values, being, roughly, about 1/250 of its value at 60° F.
The proper correction due to this cause would make the equivalent over
777 foot pounds, instead of 774.1. Professor Rowland's experiments,
when reduced to the same thermometer, same temperature, and same
latitude as Joule's, agreed very nearly with those of the latter,
being about 1/1000 part larger; so that the chief difference in the
ultimate values consists in the reductions for temperature and
latitude. The force of gravity being less for the lower latitudes, the
number representing the mechanical equivalent will be greater for the
latter, since the unit pound mass must fall through a greater number
of feet to equal the same work; so that the equivalent will be greater
at Paris than at Manchester. Professor Rowland also found that the
degrees on the air thermometer from 40° F. upward to above 60° F.
exceeded those on the mercurial thermometer throughout the
corresponding range, and that from 40° to 41° the degree was between
1/150 and 1/200 of a degree larger on the air thermometer than on the
mercurial. Although this fraction is too small to be observed by
ordinary means, yet, if it exists, it cannot be ignored if absolute
values are sought. Regnault employed the air thermometer in his
experiments, while Joule used the mercurial thermometer, and if
Joule's value 774.1 be increased by 1/200 of itself in order to reduce
it from the equivalent of the degree on the mercurial thermometer to
that on the air thermometer, we get 778 foot pounds, nearly. Rowland
found from his experiments that when reduced to the air thermometer
and to the latitude of Baltimore, the equivalent was nearly 783,
subject to small residual errors.
Nearly all writers upon this subject--except Rankine--have considered
that the mechanical equivalent of heat, in British units, was the
energy necessary to raise the temperature of one pound of water from
32° F. to 33° F., but Rankine defines it as the heat necessary to
increase the temperature of one pound of water one degree Fahrenheit
from that of maximum density, or from 39° F. to 40° F. For ordinary
practice it is immaterial which of these definitions is used, for the
errors resulting therefrom are much less than those resulting from
ordinary observations. But when the value is to be determined by
direct experiment at the standard temperature, Rankine's limits are
much to be preferred; for it is so very difficult to determine exact
values by observation when the substance is near the state bordering
on a change of state of aggregation, as that of changing from water to
ice. Observations made at about 60° F. were reduced by means of
Regnault's law for the specific heat of water, as has been stated,
which is expressed by the formula
4 9
c = 1 + ------ t + ------ t^{2}
10^{5} + 10^{7}
in which t denotes the temperature according to the Centigrade scale.
According to this law, the mechanical equivalent would not be 0.2 of a
foot pound greater at 5° C. (41° F.) than at 0° C. (32° F.); hence, if
this law were correct, it would make no practical difference whether
the temperature were at 0° C. or 5° C. This law makes the _computed_
value at 32° F. about 0.95 of a foot pound less than that determined
by experiment at 60° F.; whereas Rowland's experiments make it
_greater_ at 40° F. by more than four foot pounds, for the air
thermometer. In determining a _fixed_ value to be used for scientific
purposes, it is necessary to fix the place, the thermometer, and the
particular degree on the thermometer. The place may be known by its
latitude if reduced to the level of the sea. The air thermometer
agrees most nearly with that of the ideally perfect gas thermometer,
while the mercurial thermometer differs very much from it in some
cases. Thus, Regnault found that when the air thermometer indicated
630° F. above the melting point of ice (or 662° F.), the mercurial
thermometer indicated 651.9° above the same point (683.9° F.), a
difference of 22° F. It is apparent that the air thermometer furnishes
the best standard. As for the particular degree on the scale to be
used for the standard, it is apparent, from the observations above
made, that the temperature corresponding to that at or near the
maximum density of water is more desirable than that at the melting
point of ice. The fact, also, that the specific heats at constant
pressure and at constant volume are the same at the point of maximum
density, as shown by theory, is an additional argument in favor of
selecting this point for the standard. It thus appears that the
solution of this problem, which appears simple and very definite by
Mr. Hanssen's method, becomes intricate and, to a limited degree,
indeterminate when subjected to the refinements of direct experiment.
If the constants used by Hanssen are absolutely correct, then his
result must be unquestioned; but since physical constants are subject
to certain residual errors, one would as soon think of finding the
specific heat of air at constant volume, by using the value of the
mechanical equivalent as one of the elements, and trusting the result,
as he would to trust to the computed value of the mechanical
equivalent without subjecting it to the test of a direct experiment.
We will, therefore, examine the constants used to see if they are the
exact values of the quantities they represent.
He says they are universally accepted as correct; and this may be
true, when used for general purposes, and yet not be scientifically
exact. He uses 0.2377 as the specific heat of air. This is the value,
to four decimals, found by Regnault. Thus, Regnault gives for the mean
value of the specific heat of air
Between -30° C and + 10° C. 0.23771
" 0° C " 100° C. 0.23741
" 0° C " 200° C. 0.23751
And we know of no reason why one of these values should be used rather
than another, except that the mean of a large range of temperatures
may be more nearly correct than that of any other; and if this reason
determines our choice, the number 0.2375 would be used instead of
0.2377. Although this difference is small, yet the former value would
have reduced his result about 0.7 of a foot pound.
Again, he uses 0.1686 for the specific heat of air at constant volume.
The value of this constant has never been found to any degree of
accuracy by direct experiment, and we are still dependent upon the
method established by La Place and Poisson, according to which the
constant ratio of the specific heat of a gas at constant pressure to
that at constant volume is found by means of the velocity of sound in
the gas. The value of the ratio for air, as found in the days of La
Place, was 1.41, and we have 0.2377 ÷ 1.41 = 0.1686, the value used by
Clausius, Hanssen, and many others. But this ratio is not definitely
known. Rankine in his later writings used 1.408, and Tait in a recent
work gives 1.404, while some experiments give less than 1.4, and
others more than 1.41.
An error of one foot in a thousand in determining the velocity of
sound will affect the third decimal figure one or two units. A small
difference in the assumed weight of a cubic foot of air also affects
the result. M. Hanssen gives 0.080743 pound as the weight at 32° F.
under the pressure of one atmosphere; while Rankine gives 0.080728
pound. In my own computations I use 1.406 as a more probable value of
the constant sought. This will give for the specific heat of air at
constant pressure
0.2375 ÷ 1.406 = 0.1689
This is only 0.0003 of a unit greater than the value used by Hanssen,
but it would have given him nearly 775, instead of 771.89.
Again, he uses 491.4° F. for the absolute temperature of melting ice.
The exact value of this constant is unknown; but the mean value as
determined by Joule and Thomson, in their celebrated experiments with
porous plugs, was 492.66° F. This value would slightly change his
result. It will be seen from the above that a small change in the
constants used may affect by several units the computed value of the
mechanical equivalent. I have computed it, using 1.406 for the ratio
of the specific heat of air at constant pressure to that at constant
volume, 491.13° F. as the temperature of melting ice above the zero of
the _air_ thermometer, 26,214 feet for the height of a homogeneous
atmosphere, and 0.2375 for the specific heat of air, and I find, by
means of these constants, 778. If computed from the zero of the
absolute scale, 492.66° F., I find 777 to the nearest integer.
Recently I have used 778. If the value given by Rowland, about 783
according to the air thermometer at 39° F., should prove to be
correct, it seems probable that the constant 1.406 used above would be
reduced to about 1.403, or that the other constants must be changed by
a small amount. The height of the homogeneous atmosphere used above,
26,214 feet, is the value used by Rankine as deduced from Regnault's
figures, and only one foot less than the value used by Sir William
Thomson; but the figures used by Mr. Hanssen give 26,210½ feet.
The method above called Hanssen's is really that of Dr. Mayer (the
German professor), who in 1842 used it for determining the mechanical
equivalent; but on account of erroneous data, the value found by him
was much too small.
* * * * *
ECONOMY TRIALS OF A NON-CONDENSING STEAM ENGINE--SIMPLE, COMPOUND, AND
TRIPLE.[1]
[Footnote 1: Abstract of paper read before the Institution of
Civil Engineers, March 13.]