CCCXLIV.

FORMULAS FOR SATURATED AND SUPERHEATED

VAPORS.

BY DE VOLSON WOOD, HOBOKEN, N. J.

(Member of the Society.)

SOME fifty formulas or more have been proposed by different writers to represent the relation between the temperature and pressure of saturated steam. Regnault used the general form

for all the vapors experimented upon by him, and Rankine used

A few formulas have been proposed to represent superheated vapors that shall hold good down to the state of saturation; among which the most celebrated is one deduced by Zeuner, of the form

This gives values agreeing remarkably well with those found by observation and experiment.

Vapors, when considerably superheated, approximate so nearly to the condition of a perfect gas that it is questionable whether there is any advantage in any formula that may be devised over that of the well-known one for perfect gases,

Thus, to illustrate, M. Hirn found that the specific volume of steam at 200° C. under a pressure of three atmospheres was 0.697, and at the same temperature under a pressure of four atmospheres was 0.522; and if the steam were a perfect gas the latter

The

should be three fourths the former; or × 697 = 0.5227. agreement is as near as could be expected. (The values for superheated steam are taken from Röntgen's Thermodynamics, Du Bois's translation, p. 280 of old edition, 570 of the new.) At one atmosphere and 141° C. the specific volume is, according to Hirn, 1.85, and according to the law of perfect gases it should be, at five atmospheres and 205° C.,

but it was observed to be 0.422, an error of about three per cent. A greater error would naturally be expected in this case than in the former one, since the lower pressure and temperature were so low there would be comparatively little superheating. A comparison of the examples above one atmosphere will show that they agree more nearly with the law of perfect gases.

The specific volume of saturated steam is given with considerable accuracy by the empirical formula

Since the law of change between the state of saturation and that of a highly saturated vapor is not known, any formula representing the law of change will be more or less empirical.

It may be considered as an imperfect fluid, in which case, if Rankine's formula for imperfect fluids be accepted, the equation of the gas would be

Po Vo

in which R

=

T

, ao, a, etc., are inverse functions of the specific

volumes; p the pressure of one atmosphere, 7 the absolute temperature of melting ice, and v the corresponding specific volume. The law of change in this formula in the terms after RT depends upon an inverse function of e, whereas in Zeuner's the third term is a direct function of p. It would therefore appear, if Zeuner's equation is correct, or the nearest correct, that Rankine's formula must be erroneous, and the hypothesis upon which it is founded that of "Molecular Vortices"-will be of questionable

validity. It is admitted at the outset that this theory or hypothesis is not an accepted part of science; but Rankine's formulas have, generally, such a wide application, and as this one equation (ƒ), in his opinion, represents the results of Regnault's experiments, I have desired to see how well it could be made to represent the states of saturation and superheating. Aside from the interest involved in testing Rankine's hypothesis, the use that can be made of such an equation, if well established, may be seen from the study of M. Ledoux in determining the probable latent heat of ammonia by the use of Zeuner's equation.

I have spent much labor upon this problem, and, although the work is not yet complete, I desire to place on record some of my results.

An exact coincidence of results between theory and experiment is not to be expected. There are always errors in experiments, which, though small, exclude the possibility of expressing the law representing those results exactly. Then, too, the constants entering our theoretical equations are not known exactly, though the limits of uncertainty are comparatively small. Thus, the mechanical equivalent of heat used in Rankine and Zeuner's time was 772; now 778 is known to be nearer correct, and the absolute zero then used was 461.2° Fahr., but now 460.66° Fahr., below 0°F., is believed to be nearer correct. These differences are of the slightest importance in ordinary practice, but are not to be ignored in a critical study of the subject. Prior to the publication of Professor Peabody's Steam Tables, I compnted the specific volumes of steam in the cases where I wished to use them, by means of Rankine's equations, using the constants above given, and later compared the results with Peabody's tables, and found that the greatest discrepancy was less than 0.02 of a cubic foot; and had I used as many decimals as he did, I cannot say but there would have been even less discrepancy. Considering that he used Regnault's formulas, such an argument could hardly have been anticipated, and the result not only confirms the correctness of Rankine's formula, but shows it to be the more desirable, since it is the more simple. I have used Peabody's tables in all the following computations. If the new constants were used, the results from Zeuner's equation would not agree so closely with the results obtained from the mechanical theory as Zeuner's computations seemed to make them, although they would even then be sufficiently accurate for practical purposes.

Selecting three points on the curve of saturation of known pressure, temperature, and volume, I found

and, changing the value of b somewhat arbitrarily, the equation became

in which p is in pounds per square foot, v the volume in cubic feet, and the absolute temperature on the Fahrenheit scale. By means of this equation the following table was computed for saturated steam :

The results are fair, the greatest error being about 1.2 per cent., some being greater and some less than the tabular values of the pressures corresponding to the specific volumes. The volumes

here assumed are those found from assumed values of temperature and pressure according to the mechanical theory.

The following table was computed from the same formula for superheated steam:

These errors are not only larger than those in Table I., but err all in the same sense, the computed values being too large.

Next, the exponent n was assumed as 4, and a new determination of the constants R and b was made, involving some conditions for superheated steam, resulting in the equation

The two following tables have been computed from this formula