θ′ and θ″ we see that Rankine’s result follows immediately, provided that p(v−b) is equal to (S−s) or Rθ/m, which is approximately true for gases and vapours when v is very large compared with b. We may observe that the equation (11) is accurately true for an ideal vapour, for which pv＝(S−s)θ, provided that the total heat is defined as equal to the change of the function (E+pv) between the given limits. Adopting this definition, without restriction to the case of an ideal vapour or to saturation-pressure, the rate of variation of the total heat with temperature (dH/dθ) at constant, pressure is equal to S under all conditions, whether S is constant, or varies both with p and θ. (See Thermodynamics, § 7.)
11. Specific Heat of Vapours.—The question of the measurement of the specific heat of a vapour possesses special interest on account of this simple theoretical relation between the specific heat and the variation of the latent and total heats. The first accurate calculations of the specific heats of air and gases were made by Rankine in a continuation of the paper already quoted. Employing Joule’s value of the mechanical equivalent of heat, then recently published, in connexion with the value of the ratio of the specific heats of air S/s＝1·40 deduced from the velocity of sound, Rankine found for air S＝·240, which was much smaller than the best previous determinations (e.g. Delaroche and Berard, S＝·267), but agreed very dosely with the value S＝·238, found by Regnault at a later date. Adopting for steam the same value of the ratio of the specific heats, viz. 1·40, Rankine found S＝·385, a value which he used, in default of a better, in calculating some of the properties of steam, although he observed that it was much larger than the coefficient ·305 in Regnault’s formula for the variation of the total heat. The specific heat of steam was determined shortly afterwards by Regnault (Comptes Rendus, 36, p. 676) by condensing superheated steam at two different temperatures (about 125° and 225° C.) successively in the same calorimeter at atmospheric pressure, and taking the difference of the total heats observed. The result found in this manner, viz. S＝·475, greatly increased the apparent discrepancy between Regnault’s and Rankine’s formulae for the total heat. The discrepancy was also noticed by G. R. Kirchhoff, who rediscovered Rankine’s formula (Pogg. Ann. 103, p. 185, 1858). He suggested that the high value for S found by Regnault might be due to the presence of damp in his superheated steam, or, on the other hand, that the assumption that steam at low temperatures followed the law pv＝Rθ might be erroneous. These suggestions have been frequently repeated, but it is probable that neither is correct. G. A. Zeuner, at a later date (La Chaleur, p. 441), employing the empirical formula pv＝Bθ+Cp.25 for saturated steam, found the value S＝·568, which further increased the discrepancy. G. A. Hirn and A. A. Cazin (Ann. Chim. Phys. iv. 10, p. 349, 1867) investigated the form of the adiabatic for steam passing through the state p＝760 mm., θ＝373° Abs., by observing the pressure of superheated steam at any temperature which just failed to produce a cloud on sudden expansion to atmospheric pressure. Assuming an equation of the form log (p/760)＝a log (θ/373), their results give a＝S/R＝4·305, or S＝0·474, which agrees very perfectly with Regnault’s value. It must be observed, however, that the agreement is rather more perfect than the comparative roughness of the method would appear to warrant. More recently, Macfarlane Gray (Proc. Inst. Mech. Eng. 1889), who has devoted minute attention to the reduction of Regnault’s observations, assuming S/s＝1·400 as the theoretical ratio of specific heats of all vapours on his " aether-pressure theory," has calculated the properties of steam on the assumption S＝0·384. He endeavours to support this value by reference to sixteen of Regnault’s observations on the total heat of steam at atmospheric pressure with only 19° to 28° of superheat. These observations give values for S ranging from 0·30 to 0·46, with a mean value 0·3778. But it must be remarked that the superheat of the steam in these experiments is only 1 or 2 % of the total heat measured. A similar objection applies, though with less force, to Regnault’s main experiments between 125°and 225° C., giving the value S＝0·475, in which the superheat (on which the value of S depends) is only one-sixteenth of the total heat measured. Gray explains the higher value found by Regnault over the higher range as due to the presence of particles of moisture in the steam, which he thinks “would not be evaporated up to 124° C, but would be more likely to be evaporated in the higher range of temperature.” J. Perry (Steam Engine, p. 580), assuming a characteristic equation similar to Zeuner’s (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault’s formula (10) for the total heat. This method is logically consistent, and gives values ranging from 0·305 at 0° to 0·341 at 100°C. and 0·464 at 210° C, but the difference from Regnault’s S＝0·475 cannot easily be explained.
12. Throttling Calorimeter Method.—The ideal method of determining by direct experiment the relation between the total heat and the specific heat of a vapour is that of Joule and Thomson, which is more commonly known in connexion with steam as the method of the throttling calorimeter. It was first employed in the case of steam by Peabody as a means of estimating the wetness of saturated steam, which is an important factor in testing the performance of an engine. If steam or vapour is " wire-drawn " or expanded through a porous plug or throttling aperture without external loss or gain
Fig. 2.—Throttling Calorimeter Method.
of heat, the total heat (E+pv) remains constant (Thermodynamics, § 11), provided that the experiment is arranged so that the kinetic energy of flow is the same on either side of the throttle, Thus, starting with saturated steam at a temperature θ′ and pressure p′ , as represented by the point A on the pθ diagram (fig. 2), if the point B represent the state p″θ″ after passing the throttle, the total heat at A is the same as that at B, and exceeds that at any other point D (at the same pressure p″ as at B, but at a lower temperature θ) by the amount S×(θ″−θ), which would be required to raise the temperature from D to B at constant pressure. We have therefore the simple relation between the total heats at A and D—
| HA−HD＝S(θ″−θ). | (12) |
If the steam at A contains a fraction z of suspended moisture, the total heat HA is less than the value for dry saturated steam at A by the amount zL. If the steam at A were dry and saturated, we should have, assuming Regnault’s formula (10), HA−HD＝·305 (θ′−θ), whence, if S＝·475, we have zL＝·305 (θ′−θ)−·475 (θ′−θ). It is evident that this is a very delicate method of determining the wetness z, but, since with dry saturated steam at low pressures this formula always gives negative values of the wetness, it is clear that Regnault’s numerical coefficients must be wrong.
From a different point of view, equation (12) may be applied to determine the specific heat of steam in terms of the rate of variation of the total heat. If we assume Regnault’s formula (10) for the total heat, we have evidently the simple relation S＝0·305(θ′−θ)/(θ″−θ), supposing the initial steam to be dry, or at least of the same quality as that employed by Regnault. This method was applied by J. A. Ewing (B.A. Rep. 1897) to steam near 100° C. He found the specific heat smaller than 0·475, but no numerical results were given. A very complete investigation on the same lines was carried out by J. H. Grindley (Phil. Trans. 1900) at Owens College under the direction of Osborne Reynolds. Assuming dH/dθ＝0·305 for saturated steam, he found that S was nearly independent of the pressure at constant temperature, but that it varied with the temperature from 0·387 at 100° C. to 0·665 at 160° C. Writing Q for the Joule-Thomson “cooling effect,” dθ/dp, or the slope BC/AC of the line of constant total heat, he found that Q was nearly independent of the pressure at constant temperature, a result which agrees with that of Joule and Thomson for air and CO2; but that it varied with the temperature as (1/θ)3·8 instead of (1/θ)2. These results for the variation of Q are independent of any assumption with regard to the variation of H. Employing the values of S calculated from dH/dθ＝ 0·305, he found that the product SQ was independent of both pressure and temperature for the range of his experiments. Assuming this result to hold generally, we should have S＝0·306 at 0° C, which agrees with Rankine’s view; but increasing very rapidly at higher temperatures to S＝1·043 at 200° C., and 1·315 at 220° C. The characteristic equation, if SQ＝constant, would be of the form (v+SQ)＝Rθ/p, which does not agree with the well-known behaviour of other gases and vapours. Whatever may be the objections to Regnault’s method of measuring the specific heat of a vapour, it seems impossible to reconcile so wide a range of variation of S with his value S＝0·475 between 125° and 225° C. It is also extremely unlikely that a vapour which is so stable a chemical compound as steam should show so wide a range of variation of specific heat. The experimental results of Grindley with regard to the mode of variation of Q have been independently confirmed by Callendar (Proc. R.S. 1900), who quotes the results of similar experiments made at McGill College in 1897, but gives an entirely different interpretation, based on a direct measurement of the specific heat at 100° C. by an electrical method.
The method of deducing the specific heat from Regnault’s formula for the variation of the total heat is evidently liable in a greater degree to the objections which have been urged against his method of determining the specific heat, since it makes the value of the specific heat depend on small differences of total heat observed under conditions of greater difficulty at various pressures. The more logical method of procedure is to determine the specific heat independently of the total heat, and then to deduce the variations of total heat by equation (12). The simplest method of measuring the specific heat appears to be that of supplying heat electrically to a steady current of vapour in a vacuum-jacket calorimeter, and observing the rise of temperature produced. Employing this method, Callendar finds S＝0·497 for steam at one atmosphere
