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Rh in which the constant s = 0-99666 is chosen to make h at IOOC. = 100 cals. C., or 180 B.Th.U. per Ib. at 2I2F., reckoned from 32 F. The symbol a is the factor for reducing any product of dimensions pv to heat units. When p is in Ib. per sq. in. and v in cub. ft. per Ib., the reciprocal I/a (which it is most convenient to use with a slide rule) has the value 9-722 on the Centigrade scale, and 5-401 on the Fahrenheit scale of temperature. V, and v are the volumes of the dry saturated vapour and the liquid respectively, and dp/dT is rate of increase of saturation pressure p with temperature. When taken in conjunction with Clapeyron's equation for the latent heat, formula (16) gives a very useful relation between the total heat H and the volume V for wet saturated steam in any state, H-st = aVT(dpldT)=pV/*,. . . (17).

The factor ir = p/aT(dp/dT), which varies slowly and is inde- pendent of the wetness, has been tabulated, as affording the most expeditious and accurate method of calculating either H or V when the other is known. The relation between H and V when p is given is that most commonly required in practical work. The same formula leads to a simple expression for the entropy *,
 * = 5log e (T/T )+oV(^/dT), . . . (18)

which applies to wet steam of volume V, and also to the liquid if v is substituted for V. To represents the freezing point, 273- 1 C. or 49i-6F.

Values ranging from 0-305 at oC. to 0-665 at l6oC. had been proposed by various writers in 1900 for the specific heat of steam, but the direct measurements by the continuous electric method at atmospheric pressure from 100 to i6oC. gave results but slightly exceeding those of Regnault over the range 1 24 to224C.,and showed that the limiting value So at zero pressure was probably nearly con- stant and equal to 0-477. This was confirmed by L. Holborn and H. Henning (Ann. Phys., 18, p. 739, 1905) in a qualitative manner by comparison with air over the range 110 to 82OC.

The experiments on the cooling-effect C, when combined with those of the specific heat S, showed that the product SC was a func- tion of the temperature only, and gave the simple expression for the total heat,

H=S T-SCP + B. . . (19)

for dry steam at any pressure P. The values for dry saturated steam, given by putting the saturation pressure p in this expression, while differing materially from Regnault's formula, gave good agreement with the experiments (see 27.902) of Dieterici at oC., and of Grif- fiths at 30 and 4OC., when the constant B was reduced from Joly's observations at IOOC. with the aid of the experiments on the specific heat of water. This formula was closely confirmed by the observa- tions of H. Henning (Ann. Phys., 21, p. 849, 1906) on the latent heat between 30 and iooC. His later observations (Ann. Phys., 29, p. 441, 1909) also gave good agreement with the same curve at iSo^C., but showed a discontinuity at I2OC., which may be attrib- uted to inevitable experimental errors in such difficult work. At higher temperatures, up to 26oC., equation (19) received theoreti- cal confirmation from the formula for the latent heat proposed by M. Thiesen, namely, L=Li(^-/)^, based on the vanishing of the latent heat at the critical temperature t c. As first applied by Thiesen himself (Ann. Phys., 9, p. 80, 1902) to the case of steam, with 365C. for the critical temperature, this formula gave results which were much too low for the latent heat. It was shown, how- ever, by Traube and Teichner (Ann. Phys., 13, p. 620, 1904) that the true value of t c was 374C., which brought Thiesen's formula into agreement with (19) to less than I in 1,000 all the way from o to 26oC., when the constants were properly determined from the known values at o, 100, 180 and 374, giving the result, log L = 1-9638+0-3151 log (374-0 ( 2 ). in the logarithmic form as required for practical calculations. The importance of this formula arises from the fact that direct deter- minations of H, (for dry saturated steam) become exceedingly difficult and uncertain at temperatures above i8oC., owing to errors from leakage and wetness, and that a formula of this type has been verified for many other substances in the critical region, so that it affords the best guide to the probable variation of H, between 200 and 374C.

The throttling experiments showed that there must be a consid- erable variation of S with pressure, corresponding to the variation of SC with temperature. But the experiments on the adiabatic expansion of dry steam showed that the index n + l in the equation P/T n+l = constant, was very nearly constant and equal to 13/3 over a wide range of P and! T. Since So/R = i3/3. 't followed that the total heat of dry steam must be expressible in the form,

H = (i3a/3)P(V-6)+a&P+B. . . (21) giving the convenient expression for the volume of dry steam,

V = (3/i3o)(H-B)/P + io&/i3. . . (22).

It also followed that the coaggregation volume c = Co (To/T) n in the equation

V-6 = RT/aP-c,. . . (23)

must vary with temperature according to the index n 10/3, giving for the variation of S and C, in terms of c, the formulae,

SC=a(n + i)c-o6. . . (24) S = So+an(n + i)cP/T. (25).

It was obvious that these could not apply accurately at high pres- sures in the critical region, but they afford ample accuracy for all purposes in the pressures required in steam-engine practice.

The Munich experiments (Forsch. Ver. Deut. Ing., 21, 1905) by O. Knoblauch, R. Linde and K. Klebe, on the volume of steam, proved to be quite inconsistent with the well-known equation of Zeuner, then commonly accepted, but showed the most remarkable agreement up to l8oC. with formula (23) deduced from the throt- tling experiments. Unfortunately Linde introduced an additional factor of the form (l+kP) in the expression for the coaggregation volume c, to represent the apparent curvature of the isothermals, and the probable deviations at higher pressures. His equation has often been adopted (e.g. in the tables of Marks and Davis) and fre- quently imitated, but it is of the wrong type to represent the criti- cal conditions, and leads to impossible results at comparatively low temperatures within the range of steam-engine practice. Thus it would make the value of H, a maximum at 2O7C., which should not occur till near 28oC., and it gives a value 47 B.Th.U. lower than the Thiesen formula (20) at 5Oo"F. (26oC.), both of which results are quite impossible. Moreover, it cannot be reconciled with observa- tions on the specific heat and the cooling-effect.

The variation of S with pressure given by (25), as predicted by the experiments on C, was qualitatively confirmed by the experi- ments of O. Knoblauch and M. Jakob (Forsch. Ver. Deut. Ing., 36, p. 109, 1906) extending to 8 atmospheres. But their extrapola- tion to higher pressures was clearly impossible, and was conclu- sively disproved by the experiments of C. Thomas (Amer. Spc. Mech. Eng., 29, p. 1,021, 1907), extending to 34 atmospheres, which confirmed the variation given by (25) as closely as could be desired up to 500 Ib. and 35OC. According to Knoblauch and Jakob, the specific heat So at zero pressure increased no less than 14 % between 100 and 4OOC. This was reduced to 4% by the later experiments of O. Knoblauch and H. Mollier (Forsch. Ver. Deut. Ing., 109, p. 79, 1911). G. A. Goodenough (Steam Tables, 1915) from the same observations deduces a diminution of I %, and R. C. H. Heck (Amer. Soc. Mech. Eng., 1921) an increase of 2%, over the same range. This variation is evidently much too small and uncertain to be worth considering in any equations for steam-engine work, though it becomes quite important for the internal-combustion engine at 2000 C.

The integration of Clapeyron's equation for the saturation pres- sure (see 27.903) afforded a means of testing the theory by com- parison with Regnault's observations, which showed satisfactory agreement. The observations of L. Holborn and H. Henning (Ann. Phys., 26, p. 833, 1908), extending to 2OOC. with platinum ther- mometers, showed improved agreement at 200 C. and also, at low temperatures. The theoretical equation was not originally intended for use at temperatures above 2OOC., but the experiments of L. Holborn and A. Baumann (Ann. Phys. ,31, p. 945, 1910) at higher temperatures showed that it could not be so much as IC. in error at 26oC. This would make an error of only I in 4,000 in the value of H s, which is quite beyond the limits of experimental accuracy. A great deal has been made of the uncertainty of V, as deduced from T and dp/dt by Clapeyron's equation, which greatly exagger- ates the possible error. This equation cannot be used in practical tests, in which it is always necessary to deduce the values of V, from those of H, and p by (22), so that no uncertainty of this kind can arise, provided that the values of H, are correct, as shown by equation (20), and that the equations are consistent with the adia- batic assumed. Owing to the continuity of the adiabatic on the HP diagram, the exact point at which the steam crosses the saturation line is of little importance. The state of the steam beyond this point may be either wet or supersaturated, which may make a con- siderable difference in V, but does not materially affect the result for given values of H and P. Errors may arise in academic prob- lems if only t is given and the wetness is assumed, but the state of the steam cannot be determined in practice without measuring H, preferably by throttling, and P is the easiest quantity to observe, and is always known. If P is given, and the state of the steam is known, the error in V cannot exceed \ of I % even at 650 Ib. pres- sure, if the values of H, are correct. If the variation of H is directly determined by the throttling method, the values of H, cannot be far wrong. But if the values of H, are deduced from those of V, through Clapeyron's equation for L, as is frequently done, by assuming an arbitrary empirical formula for dp/dt in conjunction with an improbable type of equation for V, it is almost inevitable that material errors should arise from the thermodynamic incon- sistencies involved in such a circuitous process. It is most essen- tial for practical purposes that the equations should be as simple as possible and exactly consistent with the laws of thermodynamics. To be of any use, the tables must agree precisely with the expres- sion employed for the adiabatic heatdrop and the discharge through a nozzle. With such limitations it would evidently be impossible to include the critical state in any consistent system of equations without intolerable complexity, but ample accuracy can be secured for the ordinary range of steam-engine practice.

Adiabatic Heatdrop. The change of total heat H in frictionless adiabatic expansion or compression is frequently of considerable interest as representing the work done by or on the fluid in the ideal case, when there is no internal friction, and when no heat is supplied