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or lost externally. If the laws of thermodynamics are summarized in the form,

dQ=Td*=dH-oWP. . . (26)

in which dQ represents heat supplied per unit mass by friction or otherwise, we observe that, in the case of isentropic flow, for which d* = o, the change of H is equal to the integral of aVdP along the adiabatic, which is readily obtained by substituting for V in terms of H and P from (22) or (17), for any given initial state and final pressure. We may also obtain the general expression for * from those for H and V by integrating d = dH/T-(aV/T) These expressions may be put in a variety of forms according to the pur- pose for which they are required. One of the most useful for dry steam is,

DH* = (H'-H")* = (H'-B-o&P')(i-T"/T')+a&(P'-P").-.(27) in which H', P', T', and H", P", T", represent the initial and final states. An exact expression for the adiabatic heatdrop DH*, in the case of wet saturated steam, is readily obtained in terms of H' and T', T". But in practice it is usually more convenient to tabu- late H and <#>, and the Gibbs' function G=T< H, which has the advantage of being a simple function of the temperature only, and is independent of the wetness for a mixture of water and steam in any proportions. From the definition of G, if is constant and equal to its initial value ", we obtain immediately the convenient expressions,

DH* = (T'-T")*'-G'+G" = H'-H".+T"(* 8 "-*'), ., . (28). The first expression is general, and is readily applied if G' and G are tabulated. The second is obtained by substituting for G' and G " in terms of H and , but is applicable only if the final state is saturated, so that H", and ", are the tabulated values for dry saturated steam.

Effects of Supersaturation. For the general theory of the beha- viour of a vapour when cooled below the saturation temperature without condensation see 27.898-^. The state of Supersaturation is very common, in rapid expansion, and has proved to be of some practical importance, as affecting the discharge through a nozzle, and the efficiency of a turbine. It appears that steam usually fol- lows the dry adiabatic, P/T 1 ** =constant, for some distance below the saturation point. The drop of temperature is about three times as rapid as along the wet adiabatic, and the volume is smaller than that of saturated steam at the same P and H. The heatdrop, and the velocity generated, are also smaller, for a given pressure drop, than in the case of steam which is assumed to remain in the equilib- rium state of saturation throughout the expansion. If the initial steam is dry saturated, it usually remains dry for some distance beyond the throat of a nozzle, so that the discharge, as given by equation (12), is obtained from the dry adiabatic, by substituting (dP/dV) = I -3P/V at the throat, which leads to values about 5 % larger than those given by the equations for wet steam. This is confirmed by experiment, and is represented by the numerical formula for the discharge M/X< in Ib. per second per sq. in. of throat, when P' is in Ib./sq. in. and V in cub. ft./lb. in the initial state,

M/X=o-3i55(P7V')l, P,/P' =0-545, - - (29) in which the small quantity b is neglected as being usually beyond the limits of possible accuracy of measurement.

The defect of heatdrop on reaching the throat is about 5 %. If the steam continued to follow the dry adiabatic to low pressures, the defect of heatdrop would often reach 20%, which would be very serious. But soon after passing the throat, the coaggregated mole- cules begin to act as condensation nuclei, according to Kelvin's equation (see 27.899). When this limit is reached, the condensation takes the form of a very thick fog of exceedingly fine particles, and is extremely rapid, owing to the enormous number of nuclei avail- able, about IO 22 per Ib. of steam. If the expansion is relatively slow, the steam is transformed into the saturated state, and remains nearly saturated for the rest of the expansion. But if the expansion is very rapid, as in an expanding nozzle at a velocity of 3 or 4000 ft. /sec., the steam will remain near the Supersaturation limit with a loss of heatdrop amounting to nearly 8 % at low pressures, involving a c<3r- responding loss of efficiency. According to Wilson's experiments at low pressures (see 27.899), the Supersaturation limit is reached when the pressure is about 8 times the normal saturation pressure corresponding to the actual temperature of the steanl. The equiva- lent wetness of the steam at this point, when transformed to the saturated state at the same P and H, would be about 3%. This appears to be confirmed by turbine tests at these pressures, but Wil- son's experiments do not afford any direct evidence with regard to the limit at which condensation starts at higher pressures. It appears on theoretical grounds that the pressure ratio corresponding to the Supersaturation limit should not be so high as 8 at high pressures, which would require an excessive increase in the drop of temperature and in the equivalent wetness of the steam at high pressures.

There is some evidence that the equivalent wetness at the super- saturation limit is the same, namely 3 %, at high as at low pres- sures. This would permit a very simple method of calculation, but more experimental tests are required to decide the point. The effect of initial superheat in improving the efficiency of a turbine cannot be satisfactorily explained on the older theory that the steam is in

the equilibrium state of saturation throughout the expansion, but is a necessary consequence of the phenomenon of Supersaturation. The loss due to Supersaturation may be entirely eliminated if the superheat is sufficient to prevent Supersaturation. In any case the loss will be greatly reduced by superheat, and the results of calcu- lation appear to indicate that the improvement of efficiency may be exactly accounted for in this way. This point has been very fully discussed by H. M. Martin, in ' A New Theory of the Steam Turbine" (Engineering, vol. 106, 1918); and also by Callendar, Properties of Steam, pp. 305-12.

Properties of Carbonic Acid. -The critical point of COz, commonly known as carbonic acid, being at a temperature a little above 31 C., the most convenient point of the scale for accurate regulation, offers special facilities for investigating the critical phenomena. These were first elucidated by T. Andrews (Phil. Trans., 1869), whose inves- tigations formed the starting point for the theories of J. Thomson, J. D. Van der Waals, J. C. Maxwell and R. Clausius. The method employed by Andrews in measuring the volume and pressure of the liquid and vapour at various temperatures reached the highest point of refinement in the researches of E. H. Amagat (Ann. Chim. Phys., 29, p. 136, 1893), whose tables of the properties of CO 2 from o to 250 C. have generally been accepted as the standard. For practical use in refrigeration the properties are also required at temperatures down to 5OC. The saturation pressures below oC. have since been determined by Kuenen and Robson (Phil. Mag., 3, p. 154, 1902), using platinum thermometers. They also determined the vapour pressures of the solid, which follow a curve cutting that of the liquid at a sharp angle at the melting point, which is at 56-2C., where the common vapour pressure is 5-2 atmos- pheres. It is found that the vapour pressures of the liquid can be represented with a fair degree of accuracy, sufficient for most prac- tical purposes, by the simple empirical formula,

log/> = i-5363+3-i57//T, (atmospheres). . . (30) from 5OC. to the critical point, but (30) gives results which are probably about 2% too high at SOC. The values of the latent heat above oC. can be deduced from Amagat's tables of p, V, and v, by means of Clapeyron's equation. They are most important below oC. for refrigeration purposes, and have since been directly measured by C. F. Jenkin and D. R. Pye (Phil. Trans., A, 213, p. 67, 1914), who also determined the variation of the total heats, H and h, of the liquid and vapour, by experiments on the specific heat S, and the cooling-effect C, over the range 30 to +30. Their observations of the latent heat are well represented by a formula of the Thiesen type,

log L = 1-1463+0-4018 log (3i-5-/),. . . (31) and those of the total heat of the liquid under saturation pressure by a formula of the same type as that employed in the case of water, namely

h-avTdpldt = H-aVTdpldt = o-42t-6-53,. . . (32) in which the constant 0-42 represents the limiting value of the specific heat of the liquid at low pressures, and the constant 6-53 the value of the term avidp/dt at o C., from which both H and h are supposed to be reckoned. It is quite possible that the specific heat of the liquid at zero pressure may vary in the same way as that of the vapour with temperature, giving a constant value for S,, s a, in place of a constant value for S. This would simplify the equation of saturation pressure, but the observations so far made do not extend over a sufficient range to decide the point. The advantage of these formulae for the total heat is that they fit most simply with Clapeyron's equation, and give a natural approach to the critical point, where both dh/dt and dH/dT become infinite, but with opposite signs.

Equations for the Volume. The equation first proposed for CO? was that of W. J. M. Rankine (Phil. Trans., 1854, p. 337), repre- senting Regnault's experiments on the deviations from the laws of gases at moderate pressures. Rankine's equation may be put in the convenient form,

aP/RT = i/V-c/V*. . . (33).

The symbol a represents the usual factor for reducing PV to cals. C. The value of R in cals./deg. is 0-0451 for CO The coaggregation volume c was found by Rankine to vary as I/T 2, with a value 3-53 c.c./gm. at oC. This equation also represented the observations of Joule and Thomson on the cooling-effect at moderate pressures, but it becomes unsatisfactory at high pressures, and fails near the critical point, giving imaginary values of V when P exceeds RT/4ac. This difficulty is removed most simply by introducing the covolume b in the first term on the right, thus,

aP/RT = i/(V-&)-c/V,. . . (34)

which transforms the equation into a cubic of the same type as that subsequently proposed by J. D. Van der Waals in his essay on the Continuity of the Liquid and Gaseous States (1873), except that c according to Van der Waals' equation would vary inversely as T (in place of T 2 ) which would not suit the properties of CO;. If the values of b and c in (34) are determined from the condition that the cubic in V must have three equal roots at the critical point, we obtain the relations,

= 8^ = 64^/27 . . . (35)