(Received August 9, 1928.) -...

460

Steam Tables and Equations, extended by Direct Experiment to4,000 Ib./sq. in. and 400° C.

By Prof. H. L. Callendajr, F.R.S.

(Received August 9, 1928.)

Summary of the Original Equations.The original Steam Tables and Equations published in 1900* represented

the first thermodynamically consistent system devised for the tabulation of the properties of water and steam. They were based on direct experimental measurements by the following methods, which were specially devised for the purpose :—

(1) The adiabatic relation between P and T for dry steam was obtained by direct observation of its changes of temperature in adiabatic expansion and compression over the range 200° to 800° P. with a very sensitive platinum thermometer, thus eliminating a variety of errors to which the old PV method was liable. The results showed the adiabatic equation to be of the simple form,

P/T13/3 = constant, (l)fand proved that the specific heat of steam at zero pressure, denoted by S0, must be very nearly constant and equal to 0*477, or 13R/3 over this range, assuming that steam approximated to the ideal state represented by the gas equation P (Y — b) = R,T at low pressures.

(2) The variation of the specific heat of water was measured over the range 32° to 212° E. by the continuous electric method devised for this purpose, and was found to be consistent above 100° F. with a thermodynamic equation for the total heat h of water at saturation, namely,

h = s{t — 32) + vL/(Y — v), (2)Jin which s is the minimum specific heat of water at 100° F., L the latent heat of vaporisation, and Y and v the volumes of saturated steam and water at t F.

(3) The specific heat S of steam was measured by a slight modification of the same method at atmospheric pressure and 108° C. and found to be 0*497, which agreed very closely with the value of S0 required by (1), allowing for

* ‘ Roy. Soc. Proc.,’ vol. 67, p. 266 (1900); ‘ Ency. Brit.,’ 1902 ; * Phil. Mag.,’ vol. 5,

p. 48 (1903).f Verified by Makower at 108° C., ‘ Phil. Mag.,’ vol. 5, p. 226 (1903).% ‘ Phil. Trans.,’ A, vol. 200, p. 147 (1902).

on May 21, 2018http://rspa.royalsocietypublishing.org/Downloaded from

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the variation of S with pressure, thus affording independent verification of equation (1).

(4) I t followed from (1), by the application of the two laws of thermodynamics that the expression for the total heat H of the vapour must be of the form

H — B « (13/3) a? (V — + (3)in which B and b are constants, and a is the factor reducing PV to heat units.

I t also followed that the expression for V must be of the general type,Y - b = RT/aP - c, (4)

in which the defect of volume c, due to molecular coaggregation, must be such that cP/T remained constant along any adiabatic, or was some function of P/T13/3. Accordingly, the first term in the expansion of c in powers of P should be of the form c0 — cx (373*1/T)10/3, where cx is the value at 100° C. This was verified by experiments with a differential throttling calorimeter, designed to eliminate errors due to wetness or heat-loss, and the value of cx was found to be 26*30 c.c./gm. (0*4213 cu. ft./lb.). I t appeared that the second term in the expansion of c was probably of the form kc3 (P/T)2, or that c was a function of the temperature only to the second order of small quantities. The first approximation would amply suffice for practical purposes at moderate pressures, or even at high initial pressures with sufficient superheat. I t would be useless to attempt a second approximation on these lines without accurate experimental data at very high pressures. Moreover, equation (4) was obviously inconsistent with the universally accepted theory of the critical state, and could not be extended to represent the phenomena in the critical region, unless the existing theory could be disproved by new experimental evidence of a very striking character.

Extension beyond the Critical Point.The required evidence has now been obtained from three distinct methods

of experiment, showing that equations (1), (2) and (3) remain accurate in the critical region without any modification, and that equation (4), when the higher degrees of coaggregation are taken into account, is capable of representing the critical phenomena with much greater accuracy than any equation of the van der Waals type. The first observations suggesting such a possibility were made in verifying the critical temperature at 374° C. (705*2° F.) as determined by Traube and Teichner by the meniscus method in a capillary tube of quartz glass. With improved apparatus on a larger scale, using very pure water free from air, it was observed that the densities of liquid and vapour did not become

Steam Tables and Equations. 461



462 H . L. C allendar.

equal at 374° C. when the meniscus vanished, but that a visible difference of density persisted beyond this point, and could be traced nearly up to 380° C. under favourable conditions. The expansion of the liquid could be determined with considerable accuracy by using different quantities of water sealed in tubes of quartz glass, and observing the temperature at which the meniscus just reached the top of the tube. The expansion could also be traced beyond the critical point, though with rapidly diminishing accuracy, by making similar observations on the line of demarcation between liquid and vapour due to difference of density. The volume of the saturated vapour was similarly determined, by using smaller quantities of liquid, and observing the temperature at which the last trace of liquid disappeared. This observation could not be extended beyond the critical point at 374° C., but it was inferred by producing the curve that the densities would become equal and the latent heat would vanish a little above 380° C. Above 374° the liquid and vapour were capable of mixing in all proportions, and appeared to be in somewhat unstable state of equilibrium, which was easily upset by small traces of impurity, such as air or gas, which promoted ebullition and caused rapid mixing.

Density of Water and Steam.The observed variation of density for water and steam near the critical point

is illustrated by the experimental curves shown in the annexed figure, 1. A portion of Amagat’s density parabola for C02, with a scale of “ corresponding ” temperatures, is included in the figure to illustrate the procedure for finding the critical density on the orthodox theory. The cardinal feature of this theory, as exemplified in the van der Waals equation, is that the densities become equal and the latent heat vanishes at the critical point, where the meniscus disappears. The observations on C02 appear to be the most accurate available for comparison with steam, but all observers have found great difficulty in obtaining stable values of the volume near the critical point by the methods usually employed, in spite of the relatively low critical pressure and temperature of C02. The usual practice has been to draw the diameter representing the mean of the densities observed at lower temperatures, and to produce it to the critical temperature to estimate the critical density. A similar method has often been applied to steam by calculating a formula of the Thiesen type to represent the latent heat, on the assumption that it vanishes at the critical point. The volume of the vapour is then obtained from Clapeyron’s equation, and the diameter of the corresponding density curve (shown dotted in the figure) is produced to the critical point. The curve thus




obtained did not agree at all well with the direct observations, and required an excessively rapid variation for both liquid and vapour in opposite directions

20-2 24 -9 29-6 3!*SPERATURES FOR CO,CORRESPONDING TEM

co2AMAGATSCRIT.PT.

OIAM . CO,

u n s t a b l eREGION

CRITICAL POINT'S*., FROM L {

SCALE OF "t. CENTIGRADE FOR STEAMF i g . 1.— Density o f W a te r and Steam, gm./c.c.

at the critical point, where their properties should be approaching identity in all respects.

In the absence of any reasonable explanation of this disagreement with




the universally accepted theory, it was impossible to publish the direct observations of the volumes, unless some confirmation could be obtained from measurements of the latent heat, more especially when the very accurate observations of Holborn and Baumann (1910) (on the saturation pressures of steam in a steel cylinder by a static method, with platinum thermometers and a deadweight gauge) verified the critical temperature at 374° C. by showing that there was no saturation pressure beyond this point, but a continuous change of pressure with volume at constant temperature, thus confirming the orthodox theory of the critical state in the case of steam.

Direct Measurements of the Total Heat of Water and Steam.The method finally employed for the direct measurement of the total heat of

water or steam at any temperature or pressure was devised about this time, but was first employed for verifying the variation of the total heat of water between 0° and 100° C., as determined by the continuous electric method.*

In applying the same method to water and steam at high pressures, the only modification required was the introduction of a throttle between the high- pressure pocket and the condenser to reduce the steam to atmospheric pressure before condensation. This does not alter the total heat to be measured, but makes it possible to employ the same condenser in all cases, whatever the initial temperature or pressure of the steam. Unfortunately, the observations were interrupted by the War at an early stage, and remained in abeyance for some years later, until the British Electrical and Allied Industries Research Association became interested in the work and undertook the greater part of the expense involved in the provision of special apparatus and assistance for the completion of the research. The progress of the work, and the evolution of the apparatus employed, has been described and illustrated in various interim reports, published in * World Power,’ May and June, 1924, June, 1925, August and September, 1926, and 4 Howard Lectures, Roy. Soc. Arts,’ November, 1926. I t remains only to summarise the results obtained in the critical region from 2000 to 4000 lb. pressure, which are of primary importance in testing the theory, and in determining the form of the equations required for representing the properties of steam and water over the whole experimental range.

The Importance of using Air-Free Water.Great pains had been taken in the experiments by the continuous electric

method (1902), and in the later experiments (1912) with the jacketed condenser * Bakerian Lecture, * Phil. Trans.,’ A, vol. 212, p. 1 (1912).




at low pressures, to use only air-free water, since tlie presence of even 1 in 10,000 by weight of air interfered seriously with the accuracy of the results. I t was thought that such a small trace of air would be immaterial at high pressures, but the reverse was soon found to be the case. With a steady flow calorimeter capable of reading to 1 in 5000 of the total heat, and subject to a heat-loss of less than 1 per cent., it was possible to keep a continuous check on the purity of the water, and to investigate systematic errors of this kind. Besides greatly increasing the tendency to corrosion at high temperatures, the presence of air appeared to retard the attainment of true equilibrium between the complex molecules at high pressures, exaggerating the errors due to time-lag and nuclear condensation in the neighbourhood of saturation. An automatic apparatus was accordingly installed for supplying the pump with air-free distilled water, and a special form of air-cushion was devised for the experiments on the total heat of water, in which the water was completely protected from contact with the compressed air. In the case of steam this complication was unnecessary, because the steam itself acted as a perfect cushion. But without this device, it was impossible to obtain steady readings of pressure with water, and, if an ordinary air cushion were employed, the air dissolved so rapidly at 3000 lb. tha t no consistent values of the total heat near saturation could be obtained. Observations taken near the critical point with ordinary distilled water, containing from a tenth to a twentieth of its volume of air at atmospheric pressure, afforded a very striking illustration. Values found for the total heat near saturation were much too low in the case of steam owing to nuclear condensation, and much too high in the case of water owing to ebullition promoted by the dissolved air. These errors were somewhat irregular, owing to accidental variations in the quantity of air in the water, but when the apparent values of H and h at saturation were plotted against P, as in Amagat’s familiar PV — P curve for C02, they gave a remarkably good parabola, with its vertex at H = 945 B.Th.U. (525 cals. C.)at a temperature of 705° F. (374° C.), and appeared to give the most complete confirmation of the orthodox theory that could be desired.

On the other hand, when air-free water was employed, the observations both of H and h became incredibly more concordant, and gave the entirely different results shown in the annexed H — P diagram, fig. 2, for steam near the critical point. Instead of the familiar condition H = with L = 0, and V = v,commonly assumed at the point where the meniscus vanished, namely, 374° C. or 705-2° F., the value found for H was 554-2 cals. C. (997-6, B.Th.U./lb.) and that for h only 481-8 cals. C. (867-3 B.Th.U. /lb.), showing




a latent heat L = 72*4 cals. C. or 130*3 B.Th.U./lb. Moreover, the saturation lines could be traced for both liquid and vapour up to 380° C. (716° F.),

CALS. CALS.CENT. CENT.

STEAM

WETSTEAM

WATER

400035003000PRESSURE P IN LB./SQ.INCH.

F i g . 2.—H — P Diagram for Steam.

where the latent heat was still appreciable. They appeared to meet at 717° F., where a very good observation was obtained, and the mixture was found to be entirely in the state of water, with all the steam condensed, and a sudden




flattening of the isothermal. This is exactly what should happen on the coaggregation theory, but is quite irreconcilable with the van der Waals equation. I t was found, in fact, tha t the observations on the total heat of air-free water showed the most minute agreement in every detail with the old observations on the saturation volumes of water and steam in quartz-glass tubes as illustrated in the previous figure. The water sealed in these tubes had been carefully deprived of air in order to prevent boiling of the liquid and disturbance of the meniscus during heating. The qualitative agreement of the two figures is obvious on inspection, but the quantitative agreement with equation (2) for the liquid is easily verified by reference to Clapeyron’s equation, which gives in conjunction with (2) the simple relation,

Y/v — (H — st) /(h — (5)Taking the values of v from the experimental curve for the liquid in fig. 1,

the values of h could easily be calculated, and were found to agree very closely with the observed values along the saturation line for water, thus verifying equation (2) up to the critical point with remarkable precision. Beyond this point the law changes, owing to the vanishing of the surface tension at 374° C., but equation (5) still appears to hold, and gives values of Y, deduced from the observed values of H and h, continuous with the experimental curve for 1/V in fig. 1. The values thus found apply to an intimate mixture of water and steam in varying proportions, starting with dry saturated steam at 374° C. and H == 554 cals, and finishing with water at 380-5° C.

Extension of the Equations.The H — P diagram, as shown in fig. 2, could be completed directly from

the observations of H and h. More than 100 complete observations were available between the limits 370° to 380° C. alone, so tha t it was comparatively easy to draw the isothermals. This applies particularly to the curves for water below the saturation line, for which no equations were available, except (2) for the saturation line itself. But it was most important from a theoretical standpoint to see whether the observed values of H and V for steam could be represented satisfactorily by equations of the type (3) and (4), consistent with the adiabatic equation (1), and to deduce the corresponding equations for the entropy and the saturation pressure, in order to make the verification complete.

Many attempts had already been made in 1908 to represent the observed values of Y near the critical point by using a series for c including the higher degrees of coaggregation. The best and most promising of these was a simple




geometrical series with first term c and common ratio Z2, the sum of the series being represented by the expression c/(l — Z2), in which Z — JccP /T, as required by the adiabatic equation (1). Retaining the original values of c and b, and taking the saturation pressure p from an empirical formula giving p = 3158 lb. at 374° C., it was found that the value 2*39 for the constant in the expression for Z gave a very good approximation to the experimental curve for V. This form of equation gave perfect agreement with the adiabatic, and reduced exactly to the original equation for V at low pressures ; but there were no experimental values of H available for the verification of equation (3) at high pressures, and equation (4) could not be reconciled with the subsequent observations of Holborn and Baumann on the saturation pressures by a static method.

The case was completely altered when the direct observations of H and h showed that the saturation line for steam extended beyond 374° (as illustrated in fig. 2), and supplied accurate measurements of saturation pressures by a steady flow method affording a continuous check on the purity of the fluid. The values of H could be determined with greater precision than those of V, and extended over a much wider range, from 1000 to 4000 lb., thus permitting a far more exact verification of any proposed formula. Retaining the same value of c as in the original equations, the value of b could be calculated as well as that of k in the expression for Z. Thus, taking the value H = 554 • 2 cals, at 374° C., and 3222 lb. as observed on the saturation line, the value of k was found to be 2*229 (in place of 2*39 as previously found at 3158 lb.), the value of b being determined from the experimental value of V at this point. There are many alternative methods of procedure, but the above is the easiest to follow.

Substituting for P (V — b) in (3) from (4), the expression for H becomes,

H - B = S0T - (13/3) acP/(l - Z2) + a&P, (6)

in which the second term on the right represents the latent heat of coaggregation, corresponding to the term c/(l — Z2) representing the coaggregation volume, which replaces the term c in the original expression (4) for V.

The expression for the entropy fl> is deduced from (4) and (6) by integrating the general relation d<£> — dH/T — a (V/T) cCP. The terms representing the entropy of coaggregation, and replacing the term (10/3)acP/T in the original expression, to which they reduce at low pressures, are as follows :

(3/130) loge (1 + Z)/(l - Z) - (13/3) (1 - Z2). (7)



Steam Tables and . 469

In taking the difference <J> — H/T to find the Gibbs’ function G/T, the second of these terms disappears, leaving only the first, which reduces to the form acP/T at low pressures, as in the original equation for G/T.

Equation of Saturation Pressure.This is obtained in the usual way by equating the value of G/T found for

the vapour to that similarly obtained for the liquid from equation (2). Using common logarithms and collecting the terms, we thus obtain the numerical formula for the logarithm of the saturation pressure on the Centigrade scale,

log p = const. — 2903-4/T — 4*7174 log T+ 0*20956 log {(1 + Z)/(l — Z)} + 0*001138 p jT, (8)

in which the constant is found from the standard value = 760 mm. at 100° C. This equation is easily solved by trial and interpolation, taking the experimental values of pas a first approximation in the small terms. But the values found at each point agreed so closely with the experimental values that a second approximation was seldom required. Equation (8) also agreed within 0*3° C. on the average over the whole range from 0° to 374° C. with the observations of Henning and of Holborn and Baumann, though it would not have been surprising if the steady flow method had shown larger systematic differences from their static methods. The closeness of this agreement affords independent confirmation of the whole system of equations based on the adiabatic (1), because all the coefficients in (8) are obtained from direct observations of H, h, and V, and are not deduced from the saturation pressures themselves.

Thus the value of the first coefficient in (8) is deduced from the experimental value of the latent heat at 100° C., and remains unaltered since none of the fundamental constants has been changed. The coefficient of log T in the next term represents ( s — S0)/R, and is also the same as in the original equation for p. The next term represents the effect of the coaggregation on the value of G/T for the vapour, and the last term represents the effect of the small constant b. The value of b could not be determined experimentally in the original equations, because it was so small as compared with V at low pressures. I t was therefore estimated on the lines of the van der Waals theory, in which it plays a very important part. I t has now been determined in accordance with equation (4) from the observed value of V, namely, 3 • 79 c.c./gm. at 374° C., and found to be only — 0*175 c.c./gm., which is much smaller than the value




1 c.c./gm. originally assumed. I t could also be determined from equation (8) by assuming p to be 3222 lb. a t 374° C., but tbis gave the same value. The exact theoretical interpretation of this constant remains doubtful, and is of little importance in the coaggregation theory, but it was most satisfactory to find that the old experimental values of V agreed so well with the new equation of saturation pressure based on observations of H and

I t would not have been very surprising if equation (8) had failed to represent the saturation pressures observed beyond 374° C. I t was found, however, that the agreement with experiment continued up to 380 • 5° C, at which point the saturation line became tangential to the isothermal and the equation automatically ceased to give any further solution for p at higher temperatures. The somewhat abrupt flattening of the isothermal of 380-5° C. beyond this point, indicating that all the single molecules of the type H aO are condensed, and that the mixture has acquired the properties of water, was verified by measurements of h at higher pressures. But since water, according to equation (2), contains its own volume of steam, this is the most natural way in which the conditions, H — h,V = v,and L = 0 can be satisfied. The complete interpretation of these equations raises many other points of interest, which must be reserved for some future occasion. Many other substances, when sufficiently stable and capable of adequate purification, show similar phenomena, but the difficulties of accurate measurement are very great, and* such cases are beyond the scope of this paper. There are many possible modes of coaggregation, and it is doubtful whether an equally simple form of adiabatic would apply to other substances.

The van der Wools Equation.To compare the results with an equation of the van der Waals type, an

equation for P as a cubic function of 1 /V has been constructed in such a way as to be consistent with the adiabatic (1). If P (V — 6)/T is constant along any adiabatic as in (4), it follows, by dividing it by P/T133 that (V — b) T10/3 will also be constant. Thus the adiabatic condition will be satisfied by equating P (V — b) /T to any function of (Y — b) T10/3, as in the following example,

P (V t - 6)/RT = 1 — c/(V — b ) + c2/3 (Y — bf , (9) provided that c varies as 1/T10/3. This makes P a cubic function of 1/(V b),and gives a critical point of the usual type defined by the conditions, V = c s=s RT/3P. If the critical pressure is 3200 lb. at 374° C., the value of c at the critical point differs little from that given by (4), and the values of H and V show a rough agreement with those of the vapour. But no equation of the van der Waals type is capable of representing the properties of the liquid




satisfactorily, as assumed in the van der Waals theory, and (9) gives hopeless results for the saturation pressure if Maxwell’s theorem is assumed. When, however, (9) is combined with equation (2) for the liquid, it gives the saturation line shown dotted in fig. 2. The agreement with observation could be improved by taking a higher value such as 3400 for the critical pressure, and a lower value such as 2 • 5 for the index of T in the expression for c. But this would upset the agreement with the adiabatic, and the equation would in any case be most inconvenient for practical calculations. I t could never show the many coincidences in points of detail which appear to follow with such effortless precision from the coaggregation theory.

The H — P diagram is employed in these comparisons, in preference to the H — <D diagram, which is more familiar to engineers, for the following reasons:—(1) All the temperature and pressure lines rim together near the critical point on the H — <D diagram so tha t no detail can be shown in this region ; (2) the H — P diagram, on the other hand, permits the observed quantities H, P,-and t to be very easily and clearly plotted ; (3) the entropy O cannot be observed directly but is merely calculated from H and V ; (4) in virtue of equation (3), the H — P diagram shows a very close correspondence with the PV — P diagram employed by Amagat and other observers in this region, and facilitates comparison with their observations and with the van der Waals theory.

Practical Applications.The chief interest of these results for steam engineers is that the properties

of steam can be represented by comparatively simple equations which are exactly consistent with the adiabatic equation (1), even at the critical point, which had previously been considered to be impossible according to the universally accepted theory of the critical state. I t follows that the formulae previously given for the adiabatic heat-drop, and for the discharge through a nozzle, which are so important in the theory of the turbine, can be applied with confidence in all cases for dry steam ; and that equation (3), giving a most useful expression for Y in terms of H and P at any point, is equally valid. All these relations are considerably simplified in practice by the discovery that the small constant b is actually more than five times smaller than was previously supposed, so that it may safely be neglected in almost all cases.

Since equation (4) satisfies the condition that P (V — b) /T is constant along any adiabatic, equation (1) may be put in either of the equivalent forms,

(Y _ b) T10'3 = const: or P (Y — 6)1'3 = const, (10)which remain valid and are useful for many purposes.

vol. cxx .—a. 2 k



472 Steam Tables and Equations.

The verification of equations (2) and (5) for water and wet steam up to and beyond the critical point confirms all the previous expressions for wet steam, including those for the entropy of water and the Gibbs’ function G, as well as those for the volume in terms of H and and for the reheat factor.

One of the chief experimental difficulties found in dealing with steam at high pressures near saturation was the comparative sluggishness of the complex molecules in reaching a state of equilibrium. Steam when rapidly heated up to a given temperature and pressure showed lower values of H than when rapidly cooled to same point. These effects were accentuated by the presence of impurities, such as air, and great pains were taken to avoid them. The results now published represent equilibrium states, as nearly as possible, and are applicable to initial states in which the steam is maintained at a steady temperature for a sufficient time. I t appears that, in very rapid expansion from a high superheat, the more complex molecules included in the series c/(l — Z2) would have no time to form. The Enlarged Steam Tables (1924) were designed for the case of rapid expansion, and are probably more accurate for this purpose than the new extension, though they would not apply satisfactorily to initial states.

Extended Tables for the equilibrium states of steam up to 4000 lb. pressure have been calculated on the basis of the new expression for the coaggregation, and have been forwarded to the British Electrical and Allied Industries Research Association, as being the body primarily interested in the research, and mainly responsible for its successful completion.

A special debt of gratitude is due to the members of the Committee on the Properties of Steam for their support and encouragement in the face of difficulties which appeared a t first insurmountable. I have also to thank the Governors of the Imperial College for providing accommodation and workshop facilities, with free use of all the valuable apparatus in the Physics Department. Many essential parts of the apparatus were constructed entirely in the Physics workshop, and owe much to the experience of the superintendent, Mr. W. J. Colebrook, and the skill of the assistant, Mr. E. White, who constructed the electric boilers and most of the accessories. My thanks are due to Dr. H. Moss for assistance in some of the intricate calculations, and to my son, G. S. Callendar, who acted as research assistant throughout the investigation, and acquired an almost uncanny facility in running the apparatus without a hitch, and in reproducing any desired conditions.



(Received August 9, 1928.) -...

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