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The vapour pressure of water
J. Chem. Thermodynamics 1972,4,755-761
The vapour D. AMBROSE
pressure
of water
and I. J. LAWRENSON
Division of ChemicalStandards,National PhysicalLaboratory, Teddington,Middlesex, U.K. (Received3 February 1972) A new equation, in the form of a Chebyshev polynomial, is given for the representation of the vapour pressure of water. In the computation of the coefficients of this equation account was taken of changes consequent upon the adoption of the International Practical Temperature Scale of 1968 and also of new values which have been published for the vapour pressure at Celsius temperatures from 25 to 100 “C. These values, measured at the National Bureau of Standards, have been known for some time but have only recently been published. The equation is applicable over the complete liquid range from the triple point to the critical point and is particularly suitable for use with digital computers. A table of values of the vapour pressure calculated from the equation is given.
1. Introduction Tables of the vapour pressure of water currently in use(l-‘) are formulated with temperatures based on the International Practical Temperature Scale of 1948 (IPTS-48). The adoption of the International Practical Temperature Scale of 1968 (IPTS-68)“’ and the publication (‘1 of precise measurements of the vapour pressure in the Celsius temperature range 25 to 100 “C have made a new correlation desirable. Although details of these measurements have been published only recently, the results have been circulated privately for some years and have been generally known as the “Cragoe-Stimson corrections”. We report here an equation in which account is taken of the new measurements and which provides a good description of the vapour pressure of water according to the new temperature scale from the triple point temperature to the critical temperature. This equation has been used to generate new vapour-pressure tables. Several different formulations have been used for the representation of the water vapour-pressure curve. The most widely used was that derived by Gerry (and reported by Osborne, Stimson and Ginnings) (lo) from earlier measurements in the range 0 to loo OC,(ll) combined with that given by Osborne and Meyers for Celsius temperatures above 100 “C. (r*) These formulae were used for calculation of the International Skeleton Tables 1963, (r) which gave values of the water vapour-pressure (with tolerances) at intervals of 10 K, and were also used in the N.E.L. Steam Tables 1964.“’ In both of these tables the pressure is expressed in bars, and the Celsius temperature in degrees Celsius. Various other formulations have been used, and these give values of the vapour pressure generally in agreement with those of the International Skeleton Table. For example, Keenan, Keyes, Hill, and Moorec4) expressed
756
D. AMBROSE
AND
I. J. LAWRENSON
the Helmholtz free energy as a fifty-nine-constant equation in temperature and density, and derived the thermodynamic properties, including vapour pressure, from this equation (Fahrenheit temperatures in their tables are given in degrees Fahrenheit and pressures in pounds-force per square inch or Celsius temperatures in degrees Celsius and pressures in bars). For the U.K. Steam Tables 1967”) (temperatures in degrees Fahrenheit and pressures in pounds-force per square inch) and the revised version using SI units (3) (Celsius temperatures in degrees Celsius and pressures in bars) vapour pressures were calculated from a nine-constant equation. Tratz(r3) used the equation presented by Osborne and Meyers (12) for the entire temperature range with the addition of an extra term to improve the agreement with the International Skeleton Tables at 25 and 40 “C, and this formulation was used for the VDIWasserdampftafeln. (R The tables published by Goff and Gratchc6’ (Fahrenheit temperatures in degrees Fahrenheit and pressures in pounds-force per square inch) are based on a six-constant equation, while those published by Bridgeman and Aldrichc7) (Celsius temperatures in degrees Celsius and pressures in conventional millimetres of mercury and atmospheres) are based on an eleven-constant equation. Wexler and Greenspan (14) have recently derived an eleven-constant equation for the vapour pressure between 0 and 100 “C from calorimetric and gas-imperfection data, and have demonstrated the remarkable consistency existing between the measured values of these properties for water. All the above equations are of considerable complexity and (except the last) are arbitrary in form.
2. The Chebyshev equation Chebyshev polynomials, which are particularly suitable for use with digital computers, have been used by Gibson and Bruges (15) for representation of the values of the vapour pressure of water given in the N.E.L. Steam Tables (i.e. on IPTS-48) and have also been used for some other substances. (r6) In the treatment adopted here the variation of the vapour pressure is expressed by
+ f w%(x), (1) r=l where T is the temperature, p is the pressure, E,(x) is the Chebyshev polynomial in T log,, P = 42
x of degree r, a,, a,, . . ., a,. are adjustable coefficients, and x is a function of temperature defined by X = I2T-(Tmax+ Tmin>~l(Tmax-LA (2) being two temperatures respectively just above and just below the Tmx and Tmin range of measured values. The constants a, in equation (1) were calculated by computer from the best available experimental data in the range from 25 “C to the critical temperature, viz. the values reported by Stimson(g) (25 to 100 “C), by Beattie and Blaisdell”7’ (96 to 103 “C) and by Osborne, Stimson, Fiock, and Ginnings(‘*) (100 “C to T,), after all the temperatures had been converted to IPTS-68. Each observation was given a weight of one in the least-squares treatment, except for the measurements by Stimson’g’ which were given a weight of ten.
THE
VAPOUR
PRESSURE
OF
757
WATER
Selection of a suitable value for the vapour pressure at or near the ice point was difficult because of the disagreements existing among values obtained from different sources, none of which is clearly superior to the others. Existing tables give a value at 0 “C based on the value measured by Scheel and Heuse 60 years ago and used in the “Warmetabellen”.(“) However, these authors gave the vapour pressures of ice and water as equal at 0 “C instead of at the triple point, and their value is therefore suspect. More recently Prytz c2’) has obtained a value of 0.6115 kPa for the triplepoint pressure and Douslin,(219 22) who used an inclined-piston gauge for measurements over the range 0 to 20 “C, a value of 0.6122 kPa. These observations therefore suggest that the triple-point pressure is in fact higher than the currently accepted value of 0.6 112 kPa. However, all of Douslin’s values appear a little high by comparison with Stimson’s work (figure 3), and the thermodynamic analysis by Wexler and Greenspan(14) and unpublished work by Bottomley and Besley’23’ lend support for the steam-table value. It therefore seemed preferable to retain the triple-point value unchanged until new definitive measurements have been made. An eleventh-order equation was finally chosen as this expresses the vapour pressure between the triple-point temperature and the critical temperature within the estimated TABLE
1. Coefficients
of the equation:
VW/K)
a0 = 2794.0144 al = 1430.6181 aa = -18.2465 Tm&X == 648
TABLE
TmIK 273.15 273.16 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360
PlkPa
GdK
0.6107 0.6111 0.6979 0.9913 1.3884 1.9192 2.6202 3.5353 4.7172 6.2281 8.1409 10.540 13.523 17.202 21.703 27.168 33.757 41.648 51.037 62.139
365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460
log,,
a3 =
(&Pa)
=
7.6875
a4 = -0.0328
a5 =
0.2728
K
2. Vapour
PlkPa 75.192 90.453 108.20 128.73 152.38 179.48 210.40 245.54 285.32 330.15 380.52 436.90 499.80 569.75 647.29 733.01 827.50 931.37 1045.3 1169.9
pressure
ad2
+ , g p,-W)
aa = 0.1371 ag = 0.0200 a7 = 0.0629 alo = 0.0117 a, = 0.0261 all = 0.0067 T m,,, = 273 K
of water
TexIK 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560
(IPTS-68)
PlkPa
TsdK
PlkPa
1305.8 1453.8 1614.7 1789.0 1977.7 2181.4 2401 .O 2637.3 2891.1 3163.3 3454.8 3766.5 4099.2 4454.0 4831.8 5233.7 5660.5 6113.4 6593.6 7102.0
565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645 647.31
7639.9 8208.5 8809.0 9442.8 10111 10815 11557 12338 13160 14025 14935 15892 16898 17959 19076 20256 21504 22106
D. AMBROSE AND I. J. LAWRENSON
758
errors of the observed values (and is the lowest order equation which will do this). The values of the coefficients a, (equation 1) and of T,,,, Tn,in (equation 2) are given in table 1 and values of the vapour pressure calculated from them for a series of temperatures are given in table 2.
3. Comparison of the equation with the observed values and with other correlations In figure 1 and figure 2 are plotted residuals A log p = log p(obs.) -log p(calc.), of the data used to calculate the values of a, as a function of temperature. Values of the vapour pressure calculated from equation (1) for the relevant temperatures differ
0 0
0
0
0
100
200
300
t/T
FIGURE 1. Residuals(A log,,p = log,,p(obs.) - log Iop(calc.)) of the data usedto calculatea, (equation1). V, NEL SteamTables1964;“) A, Stirwon;@)0, Osborneet uZ.“~)
from the values reported by Stimson by less than the standard deviations quoted,(g) and they agree with those reported by Osborne et &(‘a) to within the quoted accuracy of 3 parts in lo4 (figure 1). They also agree with those measured by Beattie and Blaisdell”) within 7 parts in 105, the residuals having both positive and negative values (figure 2). Figure 2 includes points for the values obtained by Moser,(24) by Zmaczynski and Bonhoure (corrected values of which were reported by Swietoslawski and Smith),(25’ and (except for some between 101 and 103 “C which, with residuals up to -0.00013, fall outside the range of the graph) by Michels, Blaisse, Ten Seldam, and Wouters.‘26’ Equations of the form logp = a+ b/T were fitted to the four sets of data represented in figure 2, and the slopes dp/dT at 373.15 K were calculated as 3.616 kPa K- ’ (Beattie and Blaisdell), 3.612 kPa K-l (Moser), 3.613 kPa K-r (Zmaczynski and Bonhoure) and 3.516 kPa K-’ (Michels et al.). The first value, 3.616 kPa K-‘, is in exact agreement with the value obtained from an eleventh-order Chebyshev equation fitted to the Steam-Table value for the ice point, the values reported by Stimson(g’
L
I
I
I
d
I
I
l
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n
$0.2 =4 s 3 4 -2
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.
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.e
-
c ___--_----
__-.
-0.2
-I-
.
. I
.
U n
. q
-0.4
i I
96
I
I
I
100
98
I
I
102
I
104
tpc
FIGURE 2. Residuals (A log,,p = log,,p(obs.) - loglOp(calc.)) of experimental data near 100 “C. 0, Moser;cJ4) 0, Zmaczynski and Bonhoure;@) n , Michels et of.; l , Beattie and Blaisdell.“‘) Dashed lines correspond to deviations in temperature of 0.001 K.
FIGURE 3. Residuals (Ap = p(obs.) - g(calc.)) of published values for Celsius temperatures between 0 and 100 “C. A, Stimson;og) 0, Douslin; W) I), Pryt~;(~~) curve 1, Bridgeman and Aldrich;(7) curve 2, Moser and Zmacynski;@‘) curve 3, Goff and Gratch;@) curve 4, International Skeleton Tables and NEL Steam Tables 1964;“) curve 5, UK Steam Tables in SI Units.‘3’ Dashed lines correspond to deviations in temperature of 0.005 K.
D. AMBROSE
760
AND I. J. LAWRENSON
and by Osborne et a1.,(l*) and 101.325 kPa at 373.15 K. Beattie and Blaisdell’s values were therefore preferred as input data for the equation detailed in table 1. In figure 3 are plotted residuals Ap = (p(obs.)-p(calc.)) in the range 0 to 100 “C for the values reported by Stimson,(‘) by Douslin,(21) and by Prytz.@“) In addition curves have been drawn showing the residuals obtained for the tables published by Bridgeman and Aldrich, (‘I by Moser and Zmaczynski,“‘) by Goff and Gratch,‘@ and for the International Skeleton Tables and the N.E.L. Steam Tables,“) and for the U.K. Steam Tables in SI units. (3) No entries are made for the work of Wexler and Greenspan (14) because the values of vapour pressure given by these authors in the range 0 to 100 “C are essentially the same as those calculated here.
FIGURE 4. Residuals (A log,, p = log,, p(obs.) - log,, p(dc.)) of published values for Celsius temperatures between 100 “C and the critical temperature. Curve 1, Bridgeman and Aldrich;“) curve 2, Moser and Zmacynski;(27)curve 4, International Skeleton Tables and NEL Steam Tables 1964;(l) curve 5, UK Steam Tables in SI Units. w Dashed lines correspond to deviations in temperature of 0.005 K. Residuals due to the rounding adopted for the values shown in table 2 lie within the cross-hatched area.
Figure 4 shows residuals A logp in the range 100 “C to the critical temperature for the tables published by Bridgeman and Aldrich,‘7) by Moser and Zmaczynski,(27) the International Skeleton Tables and the N.E.L. Steam Tables 1964,“) and the U.K. Steam Tables in SI Units.(3) Bruges(‘*) has made an assessment of the critical properties of water from consideration of values available in the literature. (14* 29-38) He concluded that the best values for the critical temperature and critical pressure are (647.31 t-0.05) K [ 6 (374.08 ~0.05)
“C, IPTS-48]
and 22103 kPa; at this temperature
lated from equation (1) is (22106f 13) kPa.
the critical
pressure
calcu-
THE VAPOUR
PRESSURE
OF WATER
761
REFERENCES 1. National Engineering Laboratory. Steam Tables 1964. H.M.S.O. : Edinburgh. 1964. 2. U.K. Steam Tables 1967. Edward Arnold Ltd.: London. 1967. 3. U.K. Steam Tables in SZ Units. Edward Arnold Ltd. : London. 1970. 4. Keenan, J. W.; Keyes, F. G.; Hill, P. G. ; Moore, J. G. Steam Tables, Thermodynamic Properties of Water Including Vapour, Liquid, and Solid Phases-English Units. Wiley: New York. 1969; International Edition-Metric Units. Wiley: New York. 1969. 5. VDI- Wasserdampftafeln, 6th edition. Springer-Verlag: Berlin. 1963. 6. Goff, J. A.; Gratch, S. J. Amer. Sot. Heat. Vent. Eng. 1946, 18, 125. I. Bridgeman, 0. C.; Aldrich, E. W. J. Heat Transfer 1964, 86, 279. 8. The International Practical Temperature Scale of 1968. H.M.S.O.: London. 1969. Metrologia 1969, 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
1966, 34.
35. 36.
37. 38.
5, 35.
Stimson, H. F. J. Res. Natl. Bur. Stand. 1969, 73A, 493. Osborne, N. S.; Stimson, H. F.; Ginnings, D. C. J. Res. Natl. Bur. Stand. 1939, 23,261. Wiirmetabellen. Vieweg: Braunschweig. 1919. Osborne, N. S.; Meyers, C. H. J. Res. Natl. Bur. Stand. 1934, 13, 1. Tratz, H. Brennst- W&me-Kraft 1962, 14, 379. Wexler, A.; Greenspan, L. J. Res. Natl. Bur. Stand. 1971, 75A, 213. Gibson, M. R.; Bruges, E. A. J. Mech. Engng. Sci. 1967, 9, 24. Ambrose, D.; Counsell, J. F.; Davenport, A. J. J. Chem. Thermodynamics 1970,2,283. Beattie, J. A.; Blaisdell, B. E. Proc. Amer. Acad. Arts Sci. 1937, 71, 361. Osborne, N. S.; Stimson, H. F. ; Fiock, E. F. ; Ginnings, D. C. J. Res. Natl. Bur. Stand. 1933, 10, 155. Scheel, K.; Heuse, W. Ann. Phys. 1909,29, 723. Prytz, K. Math.-fys. Meddr. 1931, 11, 1. Douslin, D. R.; Osborne, A. J. Sci. Instrum. 1965,42, 369. Douslin, D. R. J. Chem. Thermodynamics 1971,3, 187. Bottomley, G. A.; Besley, L. Personal communication. Moser, H. Ann. Phys. 1932, 14, 790. Swietoslawski, W.; Smith, E. R. J. Res. Nat/. Bur. Stand. 1938, 20, 549. Michels, A.; Blaisse, B.; Ten Seldam, C. A.; Wouters, H. M. Physica 1943, 10, 613. Moser, R.; Zmaczynski, A. Phys. Z. 1939,40, 221. Bruges, E. A. Private communication. Juza, J. An Equation of State for Water and Steam. Academia, Nakladatelstvi Ceskoslovenske Akademie Ved: Praha. 1966. Rivkin, S. L.; Akhundov, T. S. Teploenergetika 1962, (I), 57; 1963, (9), 66. Rivkin, S. L.; Akhundov, T. S.; Troyanovskaya, G. V. Teplofzika Vysokikh Temperafur 1963, 1, 329. Rivkin, S. L.; Troyanovskaya, G. V. Teploenergetika 1964, (IO), 72. Rivkin, S. L.; Akhundov, T. S.; Kremenevskaya, E. A.; Asdullaeva, N. N. Teploenergetika (4), 59.
Tanishita, I.; Watanabe, K.; Kijima, J.; Vematsu, M. Experimental study of the specific volume of water substance in critical region. Paper presented at the Seventh International Conference on the Properties of Steam, 1968. Blank, G. New determination of the critical point of ordinary and heavy water substance. Paper presented at the Seventh International Conference on the Properties of Steam, 1968. Keyes, F. G.; Keenan, J. H.; Hill, P. G.; Moore, J. G. A fundamental equation for liquid and water vaoour. Paper presented at the Seventh International Conference on the Pronerties of Steam, 1968. Bridgeman, 0. C.; Aldrich, E. W. J. Heat Transfer 1965, 87, 266. Levelt Sengers, J. M. H. Thermodynamic anomalies near the critical point of steam. Paper presented at the Seventh International Conference on the Properties of Steam, 1968.
The vapour D. AMBROSE
pressure
of water
and I. J. LAWRENSON
Division of ChemicalStandards,National PhysicalLaboratory, Teddington,Middlesex, U.K. (Received3 February 1972) A new equation, in the form of a Chebyshev polynomial, is given for the representation of the vapour pressure of water. In the computation of the coefficients of this equation account was taken of changes consequent upon the adoption of the International Practical Temperature Scale of 1968 and also of new values which have been published for the vapour pressure at Celsius temperatures from 25 to 100 “C. These values, measured at the National Bureau of Standards, have been known for some time but have only recently been published. The equation is applicable over the complete liquid range from the triple point to the critical point and is particularly suitable for use with digital computers. A table of values of the vapour pressure calculated from the equation is given.
1. Introduction Tables of the vapour pressure of water currently in use(l-‘) are formulated with temperatures based on the International Practical Temperature Scale of 1948 (IPTS-48). The adoption of the International Practical Temperature Scale of 1968 (IPTS-68)“’ and the publication (‘1 of precise measurements of the vapour pressure in the Celsius temperature range 25 to 100 “C have made a new correlation desirable. Although details of these measurements have been published only recently, the results have been circulated privately for some years and have been generally known as the “Cragoe-Stimson corrections”. We report here an equation in which account is taken of the new measurements and which provides a good description of the vapour pressure of water according to the new temperature scale from the triple point temperature to the critical temperature. This equation has been used to generate new vapour-pressure tables. Several different formulations have been used for the representation of the water vapour-pressure curve. The most widely used was that derived by Gerry (and reported by Osborne, Stimson and Ginnings) (lo) from earlier measurements in the range 0 to loo OC,(ll) combined with that given by Osborne and Meyers for Celsius temperatures above 100 “C. (r*) These formulae were used for calculation of the International Skeleton Tables 1963, (r) which gave values of the water vapour-pressure (with tolerances) at intervals of 10 K, and were also used in the N.E.L. Steam Tables 1964.“’ In both of these tables the pressure is expressed in bars, and the Celsius temperature in degrees Celsius. Various other formulations have been used, and these give values of the vapour pressure generally in agreement with those of the International Skeleton Table. For example, Keenan, Keyes, Hill, and Moorec4) expressed
756
D. AMBROSE
AND
I. J. LAWRENSON
the Helmholtz free energy as a fifty-nine-constant equation in temperature and density, and derived the thermodynamic properties, including vapour pressure, from this equation (Fahrenheit temperatures in their tables are given in degrees Fahrenheit and pressures in pounds-force per square inch or Celsius temperatures in degrees Celsius and pressures in bars). For the U.K. Steam Tables 1967”) (temperatures in degrees Fahrenheit and pressures in pounds-force per square inch) and the revised version using SI units (3) (Celsius temperatures in degrees Celsius and pressures in bars) vapour pressures were calculated from a nine-constant equation. Tratz(r3) used the equation presented by Osborne and Meyers (12) for the entire temperature range with the addition of an extra term to improve the agreement with the International Skeleton Tables at 25 and 40 “C, and this formulation was used for the VDIWasserdampftafeln. (R The tables published by Goff and Gratchc6’ (Fahrenheit temperatures in degrees Fahrenheit and pressures in pounds-force per square inch) are based on a six-constant equation, while those published by Bridgeman and Aldrichc7) (Celsius temperatures in degrees Celsius and pressures in conventional millimetres of mercury and atmospheres) are based on an eleven-constant equation. Wexler and Greenspan (14) have recently derived an eleven-constant equation for the vapour pressure between 0 and 100 “C from calorimetric and gas-imperfection data, and have demonstrated the remarkable consistency existing between the measured values of these properties for water. All the above equations are of considerable complexity and (except the last) are arbitrary in form.
2. The Chebyshev equation Chebyshev polynomials, which are particularly suitable for use with digital computers, have been used by Gibson and Bruges (15) for representation of the values of the vapour pressure of water given in the N.E.L. Steam Tables (i.e. on IPTS-48) and have also been used for some other substances. (r6) In the treatment adopted here the variation of the vapour pressure is expressed by
+ f w%(x), (1) r=l where T is the temperature, p is the pressure, E,(x) is the Chebyshev polynomial in T log,, P = 42
x of degree r, a,, a,, . . ., a,. are adjustable coefficients, and x is a function of temperature defined by X = I2T-(Tmax+ Tmin>~l(Tmax-LA (2) being two temperatures respectively just above and just below the Tmx and Tmin range of measured values. The constants a, in equation (1) were calculated by computer from the best available experimental data in the range from 25 “C to the critical temperature, viz. the values reported by Stimson(g) (25 to 100 “C), by Beattie and Blaisdell”7’ (96 to 103 “C) and by Osborne, Stimson, Fiock, and Ginnings(‘*) (100 “C to T,), after all the temperatures had been converted to IPTS-68. Each observation was given a weight of one in the least-squares treatment, except for the measurements by Stimson’g’ which were given a weight of ten.
THE
VAPOUR
PRESSURE
OF
757
WATER
Selection of a suitable value for the vapour pressure at or near the ice point was difficult because of the disagreements existing among values obtained from different sources, none of which is clearly superior to the others. Existing tables give a value at 0 “C based on the value measured by Scheel and Heuse 60 years ago and used in the “Warmetabellen”.(“) However, these authors gave the vapour pressures of ice and water as equal at 0 “C instead of at the triple point, and their value is therefore suspect. More recently Prytz c2’) has obtained a value of 0.6115 kPa for the triplepoint pressure and Douslin,(219 22) who used an inclined-piston gauge for measurements over the range 0 to 20 “C, a value of 0.6122 kPa. These observations therefore suggest that the triple-point pressure is in fact higher than the currently accepted value of 0.6 112 kPa. However, all of Douslin’s values appear a little high by comparison with Stimson’s work (figure 3), and the thermodynamic analysis by Wexler and Greenspan(14) and unpublished work by Bottomley and Besley’23’ lend support for the steam-table value. It therefore seemed preferable to retain the triple-point value unchanged until new definitive measurements have been made. An eleventh-order equation was finally chosen as this expresses the vapour pressure between the triple-point temperature and the critical temperature within the estimated TABLE
1. Coefficients
of the equation:
VW/K)
a0 = 2794.0144 al = 1430.6181 aa = -18.2465 Tm&X == 648
TABLE
TmIK 273.15 273.16 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360
PlkPa
GdK
0.6107 0.6111 0.6979 0.9913 1.3884 1.9192 2.6202 3.5353 4.7172 6.2281 8.1409 10.540 13.523 17.202 21.703 27.168 33.757 41.648 51.037 62.139
365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460
log,,
a3 =
(&Pa)
=
7.6875
a4 = -0.0328
a5 =
0.2728
K
2. Vapour
PlkPa 75.192 90.453 108.20 128.73 152.38 179.48 210.40 245.54 285.32 330.15 380.52 436.90 499.80 569.75 647.29 733.01 827.50 931.37 1045.3 1169.9
pressure
ad2
+ , g p,-W)
aa = 0.1371 ag = 0.0200 a7 = 0.0629 alo = 0.0117 a, = 0.0261 all = 0.0067 T m,,, = 273 K
of water
TexIK 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560
(IPTS-68)
PlkPa
TsdK
PlkPa
1305.8 1453.8 1614.7 1789.0 1977.7 2181.4 2401 .O 2637.3 2891.1 3163.3 3454.8 3766.5 4099.2 4454.0 4831.8 5233.7 5660.5 6113.4 6593.6 7102.0
565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645 647.31
7639.9 8208.5 8809.0 9442.8 10111 10815 11557 12338 13160 14025 14935 15892 16898 17959 19076 20256 21504 22106
D. AMBROSE AND I. J. LAWRENSON
758
errors of the observed values (and is the lowest order equation which will do this). The values of the coefficients a, (equation 1) and of T,,,, Tn,in (equation 2) are given in table 1 and values of the vapour pressure calculated from them for a series of temperatures are given in table 2.
3. Comparison of the equation with the observed values and with other correlations In figure 1 and figure 2 are plotted residuals A log p = log p(obs.) -log p(calc.), of the data used to calculate the values of a, as a function of temperature. Values of the vapour pressure calculated from equation (1) for the relevant temperatures differ
0 0
0
0
0
100
200
300
t/T
FIGURE 1. Residuals(A log,,p = log,,p(obs.) - log Iop(calc.)) of the data usedto calculatea, (equation1). V, NEL SteamTables1964;“) A, Stirwon;@)0, Osborneet uZ.“~)
from the values reported by Stimson by less than the standard deviations quoted,(g) and they agree with those reported by Osborne et &(‘a) to within the quoted accuracy of 3 parts in lo4 (figure 1). They also agree with those measured by Beattie and Blaisdell”) within 7 parts in 105, the residuals having both positive and negative values (figure 2). Figure 2 includes points for the values obtained by Moser,(24) by Zmaczynski and Bonhoure (corrected values of which were reported by Swietoslawski and Smith),(25’ and (except for some between 101 and 103 “C which, with residuals up to -0.00013, fall outside the range of the graph) by Michels, Blaisse, Ten Seldam, and Wouters.‘26’ Equations of the form logp = a+ b/T were fitted to the four sets of data represented in figure 2, and the slopes dp/dT at 373.15 K were calculated as 3.616 kPa K- ’ (Beattie and Blaisdell), 3.612 kPa K-l (Moser), 3.613 kPa K-r (Zmaczynski and Bonhoure) and 3.516 kPa K-’ (Michels et al.). The first value, 3.616 kPa K-‘, is in exact agreement with the value obtained from an eleventh-order Chebyshev equation fitted to the Steam-Table value for the ice point, the values reported by Stimson(g’
L
I
I
I
d
I
I
l
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n l
n
$0.2 =4 s 3 4 -2
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______
-------.-w-m.
.
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.
.
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.
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.
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.
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.
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c ___--_----
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96
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100
98
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tpc
FIGURE 2. Residuals (A log,,p = log,,p(obs.) - loglOp(calc.)) of experimental data near 100 “C. 0, Moser;cJ4) 0, Zmaczynski and Bonhoure;@) n , Michels et of.; l , Beattie and Blaisdell.“‘) Dashed lines correspond to deviations in temperature of 0.001 K.
FIGURE 3. Residuals (Ap = p(obs.) - g(calc.)) of published values for Celsius temperatures between 0 and 100 “C. A, Stimson;og) 0, Douslin; W) I), Pryt~;(~~) curve 1, Bridgeman and Aldrich;(7) curve 2, Moser and Zmacynski;@‘) curve 3, Goff and Gratch;@) curve 4, International Skeleton Tables and NEL Steam Tables 1964;“) curve 5, UK Steam Tables in SI Units.‘3’ Dashed lines correspond to deviations in temperature of 0.005 K.
D. AMBROSE
760
AND I. J. LAWRENSON
and by Osborne et a1.,(l*) and 101.325 kPa at 373.15 K. Beattie and Blaisdell’s values were therefore preferred as input data for the equation detailed in table 1. In figure 3 are plotted residuals Ap = (p(obs.)-p(calc.)) in the range 0 to 100 “C for the values reported by Stimson,(‘) by Douslin,(21) and by Prytz.@“) In addition curves have been drawn showing the residuals obtained for the tables published by Bridgeman and Aldrich, (‘I by Moser and Zmaczynski,“‘) by Goff and Gratch,‘@ and for the International Skeleton Tables and the N.E.L. Steam Tables,“) and for the U.K. Steam Tables in SI units. (3) No entries are made for the work of Wexler and Greenspan (14) because the values of vapour pressure given by these authors in the range 0 to 100 “C are essentially the same as those calculated here.
FIGURE 4. Residuals (A log,, p = log,, p(obs.) - log,, p(dc.)) of published values for Celsius temperatures between 100 “C and the critical temperature. Curve 1, Bridgeman and Aldrich;“) curve 2, Moser and Zmacynski;(27)curve 4, International Skeleton Tables and NEL Steam Tables 1964;(l) curve 5, UK Steam Tables in SI Units. w Dashed lines correspond to deviations in temperature of 0.005 K. Residuals due to the rounding adopted for the values shown in table 2 lie within the cross-hatched area.
Figure 4 shows residuals A logp in the range 100 “C to the critical temperature for the tables published by Bridgeman and Aldrich,‘7) by Moser and Zmaczynski,(27) the International Skeleton Tables and the N.E.L. Steam Tables 1964,“) and the U.K. Steam Tables in SI Units.(3) Bruges(‘*) has made an assessment of the critical properties of water from consideration of values available in the literature. (14* 29-38) He concluded that the best values for the critical temperature and critical pressure are (647.31 t-0.05) K [ 6 (374.08 ~0.05)
“C, IPTS-48]
and 22103 kPa; at this temperature
lated from equation (1) is (22106f 13) kPa.
the critical
pressure
calcu-
THE VAPOUR
PRESSURE
OF WATER
761
REFERENCES 1. National Engineering Laboratory. Steam Tables 1964. H.M.S.O. : Edinburgh. 1964. 2. U.K. Steam Tables 1967. Edward Arnold Ltd.: London. 1967. 3. U.K. Steam Tables in SZ Units. Edward Arnold Ltd. : London. 1970. 4. Keenan, J. W.; Keyes, F. G.; Hill, P. G. ; Moore, J. G. Steam Tables, Thermodynamic Properties of Water Including Vapour, Liquid, and Solid Phases-English Units. Wiley: New York. 1969; International Edition-Metric Units. Wiley: New York. 1969. 5. VDI- Wasserdampftafeln, 6th edition. Springer-Verlag: Berlin. 1963. 6. Goff, J. A.; Gratch, S. J. Amer. Sot. Heat. Vent. Eng. 1946, 18, 125. I. Bridgeman, 0. C.; Aldrich, E. W. J. Heat Transfer 1964, 86, 279. 8. The International Practical Temperature Scale of 1968. H.M.S.O.: London. 1969. Metrologia 1969, 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
1966, 34.
35. 36.
37. 38.
5, 35.
Stimson, H. F. J. Res. Natl. Bur. Stand. 1969, 73A, 493. Osborne, N. S.; Stimson, H. F.; Ginnings, D. C. J. Res. Natl. Bur. Stand. 1939, 23,261. Wiirmetabellen. Vieweg: Braunschweig. 1919. Osborne, N. S.; Meyers, C. H. J. Res. Natl. Bur. Stand. 1934, 13, 1. Tratz, H. Brennst- W&me-Kraft 1962, 14, 379. Wexler, A.; Greenspan, L. J. Res. Natl. Bur. Stand. 1971, 75A, 213. Gibson, M. R.; Bruges, E. A. J. Mech. Engng. Sci. 1967, 9, 24. Ambrose, D.; Counsell, J. F.; Davenport, A. J. J. Chem. Thermodynamics 1970,2,283. Beattie, J. A.; Blaisdell, B. E. Proc. Amer. Acad. Arts Sci. 1937, 71, 361. Osborne, N. S.; Stimson, H. F. ; Fiock, E. F. ; Ginnings, D. C. J. Res. Natl. Bur. Stand. 1933, 10, 155. Scheel, K.; Heuse, W. Ann. Phys. 1909,29, 723. Prytz, K. Math.-fys. Meddr. 1931, 11, 1. Douslin, D. R.; Osborne, A. J. Sci. Instrum. 1965,42, 369. Douslin, D. R. J. Chem. Thermodynamics 1971,3, 187. Bottomley, G. A.; Besley, L. Personal communication. Moser, H. Ann. Phys. 1932, 14, 790. Swietoslawski, W.; Smith, E. R. J. Res. Nat/. Bur. Stand. 1938, 20, 549. Michels, A.; Blaisse, B.; Ten Seldam, C. A.; Wouters, H. M. Physica 1943, 10, 613. Moser, R.; Zmaczynski, A. Phys. Z. 1939,40, 221. Bruges, E. A. Private communication. Juza, J. An Equation of State for Water and Steam. Academia, Nakladatelstvi Ceskoslovenske Akademie Ved: Praha. 1966. Rivkin, S. L.; Akhundov, T. S. Teploenergetika 1962, (I), 57; 1963, (9), 66. Rivkin, S. L.; Akhundov, T. S.; Troyanovskaya, G. V. Teplofzika Vysokikh Temperafur 1963, 1, 329. Rivkin, S. L.; Troyanovskaya, G. V. Teploenergetika 1964, (IO), 72. Rivkin, S. L.; Akhundov, T. S.; Kremenevskaya, E. A.; Asdullaeva, N. N. Teploenergetika (4), 59.
Tanishita, I.; Watanabe, K.; Kijima, J.; Vematsu, M. Experimental study of the specific volume of water substance in critical region. Paper presented at the Seventh International Conference on the Properties of Steam, 1968. Blank, G. New determination of the critical point of ordinary and heavy water substance. Paper presented at the Seventh International Conference on the Properties of Steam, 1968. Keyes, F. G.; Keenan, J. H.; Hill, P. G.; Moore, J. G. A fundamental equation for liquid and water vaoour. Paper presented at the Seventh International Conference on the Pronerties of Steam, 1968. Bridgeman, 0. C.; Aldrich, E. W. J. Heat Transfer 1965, 87, 266. Levelt Sengers, J. M. H. Thermodynamic anomalies near the critical point of steam. Paper presented at the Seventh International Conference on the Properties of Steam, 1968.