Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use
Fredericia, Denmark
September 1996
Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use
©1996 International Association for the Properties of Water and Steam
Publication in whole or in part is allowed in all countries provided that attribution is given to the International Association for the Properties of Water and Steam
President:
Professor K. Watanabe
Keio University
Department of Mechanical Engineering
Faculty of Science Technology
3-14-1, Hiyoshi, Kohoku-ku
Yokohama 223, Japan
Executive Secretary:
Dr. R. B. Dooley
Electric Power Research Institute
3412 Hillview Avenue
Palo Alto, California 94303, USA
This release replaces the corresponding release of 1984 and contains 18 numbered pages.
This release has been authorized by the International Association for the Properties of Water and Steam (IAPWS) at its Meeting in Fredericia, Denmark, 8-14 September 1996, for issue by its Secretariat. The members of IAPWS are Argentina, Canada, Czech Republic,
Denmark, Germany, France, Italy, Japan, Russia, the United Kingdom, and the United States of America.
The formulation provided in this release is recommended for general and scientific use; further details about the formulation can be found in an article “New International
Formulation for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use” by W. Wagner and A. Pruß [1]. This formulation provides the most accurate representation of the thermodynamic properties of the fluid phases of water substance over a wide range of conditions available at the time this release was prepared.
IAPWS also has a formulation intended for industrial use.
A minor editorial change was made to this release in September 2006. The only edits were the correction of a printing error in the calculated liquid enthalpy at the triple point given on page 4, and an update of Reference [1]. The property calculations are unchanged.
Further information about this release and other releases issued by IAPWS can be obtained from the Executive Secretary of IAPWS. 2
The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use
1 Nomenclature
Thermodynamic quantities: Superscripts o
cp Specific isobaric heat capacity Residual cv Specific isochoric heat capacity Saturated liquid state fSpecific Helmholtz free energy Saturated vapor state
BSecond virial coefficient Ideal-gas property r
′
″hSpecific enthalpy
MMolar mass pPressure Subscripts
VRSpecific gas constant ccritical point
Rm Molar gas constant saturation Specific entropy sttriple point
TAbsolute temperature uSpecific internal energy
wSpeed of sound
Es
GT
NT
P
U
Isentropic throttling coefficient
GReduced density, G = U /Uc
Isothermal throttling coefficient
IDimensionless Helmholtz free energy, I =f/(RT)
Isothermal compressibility
Joule-Thomson coefficient
Mass density
W
Inverse reduced temperature, W = T /T c
Note: T denotes absolute temperature on the International Temperature Scale of 1990. 3
2 Reference Constants
T = 647.096 K (1)
(2)
c
Uc= 322 kg m–3
R = 0.461 518 05 kJ kg–1 K–1 (3)
The numerical values for the critical temperature Tc and critical density Uc are identical to those given in the IAPWS revised release on the critical parameters of ordinary water substance [2]. The value of the specific gas constant R is derived from values of the molar gas constant Rm [3] and the molar mass M [4], which differ slightly from the accepted values of these quantities at the time this release was prepared. The use of the more recent values would yield a specific gas constant which is greater than the value given in Eq. (3) by about
1 part in 60 000. Since the value of R in Eq. (3) has been used in obtaining the coefficients in the residual part I r, Eq. (6), then this value of R must be used in obtaining property values from the formulation, Eq. (4).
Due to the use of the specific gas constant, Eq. (4) corresponds to a mass-based formulation. In order to convert values of specific properties to molar properties, a choice of the suitable value for the molar mass must be made.
3 The Formulation
The formulation is a fundamental equation for the specific Helmholtz free energy f.
This equation is expressed in dimensionless form, I = f/(RT ), and is separated into two parts, an ideal-gas part Io and a residual part I r, so that : f U,T
RT b g
I G ,W Io G ,W ꢀ I r G ,W afafaf
,(4) where G = U/Uc and W = T /T with Uc, T and R given by Eqs. (2), (1) and (3). cc
The ideal-gas part Io of the dimensionless Helmholtz free energy is obtained from an equation for the specific isobaric heat capacity in the ideal-gas state developed by J.R. Cooper
[5] and reads:
8
ꢁJ oW
M P
N Q
L O o
I lnG ꢀ no ꢀ noW ꢀ no lnW ꢀ no ln 1ꢁ e
,(5) i
º
123ii 4 where G = U/Uc and W = T /T with Uc and T according to Eqs. (2) and (1). Table 1 contains ccthe coefficients and parameters of Eq. (5). 4r
The form of the residual part I of the dimensionless Helmholtz free energy is as follows:
751 54 56
2
ꢁD G ꢁH 2 ꢁE W ꢁJ ꢁG ci di ti
i b g b g di ti
I r nG W ꢀ nG W e ꢀꢀ
nG diW ti e
n 'biG\
(6) iii
ººººiiiii 1 i 8 i 52 i 55 ai
' T 2 ꢀ Bi aG ꢁ1 2 fwith
1
T 1ꢁW ꢀ Ai aG ꢁ1 2 f
2E
ꢂꢃi
2
ꢂꢃ2
ꢁD W ꢁ1 i
\ eꢁC
i aG ꢁ1 f
,where G = U/Uc and W = T /T with Uc and T according to Eqs. (2) and (1). The coefficients ccand parameters of Eq. (6) are listed in Table 2.
Since the 5th International Conference on the Properties of Steam in London in 1956 the specific internal energy and the specific entropy of the saturated liquid at the triple point have been set equal to zero. Thus, at the triple-point temperature Tt = 273.16 K uw 0 , sw 0.
(7) tt
In order to meet this condition, the coefficients n1o and n2o in Eq. (5) have been adjusted accordingly. As a consequence, after calculating for Tt the saturated liquid density Uw via the tphase-equilibrium condition (see Table 3), Eq. (4) yields for the specific enthalpy of the saturated liquid at the triple point: hw = 0.611 782 J kg−1.
(8) t
In the liquid-water region, small changes in density along an isotherm cause large changes in pressure. For this reason, due to an accumulation of small errors, a particular computer code may fail to return the zeros in Eq. (7) for the saturated liquid density at the triple-point temperature. In order to avoid this blemish, it is advisable to readjust the constants n1o and n2o in Eq. (5) by imposing the condition uw 0, sw 0 with the desired accuracy. tt
4 Relations of Thermodynamic Properties to the Dimensionless Helmholtz Free Energy
All thermodynamic properties can be derived from Eq.(4) by using the appropriate combinations of the ideal-gas part Io, Eq. (5), and the residual part I r,
Eq. (6), of the dimensionless Helmholtz free energy and their derivatives. Relations between thermodynamic properties and Io and I rand their derivatives are summarized in Table 3. All required derivatives of the ideal-gas part and of the residual part of the Helmholtz free energy are explicitly given in Table 4 and Table 5, respectively. 5
Besides the single-phase region, the formulation also covers the liquid-vapor saturation curve. For given saturation temperature and solving simultaneously the three equations of the phase-equilibrium condition (see Table 3) by iteration, Eq. (6) yields the thermal saturation properties pV , Uwand Uww . Then, all the other properties can be derived from Eq. (4). In this way, the properties calculated on the saturation curve are thermodynamically consistent with the properties of the single-phase region.
Note: IAPWS has issued the Supplementary Release on Saturation Properties of Ordinary Water Substance [6] containing a set of simple equations which yield values for the vapor pressure as well as the density, specific enthalpy and specific entropy of the saturated vapor and liquid. The values calculated from these equations are not identical with the corresponding values derived from Eq. (4), but agree with them within the uncertainties of the simple equations for the saturation properties.
5 Range of Validity
IAPWS has tested the formulation and endorses its validity in the following way:
(1) The formulation is valid in the entire stable fluid region of H2O from the melting-pressure curve [7] to 1273 K at pressures up to 1000 MPa; the lowest temperature on the meltingpressure curve is 251.165 K ( at 209.9 MPa) [7], see Fig. 1.
In this entire region, Eq. (4) represents the experimental data available at the time the release was prepared (except for very few data points) to within their uncertainties.
Although Eq. (4) is also in satisfactory agreement with the experimental data in the critical region, the equation has some unsatisfactory features in the immediate vicinity of the critical point. These features involve second order and higher derivatives of the dimensionless Helmholtz free energy and properties obtained from them. Specifically, the isothermal compressibilityN N Uꢁ1 U/p , and the specific isobaric heat capacity b g
T i
T d
Tcp exhibit unphysical behavior which occurs in a region from Tc to 5 K above Tc for densities 0.5 % from Uc. In addition, within a temperature range from 20 mK below Tc up to T , the isochoric heat capacity cv exhibits a maximum and the speed of sound w cexhibits a minimum not at the saturation temperature T of the corresponding isochore
V
(as it should be) but in the single-phase region up to 2.5 mK aboveT .
V
(2) In the stable fluid region, the formulation can also be extrapolated beyond the limits given under item (1).
Tests show that Eq. (4) behaves reasonably when extrapolated to pressures up to about 6
100 GPa and temperatures up to about 5000 K. This holds at least for the density and enthalpy of undissociated H2O.
In the gas region at pressures below the triple-point pressure, Eq. (4) behaves reasonably when extrapolated to the sublimation-pressure curve [7] for temperatures down to 200 K.
Due to the extremely low densities in this region which go down to about 10−6 kg m−3, attention must be paid to numerical problems.
(3) As far as can be tested with experimental data, the formulation behaves reasonably when extrapolated into the metastable regions. Eq. (4) represents the currently available experimental data of the subcooled liquid (solid-liquid metastable region) and of the superheated liquid (liquid-gas metastable region) to within the experimental uncertainty.
In the case of the subcooled gas (gas-liquid metastable region), no experimental data are available. In this region, for pressures below 10 MPa, Eq. (4) produces reasonable values close to the saturation line. For calculations further away from the saturation line, an alternative equation (the so-called gas equation) is given in reference [1].
For further details see reference [1].
6 Estimates of Uncertainty
Estimates have been made of the uncertainty of the density, speed of sound, and isobaric heat capacity when calculated from the formulation, Eq. (4). These estimates were derived from comparisons with the various sets of experimental data together with the judgement of the Working Group on Thermophysical Properties of Water and Steam of IAPWS.
For the single-phase region, these tolerances are indicated in Figs. 1 to 3, which give the estimated uncertainties in various areas. As used here “tolerance” means the range of possible values as judged by IAPWS, and no statistical significance can be attached to it. With regard to the uncertainty for the speed of sound and the specific isobaric heat capacity, see
Figs. 2 and 3, it should be noted that the uncertainties for these properties increase drastically when approaching the critical point. The statement “no definitive uncertainty estimates possible” for the high-pressure region in Figs. 2 and 3 is based on the lack of experimental data in this region.
For the saturation properties, the estimates of the uncertainties of vapor pressure, saturated liquid density, and saturated vapor density are shown in Fig. 4. 7
7 Computer-Program Verification
To assist the user in computer-program verification, three tables with test values are given. Table 6 contains values of the ideal-gas part Io and the residual part I rof the dimensionless Helmholtz free energy together with the corresponding derivatives. Table 7 lists values for the pressure p, the specific isochoric heat capacity cv, the speed of sound w, and the specific entropy s calculated at selected values of temperature T and densityU . Table 8 gives values for the vapor pressure pV , values for the densityUw, specific enthalpy h′ and specific entropy s′ for the saturated liquid, and values for the densityUww , specific enthalpy h″ and specific entropy s″ for the saturated vapor. All these saturation values have been calculated with Eq. (4) by using the phase-equilibrium condition (see the corresponding comment in Section 4).
8 References
[1] W. Wagner and A. Pruß, J. Phys. Chem. Ref. Data 31, 387 (2002).
[2] IAPWS Release on the Values of Temperature, Pressure, and Density of Ordinary and Heavy Water Substances at Their Respective Critical Points (September 1992), in
Physical Chemistry of Aqueous Systems: Meeting the Needs of Industry, edited by
H. White, Jr. et al., Proceedings of the 12th International Conference on the Properties of Water and Steam, pp. A 101 - A 102, Begell House, New York, 1995.
[3] B. N. Taylor, W. H. Parker and D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).
[4] G. S. Kell, J. Phys. Chem. Ref. Data 6, 1109 (1977).
[5] J. R. Cooper, Int. J. Thermophys. 3, 35 (1982).
[6] IAPWS Supplementary Release on Saturation Properties of Ordinary Water Substance
(September 1992), in W. Wagner and A. Pruß, J. Phys. Chem. Ref. Data 22, 783 (1993), also in Physical Chemistry of Aqueous Systems: Meeting the Needs of Industry, edited by
H. White, Jr. et al., Proceedings of the 12th International Conference on the Properties of Water and Steam, pp. A 143 - A 149, Begell House, New York, 1995.
[7] IAPWS Release on the Pressure along the Melting and Sublimation Curves of Ordinary
Water Substance (September 1993), in W. Wagner, A. Saul, and A. Pruß, J. Phys. Chem.
Ref. Data 23, 515 (1994), also in Physical Chemistry of Aqueous Systems: Meeting the Needs of Industry, edited by H. White, Jr. et al., Proceedings of the 12th International
Conference on the Properties of Water and Steam, pp. A 9 - A 12, Begell House, New
York, 1995. 8
Table 1. Numerical values of the coefficients and parameters of the ideal-gas part of the dimensionless Helmholtz free energy, Eq.(5) inio i
Jio nio
Jio
1–8.320 446 482 01 –50.973 15 3.537 342 22
26.683 210 526 8 –61.279 50 7.740 737 08
33.006 32 –70.969 56 9.244 377 96
0.012 436 481.287 289 67
0.248 73 27.507 510 5 9
Table 2. Numerical values of the coefficients and parameters of the residual part of the dimensionless Helmholtz free energy, Eq.(6) i
ci di ti ni
–1
1
–11–0.5
0.125 335 479 355 23 × 10
0.789 576 347 228 28 × 10
–0.878 032 033 035 61 × 10
0.318 025 093 454 18
2–10.875
1
3–11
4–20.5
5–0.75 2
–0.261 455 338 593 58
–0.781 997 516 879 81 × 10
0.880 894 931 021 34 × 10
–0.668 565 723 079 65
61910.204 338 109 509 65
–2
–2
6–0.375 3
7–14
8114
–4
10 1112
–0.662 126 050 396 87 × 10
–0.192 327 211 560 02
5212 1–0.257 090 430 034 38
211 11
13 134
0.160 748 684 862 51
–1
–6
–5
–3
–4
–8
–6
–11
–9
14 142
–0.400 928 289 258 07 × 10
0.393 434 226 032 54 × 10
–0.759 413 770 881 44 × 10
0.562 509 793 518 88 × 10
–0.156 086 522 571 35 × 10
0.115 379 964 229 51 × 10
0.365 821 651 442 04 × 10
–0.132 511 800 746 68 × 10
–0.626 395 869 124 54 × 10
15 1413
516 19
17 173
18 194
19 110 11
20 111 4
21 113 13
22 115 1
23 217
–0.107 936 009 089 32
0.176 114 910 087 52 × 10
–1
24 221
25 229
0.221 322 951 675 46
–0.402 476 697 635 28
0.580 833 999 857 59
0.499 691 469 908 06 × 10
–0.313 587 007 125 49 × 10
26 2210
327 210
–2
–1
428 23
29 247
30 2410
–0.743 159 297 103 41
531 210
0.478 073 299 154 80
0.205 279 408 959 48 × 10
10 633 2–0.136 364 351 103 43
–1
632 26
–1
–2
–1
–1
–1
–2
–2
–3
–4
–1
–1
–1
–1
–4
–9
734 210
0.141 806 344 006 17 × 10
0.833 265 048 807 13 × 10
–0.290 523 360 095 85 × 10
0.386 150 855 742 06 × 10
–0.203 934 865 137 04 × 10
–0.165 540 500 637 34 × 10
0.199 555 719 795 41 × 10
0.158 703 083 241 57 × 10
–0.163 885 683 425 30 × 10
0.436 136 157 238 11 × 10
0.349 940 054 637 65 × 10
–0.767 881 978 446 21 × 10
0.224 462 773 320 06 × 10
–0.626 897 104 146 85 × 10
–0.557 111 185 656 45 × 10
644 49 6–0.199 057 183 544 08
935 21
36 292
37 293
38 294
39 298
40 210 6
41 210 9
42 212 8
43 3316
444 322
445 323
546 323
14 47 410
348 650
650 646 0.317 774 973 307 38
651 650 –0.118 411 824 259 81
Di Ei Ji Hi
i
ci di ti ni
2
2
52 20 150 1.21 1–30–0.313 062 603 234 35 × 10
53 20 150 1.21 1–310.315 461 402 377 81 × 10
54 20 250 1.25 1–34–0.252 131 543 416 95 × 10
4
Ei i
ai bi Bi ni
55 3.5 0.85 0.2 28 700 0.32 0.3
56 3.5 0.95 0.2 32 800 0.32 0.3 Ci Di Ai
–0.148 746 408 567 24
0.318 061 108 784 44
10
Table 3. Relations of thermodynamic properties to the ideal-gas part Ioand the residual part I r of the dimensionless Helmholtz free energy and their derivativesa
Pressure
Property Relation pa
G ,W fr
1ꢀGI
G
2
U RT p U f/U b gT
Internal energy ua
G ,W
RT for
W I ꢀ I d W i
Wu fꢁ T f/T b gU
Entropy sa
G ,W
Rfforor
W I ꢀ I ꢁI ꢁI e W j
Ws ꢁ f/T b gU ha
G ,W orr
G
Enthalpy
1ꢀW I ꢀ I ꢀGI d W i
W
RT h f ꢁ T f/T ꢀ U f/U b gU b gT c
vaG ,W f
2or
Isochoric heat capacity c u/T b gU v
Isobaric heat capacity c h/T b gp p
ꢁW Iꢀ I e WW WW j
R
2rr
1ꢀGI ꢁGWI c
paG ,W fꢀ e GW j
G
2or
ꢁW Iꢀ I e WW WW j r2 r
R
1ꢀ 2GI ꢀG I
GGG
2rr
2
1ꢀGI ꢁGWI
Speed of sound
ꢁ e GW j wa
G ,W f
Gr2 r
1ꢀ 2GI ꢀG I
1/2
GGG
2orw p/U b gs
RT
WI
ꢀ I d WW WW i r2rr
ꢁ GI ꢀG I ꢀGWI eGW j
Joule-Thomson coefficient
GGG
PRU
2
P T/ p b gh 22rrorrr
1ꢀGI ꢁGWI ꢁW Iꢀ I 1ꢀ 2GI ꢀG I eGW j e WW WW je GG j
GGrr
Isothermal throttling coefficient
1ꢀGI ꢁGWI
GGW
G U 1ꢁ
Tr2 r
1ꢀ 2GI ꢀG I
GGG
G h/p b gT
Trr
GGW
1ꢀGI ꢁGWI
Isentropic temperaturepressure coefficient
E U R s
222rrorrr
I
1ꢀGI ꢁGWI ꢁW ꢀ I 1ꢀ 2GI ꢀG I eGW j e WW WW je GG j
GG
E T/p b gs s
Second virial coefficient r
ꢂ ꢃ
B W U lim IG aG ,W fc
ꢂ ꢃ
B T lim p/ U RT /U b g chdiT
G 0
U0
Third virial coefficient
2r
ꢂ ꢃ
C W U lim IGG aG ,W fc
1
C T lim L O
22
G 0
ꢂ ꢃ
p/ U RT /U b g chej
M2 T P
U0N Qpp
rr
VV
Phase-equilibrium condition
(Maxwell criterion)
1ꢀG wIG aG w,W ;
1ꢀ G wwIG aG ww,W ff
RTUw RTUww
F 1 I F UwI p
1rr
V
ꢁꢁ ln
I G w,W ꢁI G ww,W afa f
RT GUww J GUww J
Uw
H K H K N QW N QW N Q r2 r r2 r 2 r 2
oo
L O L O
L O L O
L O L O
L O
I I I I I I
I arrrrroo
M P M P, I
M P M P
I, I , I , I , I , I
.
M P M P
M P
WWW
WWW
GGG GW
222
G W G W
W
M P M P M P M P
M P M P
G W
M P
W
N QG
N QG
N QG
N QG 11
Table 4. The ideal-gas part Io of the dimensionless Helmholtz free energy and its derivativesa
8
ꢁJ oW
H K on3o lnW
F I
IGo
IGoG
IWo
IWoW n1o
Io n2oW
in ln 1ꢁ e
=
=+++
+
lnG
ºii 4
1/G
0+00+++0
−1/G 2
00=++++00
8
ꢁ1
ꢁJ oW
H K
Lꢁ1O noJ o 1ꢁ e
F I in2o n3o /W
ºi 4
8
0=++++0ii
MP
NQ
ꢁ2 oꢁJ oW
2no J o eꢁJ W 1ꢁ e
F I iin3o /W 2
i d i i
º
0=++––00
H K i 4
IGoW
=0+0+0+0+0oo
N QG N Q
2 o 2 o 2 o
L O L O L O L O L O
N QG
I I I I I
aooooo
I, I , I , I , I
.
M P M P M P
M P M P
WWW
GGG GW
22
G W G W
G W
M P M P M P M P M P
N QW
N QW Table 5. The residual part I r of the dimensionless Helmholtz free energy and its derivativesa
751 54 56
2
ꢁD G ꢁH 2 ꢁE W ꢁJ
ꢁG ci di ti ai
i b g b g di ti
I r nG W ꢀ nG W e ꢀꢀ
nG diW ti e
n 'biG\ with
' = T 2 ꢀ Bi aG ꢁ1 2 fiii
ººººiiiii 1 i 8 i 52 i 55
1
T 1ꢁW ꢀ Ai aG ꢁ1 2 f
2Ei
ꢂꢃ
2
ꢂꢃ2
i aG ꢁ1 f
\ eꢁC ꢁDi W ꢁ1
751 54 56
2
ꢁD G ꢁH 2 ꢁE W ꢁJ ci
L
G\ O di
\ 'bi
F I ꢀ
b g b g LMG
P
Oꢀ
IGr Wꢀꢁ 2D G ꢁH
n dG d ꢁ1W t ꢀ n eꢁG G d ꢁ1 t
d ꢁ cG c
nG diW ti e
n 'bi \ ꢀG iiiiiiiii
ib g
G J d i i
ººº
º M Pi i iiiii
H K
G G
N
Qi 1 i 55 i 8 i 52
N
Q
751 54
ꢁD G ꢁH 2 ꢁE W ꢁJ
2ci
i b g b g ci
IGrG Wꢀn d d ꢁ1 G d ꢁ2W t ꢀ n eꢁG G d ꢁ2 t
d ꢁ cG c d ꢁ1ꢁ cG c ꢁ c2G
nW ti e iiiiiiiii
i ib i g d id i
ººej º iiiiiiii 1 i 8 i 52
\ 'bi \
2'bi
2
56
L
N
G\ O
P
F \ I ꢀ 2
2
F\ ꢀG I ꢀ
¬ ꢁ2Di G d ꢀ 4Di2G G ꢁH ꢁ 4diDiG d ꢁ1 G ꢁH ꢀ d d ꢁ1 G d ꢁ2 ꢀꢀG
n 'bi 2
di iiib g b g b i g
G J
i M
ºiii
GJ
G 2 G 2
H K
G G G
HK
M
P
Qi 55 bi
751 54 56
2
ꢁD G ꢁH 2 ꢁE W ꢁJ c
L
\ O
it
MW
' b g b g di di ti ꢁ1
L i
Oꢀ
IWr ꢀꢁ 2E W ꢁJ nG
n tG W t ꢁ1 ꢀ n tG W eꢁG
nG diW ti e
\ ꢀ 'bi iiiii
ib g
ººº
º M W P i i i i iiiP
Q
W
Ni 1 i 8 i 52 i 55
NQ
2
751 54
ꢁD G ꢁH 2 ꢁE W ꢁJ
2ci
L t
F i
ꢁ 2E O ti
W 2
i b g b g di
I
IWrW ꢀꢁ 2E W ꢁJ ꢁn t t ꢁ1 G W t ꢁ2 ꢀ n t t ꢁ1 G diW ti ꢁ2eꢁG
nG diW ti e
iiii
i ibi g
i ibi g
' 'bi \
W W
ib g
ºººM
i P
Qii
N
HW Ki 1 i 52 i 8 bi
22
56
N
LO
Q
\ P
º M i
ꢀnG \ ꢀ 2 ꢀ 'bi
W 2 W 2 i 55
751 54
ꢁD G ꢁH 2 ꢁE W ꢁJ
2cidtb g b g
LOL i O
IGrW WꢀWꢀꢁ 2D G ꢁH ꢁ 2E W ꢁJ
n d tG d ꢁ1 t ꢁ1
n tG d ꢁ1 t ꢁ1
d ꢁ cG c eꢁG
nG diW ti e
iiiiiiiiii
ib g b g d i i
ºººi i i i i iiiii
MG PMW P
NQN Qi 1 i 8 i 52
\ 'bi \ 'bi \
F \ I ꢀG
2'bi
2
56
N
LG\ O
P
F\ ꢀG I ꢀ
G J
H K
ꢀꢀn 'bi
ꢀG
i M
º
GG W J
W GW G W W G
HK
M
P
Qi 55 Table 5. Continued bi
Derivatives of the distance function Δ :
Derivatives of the exponential function \ :
'bi '
G G
b 'bi ꢁ1
\
G
2\ i
ꢁ2Ci G ꢁ 1 \ af
2
2'bi 2' '
RU
||
F I
ꢀ b ꢁ1 'bi ꢁ2
b 'bi ꢁ1 b i g
2
G J
i S V
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Table 6. Values for the ideal-gas part Io, Eq. (5), and for the residual part I r, Eq. (6), of the dimensionless Helmholtz free energy together with the corresponding derivativesa for T = 500 K and U = 838.025 kg m–3
Io I r
= 0.204 797 734 × 101
= – 0.342 693 206 × 101
IGo
IGr
= 0.384 236 747 = – 0.364 366 650
IGoG
IWo
IWoW
IGoW
IGrG
IWr
IWrW
IGrW
= – 0.147 637 878 = 0.856 063 701
= 0.904 611 106 × 101
= – 0.193 249 185 × 101
= – 0.581 403 435 × 101
= – 0.223 440 737 × 101
= 0 = – 0.112 176 915 × 101
ora
For the abbreviated notation of the derivatives of I and I see the footnotes of Tables 4 and 5, respectively.
Table 7. Thermodynamic property values in the single-phase region for selected values of T andU s/(kJ kg–1 K–1) p / MPa
U /(kg m–3)
300 0.996 556 0 × 103 0.992 42 × 10–1 cv /(kJ kg–1 K–1)
0.413 018 111 × 101 0.150 151 914 × 104 0.393 062 642
0.406 798 347 × 101 0.153 492 501 × 104 0.387 405 401
T/K w/(m s–1) a
0.100 530 8 × 104 0.200 022 514 × 102
0.118 820 2 × 104 0.700 004 704 × 103
0.346 135 580 × 101 0.244 357 992 × 104 0.132 609 616
0.453 200 0 × 101 0.999 938 125
0.838 025 0 × 103 0.100 003 858 × 102
0.108 456 4 × 104 0.700 000 405 × 103
647 0.358 000 0 × 103 0.220 384 756 × 102
500 0.435 000 0 0.999 679 423 × 10–1 0.150 817 541 × 101 0.548 314 253 × 103 0.794 488 271 × 101
0.166 991 025 × 101 0.535 739 001 × 103 0.682 502 725 × 101
0.322 106 219 × 101 0.127 128 441 × 104 0.256 690 918 × 101
0.307 437 693 × 101 0.241 200 877 × 104 0.203 237 509 × 101
0.618 315 728 × 101 0.252 145 078 × 103 0.432 092 307 × 101
0.100 062 559 900 0.241 000 0 0.175 890 657 × 101 0.724 027 147 × 103 0.916 653 194 × 101
0.193 510 526 × 101 0.698 445 674 × 103 0.659 070 225 × 101
0.266 422 350 × 101 0.201 933 608 × 104 0.417 223 802 × 101
0.526 150 0 × 102 0.200 000 690 × 102
0.870 769 0 × 103 0.700 000 006 × 103 a
In the liquid-water region at low pressures small changes in density along an isotherm cause large changes in pressure. For this reason, due to an accumulation of small errors, a particular computer code or a particular PC may fail to reproduce the pressure value with nine decimal figures. Thus, here only five decimal figures are given.
Table 8. Thermodynamic property values in the two-phase region for selected values of temperaturea
T = 275 K
T = 450 K
T = 625 K pV
0.698 451 167 × 10–3 0.932 203 564
0.169 082 693 × 102
0.567 090 385 × 103
0.118 290 280 × 103
0.168 626 976 × 104
0.255 071 625 × 104
0.380 194 683 × 101
0.518 506 121 × 101
Uw /(kg m–3)
Uww /(kg m–3) h′/(kJ kg–1)
0.999 887 406 × 103
0.890 341 250 × 103
0.550 664 919 × 10–2 0.481 200 360 × 101
0.775 972 200 × 101
0.250 428 995 × 104
0.749 161 585 × 103
0.277 441 078 × 104 h″/(kJ kg –1) s′/(kJ kg–1 K–1) s″/(kJ kg–1 K–1)
0.283 094 669 × 10–1 0.210 865 845 × 101
0.910 660 120 × 101
0.660 921 221 × 101 a
All these test values were calculated from the Helmholtz free energy, Eq. (4), by applying the phaseequilibrium condition (Maxwell criterion).
Fig. 1. Uncertainties in density, Δρ/ρ, estimated for Eq. (4). In the enlarged critical region (triangle), the uncertainty is given as percentage uncertainty in pressure, Δp/p. This region is bordered by the two isochores 527 kg m−3 and 144 kg m−3 and by the 30 MPa isobar. The positions of the lines separating the uncertainty regions are approximate.
Fig. 2. Uncertainties in speed of sound, Δw/w, estimated for Eq. (4). For the uncertainty in the triangle around the critical point, see the remark in Section 6; for the definition of this region, see Fig. 1. The positions of the lines separating the uncertainty regions are approximate.
Fig. 3. Uncertainties in specific isobaric heat capacity, Δcp/cp, estimated for Eq. (4). For the uncertainty in the immediate vicinity of the critical point, see the remark in Section 6; for the definition of the triangle around the critical point, see Fig. 1. The positions of the lines separating the uncertainty regions are approximate.
Fig. 4. Uncertainties in vapor pressure, Δpσ /pσ , in saturated liquid density, Δρ′/ρ′, and in saturated vapor density, Δρ″/ρ″, estimated for Eq. (4).