The International Association for the Properties of Water and Steam

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1 IAPWS G11-15 The Internatonal Assocaton for the Propertes of Water and Steam Stockholm, Sweden July 015 Gudelne on a Vral Equaton for the Fugacty of HO n Humd Ar 015 Internatonal Assocaton for the Propertes of Water and Steam Publcaton n whole or n part s allowed n all countres provded that attrbuton s gven to the Internatonal Assocaton for the Propertes of Water and Steam Please cte as: Internatonal Assocaton for the Propertes of Water and Steam, Gudelne on a Vral Equaton for the Fugacty of HO n Humd Ar (015). Ths Gudelne has been authorzed by the Internatonal Assocaton for the Propertes of Water and Steam (IAPWS) at ts meetng n Stockholm, Sweden, 8 June 3 July, 015. The members of IAPWS are: Brtan and Ireland, Canada, the Czech Republc, Germany, Japan, Russa, Scandnava (Denmark, Fnland, Norway, Sweden), and the Unted States, plus Assocate Members Argentna & Brazl, Australa, France, Greece, New Zealand, and Swtzerland. The Presdent at the tme of adopton of ths document s Dr. Davd Guzonas of Canada. Summary The correlaton provded n ths Gudelne s a vral expanson for the fugacty of water vapor n humd ar as a functon of vapor mole fracton, temperature and pressure; detals can be found n the artcle Vral Approxmaton of the TEOS-10 Equaton for the Fugacty of Water n Humd Ar by R. Festel et al. [1]. Ths equaton s consstent wth the IAPWS "Gudelne on an Equaton of State for Humd Ar n Contact wth Seawater and Ice, Consstent wth the IAPWS Formulaton 008 for the Thermodynamc Propertes of Seawater" []. Ths Gudelne contans 11 pages, ncludng ths cover page. Further nformaton about ths Gudelne and other documents ssued by IAPWS can be obtaned from the Executve Secretary of IAPWS (Dr. R.B. Dooley, bdooley@structnt.com) or from

2 Contents 1 Nomenclature Introductory Remarks 3 3 The Fugacty Equaton 4 4 Equatons for Vral Coeffcents 4 5 Range of Valdty and Bref Dscusson 8 6 Estmates of Uncertanty 9 7 Computer-Program Verfcaton 10 8 References 11 1 Nomenclature Symbol Physcal quantty Unt A55, A56 Coeffcents of the vral coeffcents B WW, C WWW, Table a0 a4 Coeffcents of the thrd vral coeffcent C AAW, Table 3 a55, a56 Coeffcents of the vral coeffcents B WW, C WWW, Table B55, B56 Coeffcents of the vral coeffcents B WW, C WWW, Table B AA, B AW, B WW Second vral coeffcents, Eqs. (5), (7), (10) m 3 mol 1 b0 b3 Coeffcents of the thrd vral coeffcent C AWW, Table 3 b55, b56 Coeffcents of the vral coeffcents B WW, C WWW, Table b* Reducng factor, b * 10 m mol m 3 mol 1 C55, C56 Coeffcents of the vral coeffcents B WW, C WWW, Table C AAA, C AAW, C AAW, C WWW Thrd vral coeffcents, Eqs. (6), (8), (9), (11) m 6 mol c1 c3 Coeffcents of the second vral coeffcent B AW, Table 3 c* 6 6 Reducng factor, c * 10 m mol m 6 mol D55, D56 Coeffcents of the vral coeffcents B WW, C WWW, Table d1 d3 Coeffcents of the second vral coeffcent B AW, Table 3 fv Fugacty of water vapor n humd ar, Eqs. (1), () Pa Summaton ndex j1 j18 Coeffcents of the vral coeffcents B AA, C AAA, Table 4 k Uncertanty coverage factor MW Molar mass of water, MW = kg mol 1 kg mol 1

3 3 Symbol Physcal quantty Unt n1 n18 Coeffcents of the vral coeffcents B AA, C AAA, Table 4 n1 n56 Coeffcents of the vral coeffcents B WW, C WWW, Tables 1, p Absolute pressure Pa R Molar gas constant, R = J mol 1 K 1 J mol 1 K 1 SBA, SBW, SCW Sets of summaton ndces, Eqs. (5), (6), (10) T Absolute temperature (ITS-90) K T Reduced temperature, T T /100K Ta Reducng temperature, Ta = K K Tc Crtcal temperature of water, Tc = K K t1 t6 Coeffcents of the vral coeffcents B WW, C WWW, Tables 1, U Expanded uncertanty, coverage factor k = x Mole fracton of water vapor n humd ar mol mol 1 x sat Mole fracton of water vapor n saturated humd ar mol mol 1 β Auxlary functon, Eqs. (), (3) m 3 mol 1 β 55, β 56 Coeffcents of the vral coeffcents B WW, C WWW, Table γ Auxlary functon, Eqs. (), (4) m 6 mol 55, 56 Auxlary functons, A 1, Eqs. (5), (6), (8) µw Chemcal potental of water n humd ar J mol 1 d W Chemcal potental of water n humd ar, deal-gas part J mol 1 a Reducng molar densty of ar, a = mol dm 3 mol dm 3 c Crtcal mass densty of water, c = 3 kg m 3 kg m 3 Reduced recprocal temperature, = Tc/T, Eqs. (5), (6) Reduced recprocal temperature, = Ta/T, Eqs. (10), (11) Introductory Remarks The equaton of state for humd ar descrbed n the IAPWS Gudelne [] s n the form of the specfc Helmholtz energy as a functon of mass densty, temperature, and mass fracton of dry ar. It only ndrectly permts the calculaton of the mole-based chemcal potental of water n d humd ar, W, and ts deal-gas lmt, W, as functons of the mole fracton of water vapor, x, pressure, p, and temperature, T. The fugacty of water vapor n humd ar, f V, expresses the resdual part of that chemcal potental, d fv W W RT ln, (1) xp

4 4 where R s the molar gas constant. In ths Gudelne, consstent wth Eq. (1), a vral expanson s provded as an explct analytcal approxmaton formula for a more convenent evaluaton of f V at a gven x between zero and saturaton, T between 80 C and 00 C, and p up to 5 MPa. The correlaton equatons reported here for the requred second and thrd vral coeffcents for waterwater, water-ar, and ar-ar nteractons are extracted from the IAPWS Gudelne []. The Gudelne s ntended to be used for the descrpton of humd ar n geophyscal, ndustral, or metrologcal applcatons. 3 The Fugacty Equaton The fugacty equaton s gven here as a truncated seres expanson wth respect to the pressure, n terms of the second and thrd vral and cross vral coeffcents of humd ar. It s expressed as a functon of water-vapor mole fracton x, temperature T, and pressure p. The temperature values are based on the temperature scale ITS-90 [3]. The fugacty of water vapor n humd ar, f V, takes the form p 1 p fv x, T, p xp exp x, T x, T. () RT RT The molar gas constant has the value R = J mol 1 K 1 [4,5]. The auxlary functons x,t x,t represent certan combnatons of vral coeffcents [1] and are defned by and and WW AW AA x, T x xb T 1 x B T B T (3) WWW x, T x 3 xc T AWW AAW AAA 1 x 6 xc T 31 xc T 1 xc T WW AW AA x B T x1 xb T 1 x B T WW AW AA x3x 4B T 1 x3 x B T 31 x B T. (4) Equaton () s a mathematcally rgorous expresson n terms of second and thrd vral and cross vral coeffcents of humd ar. The formulaton gven by Eqs. ()-(4) remans vald for DO or other sotopc compostons of water when the vral coeffcents are chosen accordngly. Wth approprate changes n vral coeffcents, t can also be appled for dfferent chemcal and sotopc compostons of dry ar, and for water n any humd gas. 4 Equatons for Vral Coeffcents The equatons gven n ths secton for the vral coeffcents appearng n Eqs. ()-(4) are taken from Ref. [] n order to be consstent wth that IAPWS Gudelne. If mproved functons

5 5 are developed n the future for any of the coeffcents, they may be substtuted wthout changng the remander of the calculaton. TABLE 1 Coeffcents of the vral coeffcents B WW (T) and C WWW (T) of water, Eqs. (5), (6) t n TABLE Coeffcents of the vral coeffcents B WW (T) and C WWW (T) of water, Eqs. (5), (6) a b B n C D A β

6 6 The second vral coeffcent of water, 56 WW M W t B n n c S BW 55 and the thrd vral coeffcent of water, B WW T, b B exp C D 1 C WWW T,, (5) 10 WWW M W t t C n n c SCW A b n C Bb Ba B expc D 1, 55 (6) of IAPWS-95 [6] are gven n [1] as analytcal expressons. Here, MW = kg mol 1 s the molar mass of water [4], the crtcal mass densty s c = 3 kg m 3, the reduced recprocal temperature s = Tc/T, and the crtcal temperature s Tc = K. The sets of ndces over whch the frst sums n Eq. (5) and Eq. (6) run, respectvely, are SBW = {1,,3,8,9,10,3} and SCW = {4,5,11,1,4,5,6}; they relate to the orgnal coeffcents n [6] and n Table 1. As an abbrevaton, A 1 s used. The coeffcents A, a, B, b, C, D, n, t, and β of Eqs. (5) and (6) are gven n Tables 1 and. TABLE 3 Coeffcents of the cross vral coeffcents B AW (T), C AAW (T) and C AWW (T), Eqs. (7)-(9) a b c d The second ar-water cross vral coeffcent, B AW (T), s gven by [7] as 3 AW d B ( T) b* ct 1. (7) The coeffcents of Eq. (7) are gven n Table 3. The reducng factor s reduced temperature s T T /100K. b * 10 6 m 3 mol 1 ; the The thrd ar-water cross vral coeffcents C AAW (T) and C AWW (T) are estmated n [8], n the form

7 7 AAW C T c* 4 at 0 (8). (9) AWW C T c*exp 3 bt 0 The coeffcents of Eqs. (8) and (9) are gven n Table 3. The reducng factor s 6 6 * 10 m mol T T / 100 K. c ; the reduced temperature s The second vral coeffcent of dry ar, B AA 1 a S and the thrd vral coeffcent of dry ar, BA AAA j11 C n4 n11 a B AA T, j n, (10) C AAA T,, (11) are gven n [1]. Here, the reducng molar densty s a = mol dm 3, the reduced recprocal temperature s = Ta/T, and the reducng temperature s Ta = K. In Eq. (10), the set of ndces over whch the sum s executed s SBA = {1,,3,11,15,18} and relates to the coeffcents n the IAPWS Gudelne [] and n Table 4, where the coeffcents j and n of Eqs. (10) and (11) are gven. TABLE 4 Coeffcents of the vral coeffcents B AA (T) and C AAA (T) of dry ar, Eqs. (10), (11) j n

8 8 5 Range of Valdty and Bref Dscusson The fugacty equaton, Eq. (), wth the vral coeffcents gven n Secton 4 s vald for humd ar wthn the temperature and pressure ranges 193 K 473 K and 0 < p 5 MPa, where the temperature range results exclusvely from the equatons for the vral coeffcents gven n Secton 4. All valdty ntervals of the vral coeffcents combned n Eq. () overlap only n ths range. The separate ranges of valdty of the ndvdual vral coeffcents are wder, for some of them sgnfcantly, as gven n Table 5. Therefore, Eq. () wll also provde reasonable results outsde of the temperature range gven above f some vral coeffcents domnate numercally n Eq. () and are evaluated wthn ther partcular ranges of valdty. Ths apples for nstance to low-pressure condtons under whch the thrd vral coeffcents are neglgble. The pressure range gven above s derved from the truncaton error of the vral approxmaton [1] employed n ths Gudelne and may dffer f more accurate vral coeffcents, or vral coeffcents for other compostons of the humd gas, are used to evaluate Eq. (). The water-vapor mole fracton x can take any value between 0 and ts saturaton value,.e., 0 x x sat (T, p). If saturaton s mpossble at the gven (T, p), then the range of valdty s 0 x 1. Equaton () also gves reasonable results when extrapolated to values of x somewhat above the saturaton value, provded that the temperature and pressure are wthn the ranges gven above. The exact value of the water vapor fracton x sat (T, p) of saturated humd ar s determned by the equlbrum condton of equalty of fugactes (or, equvalently, of chemcal potentals) between water n the vapor phase and n the condensed phase (lqud f the temperature s above the freezng pont or ce f the temperature s below the freezng pont), as descrbed n []. TABLE 5 Estmated ranges of valdty of the vral coeffcents Coeff. Valdty Range Ref. B AA K [1] B AW K [6] B WW K [1] C AAA K [1] C AAW K [7] C AWW K [7] C WWW K [1]

9 9 6 Estmates of Uncertanty Here, estmated expanded uncertantes U, coverage factor k =, are reported, correspondng to a 95 % confdence level [9]. Uncertanty estmates avalable for each of the vral coeffcents are gven n Table 6 as functons of the temperature [1]. Uncertanty estmates for the fugacty equaton, Eq. (), are obtaned from uncertanty propagaton of values reported n Table 6 and are dsplayed n Fgure 1. The uncertanty budget of Eq. () s domnated by the uncertantes estmated for B WW at low pressures and for B AW at hgher pressures [1]. TABLE 6 Estmated expanded uncertantes U of the vral coeffcents T B AA B AW B WW C AAA C AAW C AWW C WWW K cm 3 mol 1 cm 6 mol Relatve Uncertanty, % Pressure, kpa Fg. 1 Expanded (k=) uncertantes of the fugacty equaton, Eq. (), for saturated humd ar at 00 K and 50 K wth respect to ce Ih, and at 300 K and 350 K wth respect to lqud water, from propagaton of uncertantes of Table 6, as functons of the total pressure [1]. Curves at 00 K and 50 K are vald under the weak restrcton that the unknown uncertantes U(C WWW ) at those temperatures are less than cm 6 mol and cm 6 mol, U f V / f, n percent. respectvely. Shown are relatve uncertantes, V 00 K 50 K 300 K 350 K

10 10 7 Computer-Program Verfcaton To assst the user n computer-program verfcaton, Tables 7 and 8 wth test values are gven for specfed parameter values of humd ar. They contan values for the seven vral coeffcents, Eqs. (5)-(11), as well as the fugacty fv, Eq. (), and ts auxlary functons β(x, T) and γ(x, T), Eqs. (3)-(4). TABLE 7 Numercal check values for the second vral coeffcents, B AA, B AW, B WW, and for the thrd vral coeffcents, C AAA, C AAW, C AWW, C WWW, Eqs. (5)-(11), of humd ar at the temperatures 00 K, 300 K, and 400 K. Quantty Value Value Value Unt T K B AA m 3 mol 1 B AW m 3 mol 1 B WW m 3 mol 1 C AAA m 6 mol C AAW m 6 mol C AWW m 6 mol C WWW m 6 mol TABLE 8 Numercal check values for the auxlary functons, β, γ, Eqs. (3)-(4), and for the fugacty, fv, Eq. (), at the temperature 300 K, at mole fractons x of 0.1 and 0.9, and at pressures p of 10, 100, and 1000 kpa. Note that the values of x used here do not necessarly correspond to thermodynamcally stable states of humd ar. Quantty p / Pa x = 0.1 x = 0.9 Unt β m 3 mol 1 γ m 6 mol fv Pa fv Pa fv Pa

11 11 8 References [1] Festel, R., Lovell-Smth, J., and Hellmuth, O., Vral Approxmaton of the TEOS-10 Equaton for the Fugacty of Water n Humd Ar, Int. J. Thermophys. 36, 44 (015). [] IAPWS, Gudelne on an Equaton of State for Humd Ar n Contact wth Seawater and Ice, Consstent wth the IAPWS Formulaton 008 for the Thermodynamc Propertes of Seawater (010). Avalable from [3] Preston-Thomas, H., The Internatonal Temperature Scale of 1990 (ITS-90), Metrologa 7, 3 (1990). [4] IAPWS, Gudelne on the Use of Fundamental Physcal Constants and Basc Constants of Water (01). Avalable from [5] Mohr, P.J., Taylor, B.N., and Newell, D.B., CODATA Recommended Values of the Fundamental Physcal Constants: 010, J. Phys. Chem. Ref. Data 41, (01) [6] IAPWS, Revsed Release on the IAPWS Formulaton 1995 for the Thermodynamc Propertes of Ordnary Water Substance for General and Scentfc Use (014). Avalable from [7] Harvey, A.H., and Huang, P.H., Frst-Prncples Calculaton of the Ar Water Second Vral Coeffcent, Int. J. Thermophys. 8, 556 (007). [8] Hyland, R.W., and Wexler, A., Formulatons for the thermodynamc propertes of dry ar from K to K, and of saturated most ar from K to K, at pressures to 5 MPa, ASHRAE Trans. 89, 50 (1983). [9] ISO, Gude to the Expresson of Uncertanty n Measurement (Internatonal Organzaton for Standardzaton, Geneva, 1993). Avalable at