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Virial Approximation of the TEOS-10 Equation for the Fugacity of Water in Humid Air

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An Erratum to this article was published on 06 January 2015

Abstract

Fugacity is considered the proper real-gas substitute for the partial pressure commonly used to describe ideal-gas mixtures. However, in several fields such as geophysics, meteorology, or air conditioning, partial pressure is still preferred over fugacity when non-equilibrium conditions of humid air are quantified. One reason may be that for ambient air, the deviations from ideal-gas behavior are small, another that explicit correlation equations for the fugacity of water vapor in humid air are scarce in the literature. This situation has improved with the publication of the new oceanographic standard TEOS-10, the International Thermodynamic Equation of Seawater 2010, which provides highly accurate values for the chemical potential and the fugacity of water vapor in humid air over wide ranges of pressure and temperature. This paper describes the way fugacity is obtained from the fundamental equations of TEOS-10, and it derives computationally more convenient virial approximations for the fugacity, consistent with TEOS-10. Analytically extracted from the TEOS-10 equation of state of humid air, equations for the 2nd and 3rd virial coefficients are reported and compared with correlations available from the literature. The virial fugacity equation is valid in the temperature range between \({-}80\,^{\circ }\hbox {C}\) and +200 \(^{\circ }\hbox {C}\) at pressures up to 5 MPa, and between \({-}130\,^{\circ }\hbox {C}\) and +1000 \(^{\circ }\hbox {C}\) at low pressures such as those encountered in the terrestrial atmosphere at higher altitudes.

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Notes

  1. SCOR: Scientific Committee on Oceanic Research, http://www.scor-int.org.

  2. IAPSO: International Association for the Physical Sciences of the Oceans, http://iapso.iugg.org.

  3. IAPWS: International Association for the Properties of Water and Steam, http://www.iapws.org.

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Acknowledgments

The authors are grateful to Donald Gatley, Allan Harvey, Sebastian Herrmann, Jan Hruby, and Hans-Joachim Kretzschmar for various hints regarding virial equations for humid air. They also thank the two anonymous reviewers for their critical comments and helpful suggestions. This work contributes to the tasks of the IAPWS/SCOR/IAPSO Joint Committee on Seawater (JCS).

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Correspondence to Rainer Feistel.

Appendix: List of Symbols

Appendix: List of Symbols

Omitted from this list are the empirical coefficients and the reduced quantities of Sect. 4.

1.1 Subscripts and Superscripts

A:

Dry air

AV:

Humid air

V:

Water vapor

W:

Liquid water or phase-independent water property

id:

Ideal-gas property

sat:

Property at saturation

Symbol

Remark

Unit

\(A\)

Mass fraction of dry air in humid air

\(\hbox {kg}\!\cdot \!\hbox {kg}^{-1}\)

\(a\)

Auxiliary coefficient, Eqs. 2021

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(B\)

2nd virial coefficient, with or without superscripts

\(\hbox {m}^{3}\!\cdot \!\hbox {mol}^{-1}\)

\(b\)

Auxiliary coefficient, Eqs. 2021

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\!\cdot \!\hbox {Pa}^{-1}\)

\(C\)

3rd virial coefficient, with or without superscripts

\(\hbox {m}^{6}\!\cdot \! \hbox {mol}^{-2}\)

\(c\)

Auxiliary coefficient, Eqs. 2021

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\!\cdot \!\hbox {Pa}^{-2}\)

\(e\)

Saturation pressure of pure fluid water

Pa

\(f\)

Enhancement factor of humid air, \(f=x^\mathrm{sat}p/e\)

1

\(f^\mathrm{A}, f^\mathrm{V}, f^\mathrm{AV}\)

Specific Helmholtz energies of dry air, water vapor, and humid air

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(f^\mathrm{mix}\)

Air–water interaction contribution to the specific Helmholtz energy of humid air

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(f_0^\mathrm{A} , \quad f_0^\mathrm{V} \)

Functions of temperature, unspecified

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(f_\mathrm{V}\)

Fugacity of water vapor in humid air

Pa

\(f_\mathrm{W}\)

Fugacity of water in air–saturated liquid water

Pa

\(g_{0}\)

Auxiliary coefficient, Eqs. 2021

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(g^\mathrm{AV}, g^\mathrm{V}\)

Specific Gibbs energies of humid air, water vapor

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(M, M_\mathrm{A}, M_\mathrm{W}\)

Molar masses of humid air, dry air, and water

\(\hbox {kg}\!\cdot \!\hbox {mol}^{-1}\)

\(n, n^\mathrm{A}, n^\mathrm{V}\)

Molar densities of humid air, \(n=n^\mathrm{V}+n^\mathrm{A}\), dry air, \(n^\mathrm{{A}}=\rho ^\mathrm{{A}}/M_\mathrm{A} ,\) and water vapor, \(n^\mathrm{{V}}=\rho ^\mathrm{{V}}/M_\mathrm{W} \)

\(\hbox {m}^{3}\!\cdot \!\hbox {mol}^{-1}\)

\(p\)

Total pressure

Pa

\(p_\mathrm{V}\)

Partial pressure of water vapor, \(p_\mathrm{V} = xp \)

Pa

\(R\)

Molar gas constant, \(R = 8.314\,4621\, \hbox {J}\!\cdot \!\hbox {kg}^{-1}\!\cdot \!\hbox {mol}^{-1}\)

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\!\cdot \!\hbox {mol}^{-1}\)

\(R_\mathrm{W}, R_\mathrm{A}\)

Specific gas constants of water, \(R_\mathrm{W}=R/M_\mathrm{W}\), and of dry air, \(R_\mathrm{A} =R/M_\mathrm{A} \)

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\!\cdot \!\hbox {K}^{-1}\)

\(S_\mathrm{BW}, S_\mathrm{CW}, S_\mathrm{BA}\)

Sets of summation indices, Eqs. 3032

 

\(T\)

Absolute temperature

K

\(U\)

Expanded uncertainty, coverage factor \(k = 2\)

 

\(VC^{i}\)

Running variable for virial coefficients, Eq. 41

 

\(v_\mathrm{W}\)

Specific volume of liquid water

\(\hbox {m}^{3}\!\cdot \!\hbox {kg}^{-1}\)

\(x\)

Mole fraction of water vapor in humid air

\(\hbox {mol}\!\cdot \!\hbox {mol}^{-1}\)

\(x_\mathrm{W}\)

Mole fraction of water in air–saturated liquid water

\(\hbox {mol}\!\cdot \!\hbox {mol}^{-1}\)

\(\beta \left( {x,T} \right) \)

Auxiliary function, Eqs. 2426

\(\hbox {m}^{3}\!\cdot \!\hbox {mol}^{-1}\)

\(\gamma \left( {x,T} \right) \)

Auxiliary function, Eqs. 2526

\(\hbox {m}^{6}\!\cdot \!\hbox {mol}^{-2}\)

\(\varphi _\mathrm{V}\)

Fugacity coefficient of water vapor, \(\varphi _\mathrm{V} =f_\mathrm{V}/\left( {xp} \right) \)

1

\(\mu _\mathrm{V}\)

Mass-based chemical potential of water vapor in humid air

\(\hbox {J}\!\cdot \!\hbox {kg}^{-1}\)

\(\pi \)

Poynting correction factor of liquid water

1

\(\rho \)

Mass density

\(\hbox {kg}\!\cdot \!\hbox {m}^{-3}\)

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Feistel, R., Lovell-Smith, J.W. & Hellmuth, O. Virial Approximation of the TEOS-10 Equation for the Fugacity of Water in Humid Air. Int J Thermophys 36, 44–68 (2015). https://doi.org/10.1007/s10765-014-1784-0

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