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Thermodynamic properties of real moist air, dry air, steam, water, and ice (RP-1485).

INTRODUCTION

The American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE) has long been a leader in the field of psychrometric research, publications, and data. The current psychrometric tables in the 2005 ASHRAE Handbook--Fundamentals (ASHRAE 2005) are now 25 years old and are based on the Hyland-Wexler RP-216 (1983a, 1983b) and the Stewart et al. RP-257 (1983) research projects.

Research completed in recent years has resulted in new data and the new formulations listed below. The net effect is changes in the properties of moist air and in the saturation properties of [H.sub.2]O. These changes are small for air-conditioning system psychrometrics, but they are more significant at higher pressures and temperatures (e.g., the pressures and temperatures encountered in the compression stage of gas turbines and in compressed-air energy storage applications). New developments include:

* the fundamental equation of Lemmon et al. (2000) for the calculation of the dry air properties,

* Henry's constant from the International Association for the Properties of Water and Steam Guideline 2004 (IAPWS 2004) in the determination of the enhancement factor,

* the value for the universal molar gas constant from The Committee on Data for Science and Technology (CODATA) standard by Mohr and Taylor (2005),

* IAPWS Release on an Equation of State for [H.sub.2]O Ice Ih, (IAPWS 2006),

* the air-water second molar cross-virial coefficient from Harvey and Huang (2007),

* the Revised Release 2008 on the Pressure along the Melting and Sublimation Curves of Ordinary Water Substance (IAPWS 2008), and

* the value for the molar mass of dry air from Gatley et al. (2008).

The objective of this research is to update the ASHRAE real moist air psychrometric model (Hyland and Wexler 1983a, 1983b; Nelson and Sauer 2002) and incorporate the new developments listed above. The new model applies over the range of 130 K to 623.15 K (-143.15[degrees]C to 350[degrees]C) at pressures from 0.01 kPa to 10 MPa. The deliverables include new moist air and saturated [H.sub.2]O tables for the psychrometrics chapter of the 2009 ASHRAE Handbook--Fundamentals (ASHRAE 2009).

RP-1485 is the fourth major ASHRAE psychrometric research project.

UNDERLYING PROPERTIES OF DRY AIR, STEAM, WATER, AND ICE

Thermodynamic Properties of Dry Air

The thermodynamic properties of dry air are calculated using the National Institute of Standards and Technology (NIST) Lemmon et al. (2000) reference equation as a pseudo pure component. It consists of a fundamental equation for the molar Helmholtz energy ([bar.a]) as a function of density ([rho]) and temperature (T). The constants used by Lemmon et al. are given in Table 1.
Table 1. Constants and Values Used by Lemmon et al. (2000) in Their
Fundamental Equation for Dry Air

Quantity               Symbol            Value

Universal molar   [[bar.R].sup.Lem]     8.314510
gas constant                           kJ/(kmol*K)

Specific gas         [R.sup.Lem]        0.287117
constant                                kJ/(kg*K)

Molar mass           [M.sup.Lem]         28.9586
                                         kg/kmol

Maxcondentherm   [[bar.[rho]].sub.j]     10.4477
molar density                         mol/[dm.sup.3]

Maxcondentherm        [T.sub.j]        132.6312 K
temperature


The dimensionless form of the fundamental equation reads

[[[bar.a]([rho], T)]/[[[bar.R].sup.Lem]T]] = [[alpha][degrees]]([delta], [tau]) + [[alpha].sup.r]([delta], [tau]), (1)

where [alpha][degrees] is the ideal-gas contribution to the dimensionless Helmholtz energy and [[alpha].sup.r] is the residual contribution, [delta] = [bar.[rho]]/[[bar.[rho]].sub.j] is the reduced density, and [tau] = [T.sub.j]/T is the reciprocal reduced temperature. The reducing parameters [[bar.[rho]].sub.j] and [T.sub.j] are given in Table 1.

All thermodynamic properties for dry air can be derived from Equation 1 by using the appropriate combinations of the ideal-gas part [alpha][degrees] and the residual part [[alpha].sup.r] of the dimensionless Helmholtz energy and their derivatives. The derivatives of Equation 1 are described in detail in Lemmon et al. (2000) and Herrmann et al. (2008).

Thermodynamic Properties of Steam, Water, and Ice

Overview. For calculating thermodynamic properties of water and steam, the following IAPWS standards are used:

* IAPWS-95, Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use (IAPWS 1995; Wagner and Pru[??] 2002)

* IAPWS-IF97, Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam (IAPWS 2007; Wagner and Kretzschmar 2008; Parry et al. 2000)

* IAPWS-06, IAPWS Release 2006 on an Equation of State for [H.sub.2]O Ice Ih (IAPWS 2006; Feistel and Wagner 2006)

* IAPWS-08, Revised Release 2008 on the Pressure along the Melting and Sublimation Curves of Ordinary Water Substance (IAPWS 2008; Wagner et al. 2009).

The pressure-temperature diagram in Figure 1 shows in which ranges of state the certain formulation is applied.

[FIGURE 1 OMITTED]

For temperatures T[greater than or equal to]273.15 K, the industrial formulation IAPWS-IF97 (IAPWS 2007) is used both for liquid water, including saturated liquid (region 1), and for superheated steam, including saturated vapor (region 2). The saturation pressure [p.sub.s.sup.97] (T) is calculated using the IAPWS-IF97 saturation pressure equation. For practical purposes, the triple-point temperature [T.sub.t] = 273.16 K = 0.01[degrees]C = 32.018[degrees]F of water was rounded to 273.15 K = 0[degrees]C = 32[degrees]F. Instead of the triple-point pressure [p.sub.t] = 0.6117 kPa of water, the pressure p = [p.sub.s.sup.97] (273.15 K) = 0.6112 kPa is used.

For superheated steam, including saturated steam at temperatures T [less than or equal to] 273.15 K, the scientific formulation IAPWS-95 (IAPWS 1995) has to be used because the industrial formulation is only valid for temperatures T [greater than or equal to] 273.15 K.

For ice, including the saturation state, the formulation IAPWS-06 (IAPWS 2006) is applied. The sublimation pressure is calculated from the IAPWS-08 (IAPWS 2008) sublimation pressure equation [p.sub.sub.sup.08] (T). Instead of the melting line, the phase boundary between ice and liquid, the isotherm T = 273.15 K can be used in the pressure range 0.6112 kPa [less than or equal to] p [less than or equal to] 10 MPa with reasonable accuracy.

The Formulation IAPWS-95 for General and Scientific Use. To calculate the thermodynamic properties of water and steam, the international standard for general and scientific use IAPWS-95 (IAPWS 1995) can be applied. This standard contains an equation for the specific Helmholtz energy (a) as a function of density ([rho]) and temperature (T). The equation for the reduced Helmholtz energy reads

[[a([rho], T)]/[[R.sup.95]T]] = [[alpha][degrees]]([delta], [tau]) + [[alpha].sup.r]([delta],[tau]), (2)

where [alpha][degrees] is the ideal-gas contribution and [[alpha].sup.r] is the residual part, [delta] = [rho]/[[rho].sub.c] is the reduced density, and [tau] = [T.sub.c]/T is the reciprocal reduced temperature. The reducing parameters [[rho].sub.c] and [T.sub.c] and the specific gas constant [R.sup.95] of IAPWS-95 are listed in Table 2.
Table 2. Constants and Critical Values of IAPWS-95 (IAPWS 1995)

Quantity                           Symbol               Value

Universal molar gas constant  [[bar.R].sup.95]  8.314371 kJ/(kmol*K)

Specific gas constant            [R.sup.95]     0.46151805 kJ/(kg*K)

Molar mass                       [M.sup.95]       18.015268 kg/kmol

Critical mass density           [[rho].sub.c]       322 kg/[m.sup.3]

Critical temperature              [T.sub.c]             647.096 K

Critical pressure                 [p.sub.c]            22.064 MPa


All thermodynamic properties for water and steam can be derived from Equation 2 by using the appropriate combinations of the ideal-gas part [alpha][degrees] and the residual part [[alpha].sup.r] of the dimensionless Helmholtz energy and their derivatives. The derivatives of Equation 2 are described in detail in IAPWS-95 (IAPWS 1995) and in Herrmann et al. (2008).

The Industrial Formulation IAPWS-IF97. In addition to the scientific formulation IAPWS-95 (IAPWS 1995), the industrial formulation IAPWS-IF97 (IAPWS 2007) for water and steam can be used for practical calculations. The range of validity of the IAPWS-IF97 is divided into five calculation regions. The constants and critical parameters of IAPWS-IF97 are listed in Table 3.
Table 3. Constants and Critical Values of IAPWS-IF97 (IAPWS 2007)

Quantity                           Symbol               Value

Universal molar gas constant  [[bar.R].sup.97]  8.314510 kJ/(kmol*K)

Specific gas constant            [R.sup.97]      0.461526 kJ/(kg*K)

Molar mass                       [M.sup.97]       18.015257 kg/kmol

Critical mass density           [[rho].sub.c]    322.0 kg/[m.sup.3]

Critical temperature              [T.sub.c]          647.096 K

Critical pressure                 [p.sub.c]          22.064 MPa


Liquid-Water Region 1. The liquid-water region 1, including saturated liquid line x = 0, is described by an equation for the reduced Gibbs energy as a function of reduced pressure and reciprocal reduced temperature. The equation reads:

[[[g.sub.1](p, T)]/[[R.sup.97]T]] = [[gamma].sub.1]([pi], [tau]) (3)

where [pi] = p/p* and [tau] = T*/T with p* = 16.53 MPa and T* = 1386 K, and [R.sup.97] is given in Table 3.

All thermodynamic properties for water can be derived from Equation 3 by using the appropriate combinations of the dimensionless Gibbs energy [[gamma].sub.1] and its derivatives. The derivatives of Equation 3 are described in IAPWS-IF97 (IAPWS 2007) detail in Herrmann et al. (2008).

Steam Region 2. The steam region 2, including the saturated vapor line, is described by an equation for the reduced Gibbs energy as a function of pressure and temperature. The equation reads:

[[[g.sub.2](p, T)]/[[R.sup.97]T]] = [[gamma].sub.2][degrees]([pi], [tau]) + [[gamma].sub.2.sup.r]([pi], [tau]) (4)

where [[gamma].sub.2][degrees] is the ideal-gas contribution, [[gamma].sub.2.sup.r] is the residual part, [pi] = p/p* is the reduced pressure, and [tau] = T*/T is the reciprocal reduced temperature with p* = 1 MPa and T* = 540 K. [R.sup.97] is the specific gas constant of IAPWS-IF97 (IAPWS 2007) given in Table 3.

All thermodynamic properties for steam can be derived from Equation 4 by using the appropriate combinations of the ideal-gas part [[gamma].sub.2][degrees] and the residual part [[gamma].sub.2.sup.r] of the dimensionless Gibbs energy and their derivatives. The derivatives of Equation 4 are described in detail in IAPWS-IF97 (IAPWS 2007) and in Herrmann et al. (2008).

Thermodynamic Properties of Ice. The IAPWS-06 (IAPWS 2006) formulation for ice describes the thermodynamic properties of ice in the region Ih (see Figure 1) with an equation for the specific Gibbs energy as seen here:

g = g(p, T) (5)

The constants and triple-point parameters of IAPWS-06 (IAPWS 2006) are listed in Table 4.
Table 4. Constants and Values Used in IAPWS-06 (IAPWS 2006; Feistel and
Wagner 2006)

Quantity                           Symbol               Value

Universal molar gas constant  [[bar.R].sup.06]  8.314472 kJ/(kmol*K)

Specific gas constant            [R.sup.06]     0.46152364 kJ/(kg*K)

Molar mass                       [M.sup.06]       18.015268 kg/kmol

Triple-point temperature          [T.sub.t]           273.16 K

Triple-point pressure             [p.sub.t]          0.611657 kPa


All thermodynamic properties for ice can be derived from Equation 5 by using the appropriate combinations of the Gibbs energy g and its derivatives. The derivatives of Equation 5 are described in detail in IAPWS-06 (IAPWS 2006) and in Herrmann et al. (2008).

Saturation Pressure and Saturation Temperature of Water. The saturation pressure and the saturation temperature of the liquid-vapor equilibrium (see Figure 1) are calculated using the industrial formulation IAPWS-IF97 (IAPWS 2007). It contains an implicit quadratic equation that can be solved analytically with regard to both saturation pressure or saturation temperature. The solution with regard to the saturation pressure is a function of given temperature T as seen here:

[p.sub.w, s] = [p.sub.s.sup.97](T) (6)

The solution for the saturation temperature reads:

[T.sub.w, s] = [T.sub.s.sup.97](p) (7)

Both equations are valid for temperatures T [greater than or equal to] 273.15. Equations 6 and 7 are described in detail in IAPWS-IF97 (IAPWS 2007) and in Herrmann et al. (2008).

Sublimation Pressure and Sublimation Temperature. The sublimation pressure and the sublimation temperature of the solid-vapor equilibrium (see Figure 1) are calculated using the IAPWS-08 (IAPWS 2008) formulation.

The sublimation-pressure equation reads:

[p.sub.w, s] = [p.sub.[sub].sup.08](T) (8)

In this work, Equation 8 is used in the temperature range 130 K [less than or equal to] T [less than or equal to] 273.15 K (see Figure 1). Equation 8 is described in detail in IAPWS-08 (IAPWS 2008) and in Herrmann et al. (2008). The sublimation temperature ([T.sub.sub.sup.08]) is calculated for given pressure p by solving iteratively the sublimation pressure equation (Equation 8) in terms of [T.sub.sub.sup.08] = T.

PSYCHROMETRIC EQUATIONS

Methodology

The properties of moist air are calculated from the modified Hyland-Wexler model given in Herrmann et al. (2008). The modifications incorporate:

* the value for the universal molar gas constant from Mohr and Taylor (2005)

* the value for the molar mass of dry air from Gatley et al. (2008) and that of water from IAPWS-95 (IAPWS 1995; Wagner and Pru[??] 2002)

* the calculation of the ideal-gas parts of the heat capacity, enthalpy, and entropy for dry air from the fundamental Lemmon et al. (2000) equation

* the calculation of the ideal-gas parts of the heat capacity, enthalpy, and entropy for water and steam from IAPWS-IF97 (IAPWS 2007; Wagner and Kretzschmar 2008; Parry et al. 2000) for T [greater than or equal to] 273.15 K and from IAPWS-95 (IAPWS 1995; Wagner and Pru[??] 2002) for T [less than or equal to] 273.15 K

* the calculation of the vapor-pressure enhancement factor from the equation, given by Hyland and Wexler (1983a, 1983b)

* the calculation of the second and third molar virial coefficients [B.sub.aa] and [C.sub.aaa] for dry air from the fundamental Lemmon et al. (2000) equation

* the calculation of the second and third molar virial coefficients [B.sub.ww] and [C.sub.www] for water and steam from IAPWS-95 (IAPWS 1995; Wagner and Pru[??] 2002)

* the calculation of the air-water second molar cross-virial coefficient [B.sub.aw] from Harvey and Huang (2007)

* the calculation of the air-water third molar cross-virial coefficients [C.sub.aaw] and [C.sub.aww] from Nelson and Sauer (2002; Gatley 2005)

* the calculation of the saturation pressure of water from IAPWS-IF97 (IAPWS 2007; Wagner and Kretzschmar 2008; Parry et al. 2000) for T [greater than or equal to] 273.15 K and of the sublimation pressure of water from IAPWS-08 (IAPWS 2008; Wagner et al. 2009) for T [less than or equal to] 273.15 K

* the calculation of the isothermal compressibility of liquid water from IAPWS-IF97 (IAPWS 2007; Wagner and Kretzschmar 2008; Parry et al. 2000) T [greater than or equal to] 273.15 K for and that of ice from IAPWS-06 (IAPWS 2006; Feistel and Wagner 2006) for T [less than or equal to] 273.15 K in the determination of the vapor-pressure enhancement factor

* the calculation of Henry's constant from the IAPWS Guideline 2004 (IAPWS 2004; Fernandez-Prini et al. 2003) in the determination of the enhancement factor, whereas the mole fractions for the three main components of dry air were taken from Lemmon et al. (2000). Argon was not considered in the former research projects, but it is now the third component of dry air.

Virial Equation of State

The mixture moist air is calculated using the following mixing virial equation of state. The equation contains virial coefficients up to the third virial coefficient. The virial equation of state reads:

[[p[bar.v]]/[[bar.R]T]] = 1 + [[B.sub.m]/[bar.v]] + [[C.sub.m]/[[bar.v].sup.2]] (9)

where p is the total pressure of moist air, [bar.v] is the molar mixture volume, [bar.R] is the universal molar gas constant given in Table 5, [B.sub.m] is the second molar virial mixing coefficient, and [C.sub.m] is the third molar virial mixing coefficient. The molar virial coefficients with higher order than the third one are not considered in Equation 9. The molar virial coefficients [B.sub.m] and [C.sub.m] are calculated as follows:
Table 5. Constants and Values Used for the Calculation of Psychrometric
Properties of Moist Air

Quantity                         Symbol        Value      Reference

Universal molar gas constant     [bar.R]      8.314472    Mohr and
                                            kJ/(kmol*K)    Taylor
                                                           (2005)

Molar mass of dry air           [M.sub.a]      28.966     Gatley et
                                              kg/kmol       al.
                                                           (2008)

Molar mass of water             [M.sub.w]     18.015268   IAPWS-95
                                              kg/kmol      (IAPWS
                                                            1995)

Specific gas constant of dry    [R.sub.a]    [congruent
air                             = [bar.R]/  to] 0.287042
                                [M.sub.a]     kJ/(kg*K)

Specific gas constant of water  [R.sub.w]    [congruent
                                = [bar.R]   to] 0.461524
                                    /         kJ/(kg*K)
                                [M.sub.w]

Quotient: [M.sub.w] /[M.sub.a]  [epsilon]    [congruent
= [R.sub.a] /[R.sub.w]                          to]
                                              0.621945

Pressure of the reference       [p.sub.0]     0.101325
state                                           MPa

Temperature of the reference    [T.sub.0]     273.15 K
state


[B.sub.m] = [(1-[[psi].sub.w]).sup.2][B.sub.aa] + 2(1-[[psi].sub.w])[[psi].sub.w][B.sub.aw] + [[psi].sub.w.sup.2][B.sub.ww] (10)

[C.sub.m] = [(1-[[psi].sub.w]).sup.3][C.sub.aaa] + 3[(1-[[psi].sub.w]).sup.2][[psi].sub.w][C.sub.aaw] + 3(1-[[psi].sub.w])[[psi].sub.w.sup.2][C.sub.aww] + [[psi].sub.w.sup.3][C.sub.www] (11)

where [[psi].sub.w] is the mole fraction of water vapor in the mixture moist air, and 1 - [[psi].sub.w] = [[psi].sub.a] is the mole fraction of dry air.

The derivatives of the second and the third molar virial mixing coefficients, with respect to temperature, are given as follows:

[[[dB.sub.m]]/[dT]] = [(1-[[psi].sub.w]).sup.2][[[dB.sub.aa]]/[dT]] + 2(1-[[psi].sub.w])[[psi].sub.w][[[dB.sub.aw]]/[dT]] + [[psi].sub.w.sup.2][[[dB.sub.ww]]/[dT]] (12)

[[[dC.sub.m]]/[dT]] = [(1-[[psi].sub.w]).sup.3][[[dC.sub.aaa]]/[dT]] + 3[(1-[[psi].sub.w]).sup.2][[psi].sub.w][[[dC.sub.aaw]]/[dT]] + 3(1-[[psi].sub.w])[[psi].sub.w.sup.2][[[dC.sub.aww]]/[dT]] + [[psi].sub.w.sup.3][[[dC.sub.www]]/[dT]] (13)

The calculation of the molar virial coefficients [B.sub.aa], [C.sub.aaa] for dry air, [B.sub.ww], [C.sub.www] for water vapor, and the cross-virial coefficients [B.sub.aw], [C.sub.aaw], [C.sub.aww], and their derivatives are described in the next section.

The values for the universal molar gas constant ([bar.R]), the molar mass of dry air ([M.sub.a]), and the molar mass of water ([M.sub.w]) used for the calculation of the psychrometric properties in this section are given in Table 5.

The specific gas constant for dry air can be obtained using [R.sub.a] = [bar.R] / [M.sub.a] and the specific gas constant for water can be obtained using [R.sub.w] = [bar.R] / [M.sub.w]. The values given in Table 5 for [R.sub.a] and [R.sub.w] are displayed with six decimal places.

The quotient of the molar masses of water and dry air is defined as

[epsilon] = [[M.sub.w]/[M.sub.a]] = [[R.sub.a]/[R.sub.w]]. (14)

Virial and Cross-Virial Coefficients and Their Derivatives

Dry Air. The second and third molar virial coefficients of dry air and their derivatives are calculated using Lemmon et al. (2000) and read

[B.sub.aa]([rho], T) = [1/[[bar.[rho]].sub.j]][([[partial derivative][[alpha].sup.r]]/[[partial derivative][delta]]).sub.[tau]]|[.sub.[delta] = 0] (15)

[[[dB.sub.aa]([rho], T)]/[dT]] = -[1/[[[bar.[rho]].sub.j][T.sub.j]]][[tau].sup.2]([[[partial derivative].sup.2][[alpha].sup.r]]/[[partial derivative][delta][partial derivative][tau]])|[.sub.[delta] = 0] (16)

[C.sub.aaa]([rho], T) = [1/[[bar.[rho]].sub.j.sup.2]][([[[partial derivative].sup.2][[alpha].sup.r]]/[[partial derivative][[delta].sup.2]]).sub.[tau]]|[.sub.[delta] = 0] (17)

[[[dC.sub.aaa]([rho], T)]/[dT]] = -[1/[[[bar.[rho]].sub.j.sup.2][T.sub.j]]][[tau].sup.2]([[[partial derivative].sup.3][[alpha].sup.r]]/[[partial derivative][[delta].sup.2][partial derivative][tau]])|[.sub.[delta] = 0] (18)

where [[bar.[rho]].sub.j] is the molar density at Maxcondentherm and [T.sub.j] is the temperature at Maxcondentherm (both given in Table 1), [tau] is the reciprocal reduced temperature, and [delta] is the reduced density. The derivatives [([partial derivative][[alpha].sup.r]/[partial derivative][delta]).sub.[tau]], [([[partial derivative].sup.2][[alpha].sup.r]/([partial derivative][delta][partial derivative][tau])), ([[partial derivative].sup.2][[alpha].sup.r]/[partial derivative][[delta].sup.2]).sub.[tau]], and ([[partial derivative].sup.3][[alpha].sup.r]/([partial derivative][[delta].sup.2][partial derivative][tau])) are determined using the residual part of the reduced fundamental Lemmon et al. equation (Equation 1) at the limit of [delta] = 0. The fundamental Lemmon et al. equation for dry air and its derivatives are given in detail in Lemmon et al. (2000) and in Herrmann et al. (2008).

Water Vapor. The second and third molar virial coefficients of water vapor and their derivatives are calculated using IAPWS-95 (IAPWS 1995; Wagner and Pru[??] 2002) using the following equations:

[B.sub.ww]([rho], T) = [1/[[bar.[rho]].sub.c]][([[partial derivative][[alpha].sup.r]]/[[partial derivative][delta]]).sub.[tau]]|[.sub.[delta] = 0] (19)

[[[dB.sub.ww]([rho], T)]/[dT]] = -[1/[[[bar.[rho]].sub.c][T.sub.c]]][[tau].sup.2]([[[partial derivative].sup.2][[alpha].sup.r]]/[[partial derivative][delta][partial derivative][tau]])|[.sub.[delta] = 0] (20)

[C.sub.www]([rho], T) = [1/[[bar.[rho]].sub.c.sup.2]][([[[partial derivative].sup.2][[alpha].sup.r]]/[[partial derivative][[delta].sup.2]]).sub.[tau]]|[.sub.[delta] = 0] (21)

[[[dC.sub.www]([rho], T)]/[dT]] = -[1/[[[bar.[rho]].sub.c.sup.2][T.sub.c]]][[tau].sup.2]([[[partial derivative].sup.3][[alpha].sup.r]]/[[partial derivative][[delta].sup.2][partial derivative][tau]])|[.sub.[delta] = 0] (22)

where [T.sub.c] is the critical temperature, [[bar.[rho]].sub.c] is the molar critical density, [tau] is the reciprocal reduced temperature, and [delta] is the reduced density. The derivatives [([partial derivative][[alpha].sup.r]/[partial derivative][delta]).sub.[tau]], [([[partial derivative].sup.2][[alpha].sup.r]/([partial derivative][delta][partial derivative][tau])), ([[partial derivative].sup.2][[alpha].sup.r]/[partial derivative][[delta].sup.2]).sub.[tau]], and ([[partial derivative].sup.3][[alpha].sup.r]/([partial derivative][[delta].sup.2][partial derivative][tau])) are determined using the residual part of the reduced fundamental equation of IAPWS-95, Equation 2, at the limit of [delta] = 0. The fundamental equation for water and steam from IAPWS-95 and its derivatives are given in detail in IAPWS-95 (IAPWS 1995) Herrmann et al. (2008).

Air Water. The second molar virial coefficient of air-water molecule interactions ([B.sub.aw]) is calculated using the calculation proposed by Harvey and Huang (2007). They determined the temperature-dependent second molar virial coefficient of air and water ([B.sub.aw]) and its first derivative ([dB.sub.aw]/dT) to be:

[B.sub.aw](T) = [1/[[bar.[rho]]*]][3.summation over (i = 1)][a.sub.i][[theta].sup.bi] (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

where = T/T* with T* = 100 K and [bar.[rho]*] = [10.sup.3] mol/d[m.sup.3]. The coefficients [a.sub.1] ... [a.sub.3] and [b.sub.1] ... [b.sub.3] are given in Harvey and Huang (2007) and in Herrmann et al. (2008).

Nelson and Sauer (2002) determined the third molar cross-virial coefficients of air-water [C.sub.aaw] and [C.sub.aww] as follows:

[C.sub.aaw](T) = [1/[[([bar.[rho]]*)].sup.2]][5.summation over (i = 1)][c.sub.i][[theta].sup.1-i] (25)

[[[dC.sub.aaw](T)]/[dT]] = [1/[[[([bar.[rho]]*)].sup.2]T*]][5.summation over (i = 2)][c.sub.i](1-i)[[theta].sup.-i] (26)

[C.sub.aww](T) = -[1/[[([bar.[rho]]*)].sup.2]]exp([4.summation over (i = 1)][d.sub.i][[theta].sup.1-i]) (27)

[[[dC.sub.aww](T)]/[dT]] = -[1/[[[([bar.[rho]]*)].sup.2]T*]]exp([4.summation over (i = 1)][d.sub.i][[theta].sup.1-i])[4.summation over (i = 2)][d.sub.i](1-i)[[theta].sup.-i] (28)

where [theta] = T/T* with T* = 1 K and [[bar.[rho]*] = [10.sup.3] mol/d[m.sup.3]. The coefficients [c.sub.1] ... [c.sub.5] and [d.sub.1] ... [d.sub.4] are given in Herrmann et al. (2008); also see Gatley (2005).

Saturation State of Moist Air

Equation for the Saturation Partial Pressure of Water. The partial pressure of water ([p.sub.s]) in saturated moist air is calculated using the following equation:

[p.sub.s] = f[p.sub.w, s] (29)

where f = f(p,T) is the vapor-pressure enhancement factor, and [p.sub.w,s] = [p.sub.w,s](T) is the saturation pressure of pure water. Therefore, [p.sub.s] depends on total pressure (p) and temperature (T). The next section comprises the calculation of f.

The saturation pressure of pure water ([p.sub.w,s]) is calculated for given temperature T as follows:

[p.sub.w,s] = [p.sub.s.sup.97](T) for T[greater than or equal to] 273.15 K from Equation 6 (30)

[p.sub.w,s] = [p.sub.[sub].sup.08](T) for T[less than or equal to]273.15 K from Equation 8 (31)

where [p.sub.s.sup.97] (T) is the saturation-pressure equation of water obtained from IAPWS-IF97 (IAPWS 2007), Equation 6, and [p.sub.sub.sup.08] (T) is the sublimation-pressure equation of water obtained from IAPWS-08 (IAPWS 2008), Equation 8.

Using the saturation partial pressure of water ([p.sub.s]), the saturation mole fraction of water ([[psi].sub.w,s]) is calculated using the following equation:

[[psi].sub.w,s] = [[p.sub.s]/p] = [[f[p.sub.w,s]]/p]. (32)

Equation for the Enhancement Factor. The vapor-pressure enhancement factor (f) describes the enhancement of the saturation pressure of water in the air atmosphere under elevated total pressure. The calculation of the enhancement factor as a function of total pressure (p) and temperature (T) is given by this equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

where T is the temperature of the mixture moist air, p is the total pressure, [[psi].sub.w,s] is the mole fraction of water in saturated moist air, and [bar.R] is the current value of the universal molar gas constant from Mohr and Taylor (2005) given in Table 5.

Furthermore, [p.sub.w,s] = [p.sub.s.sup.97] (T) is the saturation pressure for temperatures T [greater than or equal to] 273.15 K obtained using Equation 6 of IAPWS-IF97 (IAPWS 2007). For temperatures T [less than or equal to] 273.75 K, [p.sub.w,s] = [p.sub.sub.sup.08] (T) is the sublimation pressure from Equation 8 of IAPWS-08 (IAPWS 2008). (For practical purposes, the triple-point temperature [T.sub.t] = 273.16 K = 0.01[degrees]C = 32.018[degrees]F of water was rounded to 273.15 K = 0[degrees]C = 32[degrees]F. Therefore, instead of the triple-point pressure [p.sub.t] = 0.6117 kPa, the pressure p = [p.sub.s.sup.97](273.15 K) = 0.6112 kPa is used for water.)

For temperatures T [greater than or equal to] 273.15 K, the molar volume of saturated liquid water is calculated using IAPWS-IF97 (IAPWS 2007) using the following equation:

[[bar.v].sub.w,s] = [[[[bar.R].sup.97]T]/[p.sub.w,s]][pi][([[partial derivative][[gamma].sub.1]]/[[partial derivative][pi]]).sub.[tau]] (34)

where [pi] is the reduced pressure, and [tau] is the reciprocal reduced temperature (see Equation 3). The saturation pressure of water [p.sub.w,s] = [p.sub.s.sup.97](T) is obtained using Equation 6, and [[bar.r].sup.97] is the molar gas constant of IAPWS-IF97, given in Table 3. The derivative [([partial derivative][[gamma].sub.1]/[partial derivative][pi]).sub.[tau]] is formed from the reduced Gibbs equation of IAPWS-IF97, Equation 3. More details about this fundamental equation and its derivatives are given in Herrmann et al. (2008).

For temperatures T [less than or equal to] 273.15 K, the molar volume of saturated ice is calculated using IAPWS-06 (IAPWS 2006) using the following equation

[[bar.v].sub.w,s] = [M.sup.06][([[partial derivative]g]/[[partial derivative]p]).sub.T] (35)

where [([partial derivative]g/[partial derivative]p).sub.T] is the derivative of the specific Gibbs equation of IAPWS-06, Equation 5, and [M.sup.06] is the molar mass of IAPWS-06, given in Table 4. Details about this Gibbs equation and its derivatives are given in Herrmann et al. (2008).

The calculation of the second molar virial coefficients [B.sub.aa], [B.sub.aw], [B.sub.ww], and of the third molar virial coefficients [C.sub.aaa], [C.sub.aaw], [C.sub.aww], [C.sub.www] is described in the previous section.

The quantity [[kappa].sub.T] in Equation 33 is the isothermal compressibility of saturated liquid water for temperatures T [greater than or equal to] 273.15 K and that of saturated ice for temperatures T [less than or equal to] 273.15 K. The following section contains the algorithms for both cases of temperatures.

In Equation 33, [[beta].sub.H] is the Henry's law constant. Equation 38 contains its calculation.

Using Equation 33, the enhancement factor (f) has to be calculated iteratively because the saturation mole fraction of water ([[psi].sub.w,s]) in Equation 33 depends on f via Equation 32.

Isothermal Compressibility. For temperatures T [greater than or equal to] 273.15 K, the isothermal compressibility ([[kappa].sub.T]) of liquid water is calculated for total pressure and temperature using the expression

[[kappa].sub.T] = -[1/p][pi][([[[partial derivative].sup.2][[gamma].sub.1]]/[[partial derivative][[pi].sup.2]]).sub.[tau]][[([[partial derivative][[gamma].sub.1]]/[[partial derivative][pi]])].sub.[tau].sup.[-1]], (36)

where [pi] is the reduced pressure, and [tau] is the reciprocal reduced temperature (see Equation 3). The derivatives [([partial derivative][[gamma].sub.1]/[partial derivative][pi]).sub.[tau]] and [([[partial derivative].sup.2][[gamma].sub.1]/[partial derivative][[pi].sup.2]).sub.[tau]] are formed using the reduced Gibbs equation of IAPWS-IF97 (IAPWS 2007) region 1, Equation 3. Details about this fundamental equation and its derivatives are given in Herrmann et al. (2008).

For temperatures, T [less than or equal to] 273.15 K is determined using IAPWS-06 (IAPWS 2006) for ice as seen here:

[[kappa].sub.T] = -[([[[partial derivative].sup.2]g]/[[partial derivative][p.sup.2]]).sub.T][[([[partial derivative]g]/[[partial derivative]p])].sub.T.sup.[-1]], (37)

where [([partial derivative]g/[partial derivative]p).sub.T] and [([[partial derivative].sup.2]g/[partial derivative][p.sup.2]).sub.T] are the derivatives of the specific Gibbs equation, Equation 5. Details about this fundamental equation and its derivatives are given in Herrmann et al. (2008).

In the iteration process of the enhancement factor via Equation 33, [[kappa].sub.T] is set to zero for [p.sub.w,s](T) > p.

Henry's Law Constant. Henry's law constant [[beta].sub.H] is calculated using IAPWS-04 (IAPWS 2004; Fernandez-Prini et al. 2003) for the three main components of dry air:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [[psi].sub.Ar] are the mole fractions of nitrogen, oxygen, and argon in dry air. The calculation of the three terms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [[beta].sub.Ar] for the Henry's law constant for each solvent are described in detail in Herrmann et al. (2008).

[[beta].sub.H] is set to zero for T [less than or equal to] 273.15 K or T > [T.sub.w,s] (p), where [T.sub.w,s] is the saturation temperature of pure water calculated using Equation 7.

Graphical Representation of the Enhancement Factor. Values of the vapor-pressure enhancement factor f calculated using Equation 33 for several total pressures are plotted over temperature t for t > 0[degrees]C in Figure 2. In the figure, the enhancement increases with increasing total pressure (p). Each curve p = const starts at [Florin] = 1.0 and the saturation temperature of pure water. The vapor-pressure enhancement factor at 0.101325 MPa is 1.0041 at 0[degrees]C and is often disregarded in air-conditioning calculations. The vapor-pressure enhancement factor must be included for many industrial process calculations (e.g., at a total pressure of 10 MPa the vapor-pressure enhancement factor is 1.4638).

[FIGURE 2 OMITTED]

Psychrometric Properties of Moist Air

Humidity Ratio. The humidity ratio (W) is defined as the quotient of the mass of water ([m.sub.w]) divided by the mass of dry air ([m.sub.a]) in the mixture moist air as seen here:

W = [[m.sub.w]/[m.sub.a]] = [[m.sub.w]/[(m-[m.sub.w])]] (39)

where m is the mass of the mixture moist air.

With the mole fraction [[psi].sub.w] of water one obtains

W = [epsilon][[[psi].sub.w]/[(1-[[psi].sub.w])]], (40)

where [epsilon] is the quotient of the molar masses of dry air and water, Equation 14. By solving Equation 40 in terms of [[psi].sub.w] one obtains

[[psi].sub.w] = [W/[([epsilon] + W)]]. (41)

Using [[psi].sub.w], the molar mass of moist air (M) is calculated using the equation

M = (1-[[psi].sub.w])[M.sub.a] + [[psi].sub.w][M.sub.w] (42)

where [M.sub.a] and [M.sub.w] are the molar masses of dry air and water, given in Table 5.

Relative Humidity of Unsaturated or Saturated Moist Air. In addition, for unsaturated and saturated moist air, the water content can be specified by the quantity relative humidity:

[phiv] = [[[psi].sub.w]/[[psi].sub.w,s]] (43)

with the definition range 0 [less than or equal to] [phiv] [less than or equal to] 1, the value [phiv] = 0 for dry air, and the value [phiv] = 1 for saturated moist air.

With the partial pressure of water vapor [p.sub.w] = [[psi].sub.w]p and the saturation partial pressure of water [p.sub.s] = [[psi].sub.w,s]p, according Equation 29 one obtains

[phiv] = [[p.sub.w]/[p.sub.s]]. (44)

This equation is valid for [p.sub.w] [less than or equal to] [p.sub.s].

Using the previous equations the following equation is obtained:

W = [epsilon][[[phiv][p.sub.s]]/[(p-[phiv][p.sub.s])]] (45)

with [epsilon] as quotient of the molar masses of water and dry air, Equation 14, and [p.sub.s] calculated using Equation 29.

Saturation State. The saturation state of moist air is characterized by

[p.sub.w] = [p.sub.s](p, T) according to Equation 29,

[[psi].sub.w] = [[psi].sub.w,s] according to Equation 32,

[phiv] = 1,

and therefore by

[W.sub.s] = [epsilon][[[psi].sub.w,s]/[(1-[[psi].sub.w,s])]] (46)

where [[psi].sub.w,s] is the mole fraction of water vapor in saturated moist air, and is given in Equation 14.

Molar and Air-Specific Volume. The molar volume ([bar.v]) (or molar density [bar.[rho]] = 1/[bar.v]) for moist air can be calculated iteratively for given pressure p, given temperature T, and mole fraction of water vapor [[psi].sub.w] using the expression

p(T,[bar.v], [[psi].sub.w]) = [[[bar.R]T]/[bar.v]](1 + [[B.sub.m]/[bar.v]] + [C.sub.m]/[[bar.v].sup.2]), (47)

where [bar.R] is the universal molar gas constant given in Table 5, and [B.sub.m] and [C.sub.m] are the second and third molar virial coefficients of the mixture given by Equations 10 and 11. The mole fraction of water vapor [[psi].sub.w] can be calculated from given humidity ratio W using Equation 41. Herrmann et al. (2008) comprises a detailed description of the algorithm.

In the case of [[psi].sub.w] = 0, the molar volume of dry air is obtained from Equation 47. (The molar volume for dry air obtained from Equation 47 differs from that of the fundamental Lemmon et al. (2000) equation, Equation 1, because only the second and the third virial coefficients are used.)

The air-specific volume (v) or specific density ([rho]) can be obtained using the iteratively calculated molar volume ([bar.v]) as follows:

v = [[(1 + W)[bar.v]]/M] and [rho] = [[(1 + W)]/v] (48)

where M is the molar mass of the mixture calculated using Equation 42, and W is the humidity ratio, Equation 40.

Molar and Air-Specific Enthalpy. The molar enthalpy ([bar.h]) for moist air is calculated using the ideal-gas parts of dry air and water vapor and the real-gas correction from the virial equation for moist air. Equation 49 can be used for this calculation:

[bar.h](T, [bar.v], [[psi].sub.w]) = [[bar.h].sub.0] + (1-[[psi].sub.w])[[bar.h].sub.a.sup.o] + [[psi].sub.w][[bar.h].sub.w.sup.o] + [bar.R]T[([B.sub.m]-T[d[B.sub.m]]/[dT])[1/[bar.v]] + ([C.sub.m]-[T/2][d[C.sub.m]]/[dT])1/[[bar.v].sup.2]] (49)

where [[psi].sub.w] is the mole fraction of water vapor, [[bar.h].sub.a.sup.o] is the ideal-gas molar enthalpy of dry air and [[bar.h].sub.w.sup.o] is the ideal-gas molar enthalpy of water vapor--both described below, [bar.v] is the molar volume of the mixture moist air calculated iteratively using Equation 47, [B.sub.m] and [C.sub.m] are calculated using Equations 10 and 11, d[B.sub.m]/dT and d[C.sub.m]/dT are calculated using Equations 12 and 13, and [bar.R] is the universal molar gas constant given in Table 5. The mole fraction of water vapor ([[psi].sub.w]) can be calculated from given humidity ratio W using Equation 41. Herrmann et al. (2008) comprises a detailed description of the algorithm. The value of [[bar.h].sub.0] results from the adjustment of the molar enthalpy to zero at [p.sub.0] and [T.sub.0], given in Table 5. The value is [[bar.h].sub.o] = 2.924425468 kJ/kmol.

Using the fundamental Lemmon et al. (2000) equation, the molar ideal-gas enthalpy for dry air is calculated as follows:

[[bar.h].sub.a.sup.o] = [[bar.h].sub.0.sup.Lem] + [[bar.R].sup.Lem]T[1 + [tau][([[partial derivative][[alpha].sup.o]]/[[partial derivative][tau]]).sub.[delta]]] (50)

where [[bar.R].sup.Lem] is the universal molar gas constant used by Lemmon et al. (and given in Table 1), [tau] is the reciprocal reduced temperature, and [delta] is the reduced density. The derivative [([partial derivative][[alpha].sup.o]/[partial derivative][tau]).sub.[delta]] is determined using the ideal-gas part of the reduced fundamental Lemmon et al. equation, Equation 1, and described in detail in Herrmann et al. (2008). The value [[bar.h].sub.0.sup.Lem] =7914.149298 kJ/kmol results from shifting the reference state to [T.sub.0] = 273.15 K used for moist air.

The IAPWS-IF97 (IAPWS 2007) formulation is used to calculate the molar ideal-gas enthalpy for water vapor at temperatures T[greater than or equal to] 273.15 K:

[[bar.h].sub.w.sup.o] = [[bar.h].sub.0.sup.97] + [[bar.R].sup.97]T[tau][([[partial derivative][[gamma].sub.2.sup.o]]/[[partial derivative][tau]]).sub.[pi]] (51)

where [[bar.R].sup.97] is the molar gas constant of IAPWS-IF97, given in Table 3, [pi] is the reduced pressure, and [tau] is the reciprocal reduced temperature. The derivative [([[partial derivative][gamma].sub.2.sup.o]/[partial derivative][tau]).sub.[pi]] is determined using the ideal-gas part of the reduced fundamental equation of IAPWS-IF97, Equation 4, and is described in detail in Herrmann et al. (2008). The value [[bar.h].sub.0.sup.97] = -0.01102142797 kJ/kmol results from shifting the reference state.

The IAPWS-95 (IAPWS 1995) formulation is used to calculate the molar ideal-gas enthalpy for water vapor at temperatures T [less than or equal to] 273.15 K as follows:

[[bar.h].sub.w.sup.o] = [[bar.h].sub.0.sup.95] + [[bar.R].sup.95]T[1 + [tau][([[partial derivative][[alpha].sup.o]]/[[partial derivative][tau]]).sub.[delta]]] (52)

where [[bar.R].sup.95] is the molar gas constant of IAPWS-95, given in Table 2, [tau] is the reciprocal reduced temperature, and [delta] is the reduced density. The derivative [([[partial derivative][gamma].sub.2.sup.o]/[partial derivative][tau]).sub.[pi]] is determined using the ideal-gas part of the reduced fundamental equation of IAPWS-95, Equation 2, and described in detail in the report of Herrmann et al. (2008). The value [[bar.h].sub.0.sup.95] = -0.01102303806 kJ/kmol results from shifting the reference state.

The air-specific enthalpy (h) can be obtained using the molar enthalpy ([bar.h]) calculated using Equation 49 via h = [bar.h](1+W)/M, where M is the molar mass of the mixture calculated using Equation 42, and W is the humidity ratio, Equation 40.

Molar and Air-Specific Entropy. The molar entropy [bar.s] for moist air is calculated using the ideal-gas parts of dry air and water vapor and the real-gas correction from the virial equation for moist air. The equation reads

[bar.s](T, [bar.v], [[psi].sub.w]) = [[bar.s].sub.0] + (1-[[psi].sub.w])[[bar.s].sub.a.sup.o] + [[psi].sub.w][[bar.s].sub.w.sup.o] - [bar.R][([B.sub.m] + T[d[B.sub.m]]/[dT])[1/[bar.v]] + ([C.sub.m] + T[d[C.sub.m]]/[dT])[1/[2[[bar.v].sup.2]]] + (1-[[psi].sub.w])ln(1-[[psi].sub.w]) + [[psi].sub.w]ln([[psi].sub.w])] (53)

where [[psi].sub.w] is the mole fraction of water vapor, [[bar.s].sub.a.sup.o] is the ideal-gas molar entropy of dry air [[bar.s].sub.w.sup.o] is the ideal-gas molar entropy of water vapor--both described below, [bar.v] is the molar volume of moist air calculated iteratively using Equation 47, [B.sub.m] and [C.sub.m] are calculated using Equations 10 and 11, d[B.sub.m]/dT and d[C.sub.m]/dT are calculated using Equations 12 and 13, and [bar.R] is the universal molar gas constant, given in Table 5. The mole fraction of water vapor ([[psi].sub.w]) can be calculated from given humidity ratio W using Equation 41. Herrmann et al. (2008) comprises a detailed description of the algorithm. The value of [[bar.s].sub.0] results from the adjustment of the molar entropy to zero at [p.sub.0] and [T.sub.0], given in Table 5. The value is [[bar.s].sub.0] = 0.02366427495 kJ/(kmol * K).

Using the fundamental Lemmon et al. (2000) equation, the molar ideal-gas entropy for dry air is calculated as follows:

[[bar.s].sub.a.sup.o] = [[bar.s].sub.0.sup.Lem] + [[bar.R].sup.Lem][[tau][([[partial derivative][[alpha].sup.o]]/[[partial derivative][tau]]).sub.[delta]]-[[alpha].sup.o]] + [[bar.R].sup.Lem]ln([[bar.v].sub.a]/[[bar.v].sub.a.sup.o]) (54)

where [[bar.R].sup.Lem] is the universal molar gas constant used by Lemmon et al., given in Table 1, [tau] is the reciprocal reduced temperature, [delta] is the reduced density, [[alpha].sup.o] is the ideal-gas part of the reduced fundamental Lemmon et al. equation, Equation 1, and [([partial derivative][[alpha].sup.o]/[partial derivative][tau]).sub.[delta]] is its derivative, described in detail in Herrmann et al. (2008). The molar volume of dry air ([[bar.v].sub.a]) is calculated iteratively for given total pressure p and temperature T from the following equation:

p = [[[[bar.R].sup.Lem]T]/[[bar.v].sub.a]](1 + [[B.sub.aa]/[[bar.v].sub.a]] + [C.sub.aaa]/[[bar.v].sub.a.sup.2]) (55)

where [B.sub.aa] and [C.sub.aaa] are the second and third molar virial coefficients of dry air calculated using Equations 15 and 17. The ideal-gas molar volume of dry air [[bar.v].sub.a.sup.o] results from [[bar.v].sub.a.sup.o] = [[bar.R].sup.Lem][T.sub.0]/[p.sub.0], where [p.sub.0] and [T.sub.0] are the values at the reference state given in Table 5. The value [[bar.s].sub.0.sup.Lem] = -196.1375815 kJ /(kmol * K) results from shifting the reference state used for moist air.

The IAPWS-IF97 (IAPWS 2007) formulation is used to calculate the molar ideal-gas entropy for water vapor at temperatures T [greater than or equal to] 273.15 K:

[[bar.s].sub.w.sup.o] = [[bar.R].sup.97][[tau][([[partial derivative][[gamma].sub.2.sup.o]]/[[partial derivative][tau]]).sub.[pi]]-[[gamma].sub.2.sup.o]] (56)

where [[bar.R].sup.97] is the molar gas constant of IAPWS-IF97 given in Table 3, [pi] is the reduced pressure, [tau] is the reciprocal reduced temperature, [[gamma].sub.2.sup.o] is the ideal-gas part of the reduced fundamental equation of IAPWS-IF97, Equation 4, and [([partial derivative][[gamma].sub.2.sup.o]/[partial derivative][tau]).sub.[pi]] is its derivative, described in detail in Herrmann et al. (2008).

The IAPWS-95 (IAPWS 1995) formulation is used to calculate the molar ideal-gas entropy for water vapor at temperatures T [less than or equal to] 273.15 K as follows:

[[bar.s].sub.w.sup.o] = [[bar.R].sup.95][[tau][([[partial derivative][[alpha].sup.o]]/[[partial derivative][tau]]).sub.[delta]]-[[alpha].sup.o]], (57)

where [[bar.R].sup.95] is the molar gas constant of IAPWS-95, given in Table 2, [[alpha].sup.o] is the ideal-gas part of the reduced fundamental equation of IAPWS-95, Equation 2, [([partial derivative][[alpha].sup.o]/[partial derivative][tau]).sub.[pi]] is its derivative, and [tau] is the reciprocal reduced temperature. The reduced density [delta] = [[bar.[rho]].sub.w.sup.o]/[[bar.[rho]].sub.c] is calculated using [[bar.[rho]].sub.w.sup.o] = [p.sub.0]/([[bar.R].sup.95]T) with [p.sub.0], given in Table 5, and [[bar.[rho]].sub.c] = [[rho].sub.c]/[M.sup.95] with [[rho].sub.c] and [M.sup.95], given in Table 2. Herrmann et al. (2008) comprises a detailed description of the algorithm.

The air-specific entropy s can be obtained from the molar entropy [bar.s] calculated from Equation 53 using s = [bar.s](1+W)/M.

Compressibility Factor. The compressibility factor Z = [bar.p]v/[bar.R]T for moist air results from the following equation:

Z(T, [bar.v], [[psi].sub.w]) = 1 + [[B.sub.m]/[bar.v]] + [[C.sub.m]/[[bar.v].sup.2]] (58)

where [[psi].sub.w] is the mole fraction of water vapor, [bar.v] is the molar volume of moist air calculated iteratively from Equation 47, [B.sub.m] and [C.sub.m] are the second and third molar virial coefficients of the mixture, given by Equations 10 and 11. The mole fraction of water vapor ([[psi].sub.w]) can be calculated from given humidity ratio W using Equation 41. Herrmann et al. (2008) comprises a detailed description of the algorithm.

Dew-Point Temperature and Frost-Point Temperature. The dew-point temperature (T [greater than or equal to] 273.15 K) or the frost-point temperature (T [less than or equal to] 273.15 K) is calculated iteratively using the following equation:

[p.sub.w] = f(p, [T.sub.d])[p.sub.w, s]([T.sub.d]) (59)

where f(p,[T.sub.d]) is the vapor-pressure enhancement factor calculated using Equation 33 for given total pressure p and dew-point or frost-point temperature [T.sub.d]. The quantity [p.sub.w,s] is the saturation pressure of pure water, calculated for T = [T.sub.d] using Equation 30 or Equation 31, respectively. In Equation 59, the partial pressure of water vapor in moist air ([p.sub.w]) is calculated for total pressure p and humidity ratio W or the mole fraction of water vapor ([[psi].sub.w]) using the following equation:

[p.sub.w] = [[psi].sub.w]p = [W/[[epsilon] + W]]p, (60)

where [epsilon] is the quotient of the molar masses of dry air and water, Equation 14. Herrmann et al. (2008) comprises a detailed description of the algorithm.

For calculating dew-point or frost-point temperature [T.sub.d] for given total pressure p and given humidity ratio W or given mole fraction of water vapor [[psi].sub.w], Equations 59 and 60 in connection with Equations 33, 30, or 31 have to be solved iteratively. The iterative calculation is also required, when determining humidity ratio W or mole fraction of water vapor [[psi].sub.w] for given total pressure p and given dew-point temperature or frost-point temperature [T.sub.d].

Wet-Bulb Temperature and Ice-Bulb Temperature. The wet-bulb temperature (T [greater than or equal to] 273.15 K) or ice-bulb temperature (T [less than or equal to] 273.15 K) is calculated using the following equation:

h(p, T, W) = [h.sub.wb, s](p, [T.sub.wb], [W.sub.wb, s]) + (W-[W.sub.wb, s])[h.sub.w](p, [T.sub.wb]) (61)

where h is the air-specific enthalpy of unsaturated moist air at total pressure p, dry-bulb temperature T, and humidity ratio W. The quantity [h.sub.wb,s] is the air-specific enthalpy of saturated moist air at wet-bulb or ice-bulb temperature [T.sub.wb], and [h.sub.w] is the specific enthalpy of liquid water or ice at wet-bulb or ice-bulb temperature. The humidity ratio ([W.sub.wb,s]) of saturated moist air at wet-bulb temperature can be calculated using the following equation:

[W.sub.wb, s] = [epsilon][[p.sub.wb, s]/[(p-[p.sub.wb, s])]] (62)

with

[p.sub.wb, s] = [f.sub.wb][p.sub.wb, w, s] (63)

where [epsilon] is the quotient of the molar masses of dry air and water, Equation 14, [f.sub.wb](p,[T.sub.wb]) is the vapor-pressure enhancement factor calculated using Equation 33 for given total pressure p and temperature [T.sub.wb], and [p.sub.wb,w,s] is the saturation pressure of pure water, calculated for T = [T.sub.wb] using Equation 30 or Equation 31, respectively. Herrmann et al. (2008) comprises a detailed description of the algorithm.

For calculating wet-bulb or ice-bulb temperature [T.sub.wb] for given total pressure p, dry-bulb temperature T, and humidity ratio W, Equation 61 in connection with Equations 62, 63, 33, 30, or 31 have to be solved iteratively. This is also true when calculating humidity ratio W of dry-bulb state for given total pressure p, dry-bulb temperature T, and wet-bulb or ice-bulb temperature [T.sub.wb].

COMPARISON OF THE NEW ALGORITHM WITH EXPERIMENTAL DATA

This section contains comparisons that were carried out between experimental data for the partial pressure of water vapor of saturated moist air available in the literature (not from this paper) to values calculated from the algorithm developed in this work (abbreviated as HKG) for the thermodynamic properties of moist air. In addition, models developed by Hyland and Wexler (1983a, 1983b), Rabinovich and Beketov (1995), and Yan et al. (Ji et al. 2003a; Ji and Yan 2003b; Ji and Yan 2006) are included in these comparisons.

In Figure 3, the deviations of values calculated using the above-mentioned models, except the model of Yan (YAN), in comparison with the experimental data from Pollitzer and Strebel (1924) at 323 K as well as from Webster (1950) at 273 K and 288 K are plotted over total pressure. The deviation of the new algorithm (HKG) is smaller than that of the Hyland and Wexler model (HW). The deviations of the values obtained from the Rabinovich and Beketov model (RB) compared with the experimental data are similar to that of the HKG model, except at 273 K.

[FIGURE 3 OMITTED]

Figure 4 shows deviations of values calculated using the considered models compared with experimental data from Hyland and Wexler (1973) at 303 K, 313 K, and 323 K, as well as from Hyland (1975), at 343 K, plotted over total pressure. It is obvious that the new model agrees with the experimental data for pressures below 6 MPa. Above this pressure, the values for the saturation partial pressure of water calculated using the new model and using the model by Hyland and Wexler are greater than the experimental data, while the Rabinovich and Beketov and Yan (YAN) models do not follow this trend.

[FIGURE 4 OMITTED]

Figure 5 illustrates the deviations of values calculated using the models included in these comparisons to experimental data from Wylie and Fisher (1996) at 293 K, 323 K, and 348 K plotted over total pressure. All models show good agreement with the experimental data. Again, at high pressures the model presented in this work calculates values for the partial pressures, which are greater than the experimental data. The largest deviation shows Rabinovich and Beketov's model at 293 K and 348 K.

[FIGURE 5 OMITTED]

COMPARISON OF THE NEW ALGORITHM WITH OTHER MODELS

Figure 6 shows deviations of values for the density of moist air calculated from different models compared with density values calculated from the model presented in this work. The deviations were plotted for several total pressures from 0.1 to 10 MPa over mole fractions from 0 to 0.5 mol/mol. The models and algorithms listed in Table 6 are considered in the comparisons. The zero lines in the diagrams of Figure 6 represent the model proposed in this work.
Table 6. Models and Algorithms for Calculating Thermodynamic Properties
of Moist Air Used in the Comparisons Carried Out in This Work

Abbr.            Model                      References

HW       Virial equation for the     Hyland and Wexler (1983a,
                 mixture                       1983b)

RB       Virial equation for the   Rabinovich and Beketov (1995)
                 mixture

YAN      Modified Redlich-Kwong    Yan (Ji et al. 2003a; Ji and
       equation of state for the    Yan 2003b; Ji and Yan 2006)
                 mixture

HuAir  Ideal mixture of the real     Kretzschmar et al. (2005)
       fluids, dry air, and water
       and Nelson-Sauer model for
          the saturation state

SKU    Modified ideal mixture of       Herrmann et al. (2009)
       the real fluids, dry air,
       and water and Nelson-Sauer
        model for the saturation
                 state


As can be seen, the YAN model is not able to describe the density of dry air ([[psi].sub.w] = 0) accurately. At ambient pressure, no model differs more than -0.03% from the model presented in this work. The new model shows only small deviations compared with the HW model, because both models are based on the virial approach. The YAN and RB models show different behaviors compared to the new model, especially at higher pressures. While YAN deviates up to 1.2% at 10 MPa and [[psi].sub.w] = 0, the RB model deviates up to -2.1% at 10 MPa and [[psi].sub.w] = 0. With increasing pressure, the deviations of the ideal-mixture models HuAir and SKU to the new model increase up to -1% at [[psi].sub.w] = 0. The model for calculating thermodynamic properties of moist air proposed in this work shows reasonable behavior when compared to other models.

In addition, ASHRAE RP-1485 (Herrmann et al. 2008) comprises detailed results for comparisons of the developed algorithm with the model of Hyland and Wexler model (1983a, 1983b) for the thermodynamic properties enthalpy, entropy, dew-point temperature, and wet-bulb temperature of moist air.

THE MODEL'S RANGE OF VALIDITY

Table 7 shows the model's range of validity for the thermodynamic properties of moist air described in this section.
Table 7. Range of Validity of the Presented Model

Property                    Range of Validity

Pressure     0.00001  [less        p        [less    10
                      than                  than
                      or                    or
                      equal                 equal
                      to]                   to]

Temperature  -143.15  [less        t        [less    350
                      than                  than
                      or                    or
                      equal                 equal
                      to]                   to]

Humidity        0     [less        W        [less    10
ratio                 than                  than
                      or                    or
                      equal                 equal
                      to]                   to]

Mole            0     [less  [[psi].sub.w]  [less  0.94145
fraction of           than                  than
water vapor           or                    or
                      equal                 equal
                      to]                   to]

Relative        0     [less      [phiv]     [less     1
humidity              than                  than
                      or                    or
                      equal                 equal
                      to]                   to]

Dew-point    -143.15  [less        t        [less    350
temperature           than                  than
                      or                    or
                      equal                 equal
                      to]                   to]

Wet-bulb     -143.15  [less        t        [less    350
temperature           than                  than
                      or                    or
                      equal                 equal
                      to]                   to]

Property

Pressure              MPa

Temperature        [degrees]C

Humidity     [kg.sub.w]/[kg.sub.a]
ratio

Mole           [kmol.sub.w]/kmol
fraction of
water vapor

Relative        (decimal ratio)
humidity

Dew-point          [degrees]C
temperature

Wet-bulb           [degrees]C
temperature


UNCERTAINTY OF THE MODEL

The model presented in this work utilizes improved correlations for the properties of dry air and water applied to the basic underlying virial model of Hyland and Wexler (1973, 1983a, 1983b). By using improved correlations, the uncertainties of the new model are slightly less than the Hyland and Wexler model uncertainties.

The accuracy of the presented model is greater than that of Nelson and Sauer (2002) because the fitted polynomials for the ideal-gas parts of heat capacity, enthalpy, and entropy for dry air have been replaced by the fundamental equation of Lemmon et al. (2000). Instead of the fitted polynomials for virial coefficients for dry air and steam of Nelson and Sauer, the fundamental equations of Lemmon et al. and of IAPWS-95 (IAPWS 1995) are used in the ASHRAE RP-1485 model. In addition, the equation for the second Nelson and Sauer cross-virial coefficient ([B.sub.aw]) has been replaced with the more accurate Harvey and Huang equation (2007). Furthermore, the fitted polynomials for the isothermal compressibility of liquid water and for Henry's constant in the calculation of the enhancement factor have been replaced by the IAPWS-IF97 (IAPWS 2007) and the IAPWS Guideline 2004 (IAPWS 2004) equations. Finally, the current values for the universal molar gas constant and the molar masses for dry air and water are used in this work.

The uncertainties of selected properties calculated using the presented model are listed in Table 8.
Table 8. Uncertainty of the Presented Model

Property                          Uncertainty

Density   0.1%   [less than or equal  0.3% (moist air with W
          (dry           to]                   = 0.1
          air)   [DELTA][rho]/[rho]   [kg.sub.w]/[kg.sub.a])
                 [less than or equal
                         to]

Isobaric  1.0%   [less than or equal  3.0% (moist air with W
heat      (dry      to] [DELTA]                = 0.1
capacity  air)   [c.sub.p]/[c.sub.p]  [kg.sub.w]/[kg.sub.a])
                 [less than or equal
                         to]

Enthalpy    2    [less than or equal  kJ/kg (moist air with W
          kJ/kg  to] [DELTA]h [less            = 0.1
          (dry   than or equal to] 3  [kg.sub.w]/[kg.sub.a])
          air)


For moist air with a higher humidity ratio than 0.1 [kg.sub.w]/[kg.sub.a] uncertainties cannot be given because no experimental data are available.

TABLES OF PROPERTY VALUES CALCULATED USING THE MODEL

Tables including the following thermodynamic properties calculated using the algorithms of this paper are available at www.thermodynamics-zittau.de and in "Publications":

* Saturation properties [W.sub.s], [v.sub.s], [h.sub.s], and [s.sub.s] of moist air for temperatures from -60[degrees]C to 90[degrees]C at atmospheric pressure (0.101325 MPa)

* Properties [t.sub.wb], v, h, and s of moist air for humidity ratio from 0 to 1 [kg.sub.w]/[kg.sub.a] at 200[degrees]C and 320[degrees]C for pressures 0.101325, 1, 2, 5, and 10 MPa.

SUMMARY AND CONCLUSION

This research updates the modeling of moist air as a real gas using the virial equation of state. All of the latest IAPWS standards and NIST publications for dry air, steam, water, ice, and calculating virial coefficients for moist air were used. Therefore, the accuracy of the new model is improved compared to the Hyland-Wexler (1973, 1983a, 1983b) and Nelson-Sauer (2002) models. The range of validity of the new model is in pressure from 0.01 kPa up to 10 MPa, in temperature from 130 K up to 623.15 K, and in humidity ratio from 0 up to 10 [kg.sub.w]/[kg.sub.a].

Tables containing values for thermodynamic properties calculated using the algorithm of this paper are available at www.thermodynamics-zittau.de under the "Publications" tab.

The new model was used to produce moist air and [H.sub.2]O saturation property tables for the psychrometric chapter in the 2009 ASHRAE Handbook--Fundamentals (ASHRAE 2009), which is the first update of these tables since 1985. This new moist air table is close to the 1985 table, because moist air at ambient pressure behaves essentially as an ideal gas; also the underlying data used by Hyland and Wexler was quite accurate. Greater deviations to the former models occur at higher pressures and temperatures.

The following subjects are of interest for further investigations. The third cross-virial coefficients for air-water interactions can be improved. The GERG-2004 equation (Kunz et al. 2007) should be considered in the development of new algorithms for the thermodynamic properties of moist air.

The algorithms for calculating thermodynamic properties of moist air proposed in this paper are implemented in the property library LibHuAirProp, which can be requested from the authors (see www.thermodynamics-zittau.de).

ACKNOWLEDGMENTS

The authors would like to thank ASHRAE, the sponsor of RP-1485, and ASHRAE Technical Committee 1.1, which approved the project. Particular acknowledgment should also be given to the members of the project monitoring subcommittee: R.M. Nelson (Chairman), R. Crawford, A. Jacobi, T.H. Kuehn, and V.W. Peppers.

NOMENCLATURE

a = specific Helmholtz energy

B = second molar virial coefficient

C = third molar virial coefficient

[c.sub.p] = isobaric heat capacity

f = vapor-pressure enhancement factor

g = specific Gibbs energy

h = specific enthalpy, air-specific enthalpy

m = mass

M = molar mass, molar mass of the mixture

p = pressure

R = specific gas constant

[bar.R] = universal molar gas constant

s = specific entropy, air-specific entropy

t = Celsius temperature

T = Kelvin temperature

v = specific volume, air-specific volume

W = humidity ratio

Z = compressibility factor (real gas factor)

[alpha] = reduced Helmholtz energy

[[beta].sub.H] = Henry's law constant

[gamma] = reduced Gibbs energy

[delta] = reduced density

[epsilon] = ratio of the molar mass of water and dry air

[theta] = reduced temperature

[[kappa].sub.T] = isothermal compressibility

[pi] = reduced pressure

[rho] = mass density

[tau] = reciprocal reduced temperature

[phiv] = relative humidity

[psi] = mole fraction

Superscripts

Lem = value taken from Lemmon et al. (2000)

o = ideal-gas part

r = residual part

95 = value taken from the scientific formulation IAPWS-95 (IAPWS 1995)

97 = value taken from the industrial formulation IAPWS-IF97 (IAPWS 2007)

06 = value taken from the IAPWS-06 (IPAPWS 2006) formulation for ice

08 = value taken from the IAPWS-08 (IAPWS 2008) formulation for ordinary water substance

[bar] = molar property

* = reducing quantity

Subscripts

a = dry air

c = critical

d = dew point

j = maxcondentherm

m = mixture property

s = saturation state

sub = sublimation state

t = triple point

w = water (water vapor, liquid water, or ice)

wb = wet-bulb

0 = reference state

1, 2 = regions of IAPWS-IF97 (IAPWS 2007)

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Sebastian Herrmann

Student Member ASHRAE

Donald P. Gatley, PE

Fellow/Life Member ASHRAE

Hans-Joachim Kretzschmar, PhD

Member ASHRAE

Sebastian Herrmann is a doctoral student and Hans-Joachim Kretzschmar is a professor at the Zittau/Goerlitz University of Applied Sciences, Zittau, Germany. Donald P. Gatley is president of Gatley & Associates, Inc., Atlanta, GA.

Received February 14, 2009; accepted May 6, 2009

This paper is based on findings resulting from ASHRAE Research Project RP-1485.
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Geographic Code:1USA
Date:Sep 1, 2009
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