W.Wagner1 J.R.Cooper2 A.Dittmann3
J.Kijima4 H.-J.Kretzschmar5
A.Kruse6
R.Maresˇ7
K.Oguchi4
H.Sato8
I.Sto¨cker5
O.Sˇifner9
Y.Takaishi4
I.Tanishita4
J.Tru¨benbach3 Th.Willkommen3The IAPWS Industrial Formulation1997for the Thermodynamic Properties of Water and Steam
In1997,the International Association for the Properties of Water and Steam(IAPWS) adopted a new formulation for the thermodynamic properties of water and steam for industrial use.This new formulation,called IAPWS Industrial Formulation1997for the Thermodynamic Properties of Water and Steam(IAPWS-IF97),replaces the previous industrial formulation,IFC-67,that had formed the basis for power-plant calculations and other applications in energy engineering since the late1960’s.IAPWS-IF97improves significantly both the accuracy and the speed of the calculation of the thermodynamic properties compared with IFC-67.The differences between IAPWS-IF97and IFC-67will require many users,particularly boiler and turbine manufacturers,to modify design and application codes.This paper summarizes the need and the requirements for such a new industrial formulation and gives the entire numerical information about the individual equations of IAPWS-IF97.Moreover,the scientific basis for the development of the equations is summarized and the achieved quality of IAPWS-IF97is presented regarding the three criterions accuracy,consistency along region boundaries,and computation speed.For comparison,corresponding results for the previous standard IFC-67are also
presented.
1Introduction
动力
环境监控In the1960s an industrial formulation for the thermodynamic properties of water and steam was developed.This was called“The 1967IFC Formulation for Industrial Use”(IFC-67)[1].IFC-67 was formally recognized for the calculation of thermodynamic properties of water and steam for official use such as performance guarantee calculations of power cycles.In addition to this,IFC-67was used for innumerable other industrial applications.However, compared with today’s requirements IFC-67contains a number of weaknesses.Moreover,because of the progress that has been achieved in mathematical methods to develop accurate equations of state,a number of reasons warranted the development of a new industrial formulation.This newly developed formulation was adopted by the Interna-tional Association for the Properties of Water and Steam(IAPWS) at its meeting in Erlangen(Germany),September1997,under the name“IAPWS Industrial Formulation1997for the Thermody-namic Properties of Water and Steam”abbreviated to“IAPWS Industrial Formulation1997”or even shorter IAPWS-IF97[2]. Since this date IAPWS-IF97has been officially valid.However, due to the need to modify design and application
codes,IAPWS has recommended an introductory period,lasting until January1, 1999,during which IAPWS-IF97should not be used for contrac-tual commitments.
This article contains details relevant to the development of IAPWS-IF97,the full numerical information on the individual equations needed for their use,details of their accuracy,consis-tency along region boundaries,and results of computing-time investigations in comparison with IFC-67.
1Ruhr-Universita¨t Bochum,Lehrstuhl fu¨r Thermodynamik,D-44780Bochum,
Germany,corresponding author
2Queen Mary and Westfield College,Department of Engineering,London,United
Kingdom
3Technische Universita¨t Dresden,Institut fu¨r Thermodynamik und Technische
Geba¨udeausru¨stung,Dresden,Germany
4Kanagawa Institute of Technology,Faculty of Engineering,Atsugi,Japan
5Hochschule Zittau/Go¨rlitz(FH),Fachgebiet Technische Thermodynamik,Zittau,
Germany
6Ruhr-Universita¨t Bochum,Lehrstuhl fu¨r Thermodynamik,Bochum,Germany
Current address:Bayern Innovativ GmbH,Nu¨rnberg,Germany
7University of West Bohemia,Department of Thermodynamics,Plzen,Czech
Republic
8Keio University,Faculty of Science&Technology,Yokohama,Japan
9Academy of Sciences of Czech Republic,Institute of Thermomechanics,Prague,
Czech Republic
150/Vol.122,JANUARY2000Transactions of the ASME
Copyright©2000by ASME
Those who are only interested in the numerical information needed to use the equations of IAPWS-IF97can find this infor-mation in compact form in several steam tables,for example [3,4,4a],or in the release on IAPWS-IF97[2].
2Need for the Development of the New Industrial Formulation IAPWS-IF97
In order to demonstrate the need for a new industrial formula-tion,the characteristics of the industrial formulation IFC-67are considered.As shown in Fig.1,the entire range of validity (0°C Յt Յ800ЊC for p Յ100MPa)was divided into five regions with separate equations.For each of the regions 1(liquid)and 2(vapor)there was an equation of the specific Gibbs free energy g as function of pressure p and temperature T ,namely g (p ,T ).Each of the regions 3and 4was covered by an equation of the specific Helmholtz free energy f as function of specific volume v and temperature T ,namely f (v ,T ).The fifth region was the saturation curve for which a saturation-pressure equation p s (T )was given.Based on the equations for the industrially most important regions 1,2,and 5,the following properties could be directly calculated with IFC-67as a function of p and T :specific volume v ,specific enthalpy h ,specific entropy s and specific isobaric heat capacity c p ,and in addition the saturation pressure p s as a function of T .If other combinations of variables were of interest,for
example the combinations v (p ,h ),T (p ,h ),s (p ,h ),h (p ,s ),T (p ,s ),and T s (p )for turbine-expansion calculations,these had to be determined via corresponding iterations.Due to these iterations in combination with a relatively complex structure of the equa-tions,calculations for the complete power cycle with the whole set of the IFC-67equations required relatively long computing times.Nowadays,with modern mathematical tools to establish effec-tive structures of such property formulations [5],the long process-calculation times with the IFC-67equations became a real weak-ness of this formulation.Besides this main disadvantage of IFC-67there were several other deficiencies which are summarized as follows:
1
印度
女警察For certain regions IFC-67no longer met the present standard of accuracy.2For some properties there were considerable inconsistencies at region boundaries.
3The technically important property speed of sound w was not incorporated in the set of the IFC-67equations.
4IFC-67was not based on the current temperature scale ITS-90[6].
5IFC-67was based on earlier data and was therefore not con-nected to the current scientific standard of IAPWS for the thermodynamic properties of ordinary water substance,the IAPWS-95formulation [7,8].More details concerning the above mentioned items can be seen in the figures of Section 5.5.3Administrative Measures of IAPWS for the Devel-opment and Examination of IAPWS-IF97
Due to the weaknesses of the IFC-67formulation listed in the previous section,at the IAPWS meeting in Buenos Aires in 1990it was decided that a set of new fast equations should be developed for industrial calculations.This new industrial formulation,later called IAPWS-IF97,should then replace the industrial formulation IFC-67.In order to develop the entire equation package in an international collaboration,the Task Group “New Industrial For-mulation”was established.It consisted of the following 12mem-bers from seven countries:W.Wagner (Chairman,Germany),A.Alexandrov (Russia),J.R.Cooper (United Kingdom),A.Ditt-mann (Germany),J.Gallagher (USA),P.G.Hill (Canada),H.-J.
Nomenclature
a ,
b ,c
ϭadjustable parameters
c p ϭspecific isobaric heat capacity c v ϭspecific isochoric heat capacity
d ϭadjustabl
e parameter
f
ϭ
specific Helmholtz free energy,f ϭu ϪTs
g ϭspecific Gibbs free energy,g ϭh ϪTs h ϭspecific enthalpy I ϭexponent i ϭserial number J ϭexponent j ϭserial number M ϭmolar mass n ϭcoefficient p ϭpressure
R ϭspecific gas constant R m
ϭ
molar gas constant
s ϭspecific entropy
T ϭthermodynamic temperature 10t ϭCelsius temperature,t ϭT Ϫ
273.15K
u ϭspecific internal energy v ϭspecific volume w ϭspeed of sound
ϭtransformed pressure,Eqs.(27a )and
(55a )
␥ϭdimensionless Gibbs free energy,
␥ϭg /(RT )
⌬ϭdifference in any quantity ␦ϭreduced density,␦ϭ/*ϭreduced enthalpy,ϭh /h *ϭreduced temperature,ϭT /T *ϭtransformed temperature,Eq.(27b )ϭreduced pressure,ϭp /p *ϭmass density
ϭreduced entropy,ϭs /s *
ϭinverse reduced temperature,ϭ
T */T
ϭdimensionless Helmholtz free en-ergy,ϭf /(RT )Superscripts o ϭideal-gas part;ideal gas r ϭresidual part
*ϭreducing quantity Јϭsaturated liquid state Љϭsaturated vapor state Subscripts
b ϭnormal boiling point
c ϭcritical point max ϭmaximum value s ϭsaturation state t ϭtriple point tol
ϭ
tolerated
10
All temperature values given in this article are temperatures according to the International Temperature Scale of 1990
(ITS-90)
Fig.1Regions and equations of IFC-67.The boundary between regions 2and 3is described by the L-function
Journal of Engineering for Gas Turbines and Power JANUARY 2000,Vol.122/151
Kretzschmar(Germany),R.Maresˇ(Czech Republic),K.Oguchi (Japan),H.Sato(Japan),O.Sˇifner(Czech Republic),and J.T.R. Watson(United Kingdom).This group was responsible for the development of IAPWS-IF97with respect to organizational ques-tions of scientific nature(structure of the formulation,selection of the individual equations,official report to IAPWS,etc.).Thefinal form of IAPWS-IF97is based on contributions and equations by the scientists who form the group of authors of this article.
The entire project was continuously supervised by the IAPWS Working Group“Industrial Calculations”of which many members are representatives of international companies involved in the power industry.This working group,chaired by B.Rukes(Ger-many),is the successor of the IAPWS“Subcommittee on Indus-trial Calculations.”The progress of the development of IAPWS-IF97was continuously discussed in joint sessions of this group with the IAPWS Working Group“Ther
mophysical Properties of Water and Steam”which was chaired by J.R.Cooper(United Kingdom).
Finally,at the IAPWS meeting in Paris in1995the Task Group “New Industrial Formulation-Evaluation”was founded.This
group was responsible for the examination of IAPWS-IF97and
consisted of the following members:K.Miyagawa,(Chairman,
previously Fuji Electric,Japan),H.W.Bradly(Bradly Associates,
United Kingdom),R.B.McClintock(previously General Electric,
USA),I.Kodl(Skoda,Czech Republic),W.T.Parry(General
Electric,USA),C.Perstrup(Elsam Project,Denmark),B.Rukes
(Siemens KWU,Germany),M.Scala(Ansaldo,Italy),P.F.Smith
(GEC Alstom Power Generation,United Kingdom),and R.C.
我回到了清朝Spencer(previously General Electric,USA).
4Requirements for IAPWS-IF97
The requirements for the industrial formulation IAPWS-IF97
are based on the proposal of the former“Subcommittee on Indus-
trial Calculations,”which was agreed with the Task Group“New
Industrial Formulation.”The main items are summarized in the
following sections.
4.1Range of Validity.The entire set of equations of
IAPWS-IF97should have the same range of validity as given for
IFC-67,which is defined by the following temperature and pres-
sure range:
0ЊCՅtՅ800ЊC pՅ100MPa.
For high-temperature applications such as in gas turbines the
following extension of the range of validity was requested:
800ЊCՅtՅ2000ЊC pՅ10MPa.
4.2Accuracy.For the properties specific volume v,specific
enthalpy h,and saturation pressure p s,IAPWS-IF97should gen-
erally meet the corresponding values from the scientific standard,
the“IAPWS Formulation1995for the Thermodynamic Properties
of Ordinary Water Substance for General and Scientific Use”[7,
8],hereafter abbreviated to IAPWS-95,within the tolerances of the
International Skeleton Tables IST-85in its version of1994[9].
Roughly summarizing,the relevant IST-85tolerances are,depen-
dent on the state range,for v betweenϮ0.01percent andϮ0.3
percent,for h betweenϮ0.1percent andϮ0.3percent,and for
p sϮ0.025percent.Based on extremely accurate experimental data,in the liquid region for pϽ1MPa the IST-85tolerances[9]
for v and h and on the saturation curve for tϽ100ЊC the
tolerances for p s are extraordinarily small;here,the smallest
tolerances areϮ0.001percent in v,Ϯ0.03percent in h,and Ϯ0.002percent in p s.However,in view of the technical demands in this range,the permitted tolerances to the IAPWS-95values were increased toϮ0.01percent in v,toϮ0.1percent in h,and to Ϯ0.025percent in p s.For the specific isobaric heat capacity c p and the speed of sound w,IAPWS-IF97should represent the values from IAPWS-95to withinϮ1percent except for the range very near the critical point where clearly larger deviations were al-lowed.By taking as reference the IAPWS-95formulation[7,8]the agreement between the industrial formulation and the scientific formulation of IAPWS is ensured.兖州
地震 Besides the representation of v,h,c p,and w for the(stable) homogeneous regions including saturatio
n,the specification in-cluded the requirement that the equations should also yield rea-sonable values for metastable states close to the stable regions.
4.3Maximum Inconsistencies at Region Boundaries. With regard to the continuity at the region boundaries(see Fig.2), reference is made to the so-called Prague values[10].These Prague values,established for IFC-67,give permissible differences in the property values along the region boundaries when calculat-ing these properties from all equations valid at the corresponding boundary.The continuity requirements are as follows:
(a)Single Phase.
Specific volume:⌬vϭϮ0.05percent
Enthalpy:⌬hϭϮ0.2kJ kgϪ1
Heat capacity:⌬c pϭϮ1percent
Entropy:⌬sϭϮ0.2J kgϪ1KϪ1
Gibbs free energy:⌬gϭϮ0.2kJ kgϪ1
Speed of sound:⌬wϭϮ1percent
(b)Saturation.
Saturation pressure:⌬p sϭϮ0.05percent
Saturation temperature:⌬T sϭϮ0.02percent
Gibbs free energy:⌬gϭϮ0.2kJ kgϪ1
4.4Increase of Computation Speed.This item was the most important demand for IAPWS-IF97.The main requirement regarding the computation speed was that the calculation of all property functions listed in Table1for regions1,2,and4should be altogether three times faster than with IFC-67;for the definition of the individual regions of IAPWS-IF97see Fig.2.Table1is based on a survey made by the“Subcommittee on Industrial Calculations”among the international power-cycle companies and related industries.In addition to the most important property functions for these regions the table also gives the average fre-quencies of use of the corresponding functions.
For regions3and5of IAPWS-IF97the computation-speed requirements only related to a few functions(see Table18),where these functions are not combined with frequency-of-use values. For re
gion3,corresponding to regions3and4of IFC-67,it was only necessary that IAPWS-IF97was not slower than with IFC-67. For region5,the computing-time requirements related only to 1073.15K,the maximum temperature for which IFC-67was valid. For this isotherm IAPWS-IF97should be three times faster than with
IFC-67.
Fig.2Regions and equations of IAPWS-IF97.The boundary between regions2and3is described by the B23-equation,see section5.3
152/Vol.122,JANUARY2000Transactions of the ASME
In order to perform all these computing-time investigations, special benchmark programs for a specified PC and compiler were developed.These programs took into account the frequencies of use(if any)of the corresponding property functions.
5The IAPWS Industrial Formulation1997
This section gives full information about the IAPWS Industrial Formulation1997(IAPWS-IF97)covering all numerical details needed for the use of the individual equations,statements on their development and details concerning accuracy,consistency along the region boundaries,and computation speed of IAPWS-IF97 compared with the previous industrial standard IFC-67.
5.1Concept and Structure of IAPWS-IF97.The IAPWS Industrial Formulation1997consists of a set of equations for different regions which cover the following range of validity: 273.15KՅTՅ1073.15K pՅ
100MPa
1073.15KՅTՅ2273.15K pՅ10MPa.
Figure2shows in which way the entire range of validity of IAPWS-IF97is covered by its equations.The division into indi-vidual regions is very similar to IFC-67.One difference is that the middle density range is covered by only one region,namely by region 3.The other difference is that there is additionally a high-temperature region,region5.Region4corresponds to the saturation curve.The boundaries of the regions can be directly taken from Fig.2except for the boundary between regions2and 3;this boundary is defined by the so-called B23-equation given in Section5.3.Both regions1and2are individually covered by a fundamental equation for the specific Gibbs free energy g(p,T), region3by a fundamental equation for the specific Helmholtz free energy f(,T),and the saturation curve,corresponding to region 4,by a saturation-pressure equation p s(T).The high-temperature region5is also covered by a g(p,T)equation.Thesefive equations,shown in rectangular boxes in Fig.2,form the so-called basic equations.
In order to meet the main requirement of a short computing time,the entire set of the IAPWS-IF97equations was developed based on the following two-step concept:
1Afterfinding convenient functional terms,the structure of the four basic equations for the homogeneous regions was opti-mized using the method by Setzmann and Wagner[5]in such
a way that the requirements regarding accuracy and consis-
tency along region boundaries were met with equation struc-tures allowing short computing times.In this optimization process,the equations werefitted to input values calculated from the IAPWS-95formulation[7,8].In this way IAPWS-IF97was coupled with the current scientific standard IAPWS-
95.
2All those thermodynamic properties which are not direct func-tions of the independent variables of the basic equations are not found by iteration from the basic equations.Instead of this, so-called backward equations were developed,namely equa-tions T(p,h)and T(p,s)for regions1and2and T s(p)for the saturation curve.With these backward equations,shown in Fig.
2for the corresponding regions,all the functions shown in rectangular boxes in Table1can be calculated without any iteration.For example,if h(p,s)is to be calculated in region 2,first the temperature T is calculated from the backward equation T(p,s)and then h(p,T)can be directly obtaine
d from the corresponding basic equation g(p,T). However,this entire concept required that the numerical con-sistency between the backward and the basic equations was ex-tremely good.Otherwise it would have caused numerical problems when“jumping”back and forth between the basic and the back-ward equations,for example,when calculating the turbine-expansion line of a power-cycle process.Based on test calculations with characteristic power cycles via iterations with IFC-67,the following numerical consistency requirements werefinally set up: (a)The temperature determined from the backward equation
T(p,h)for given values of p and h had to agree with the
temperature value calculated for the same p and h from the
corresponding basic equation g(p,T)within a tolerated
temperature difference⌬T tol.This⌬T tol value amounts to
Ϯ25mK for the entire region1and for region2at entropy
values not greater than5.85kJ kgϪ1KϪ1.For region2at
entropy values greater than5.85kJ kgϪ1KϪ1,the permis-
sible⌬T tol value amounts toϮ10mK;the smaller⌬T tol
inconsistency value in this part of region2(turbine expan-
sion)is particularly important for the power industry. (b)The temperature determined from the backward equation
T(p,s)for given values of p and s had to agree with the
temperature calculated for the same p and s from the
corresponding basic equation g(p,T)within a tolerated
temperature difference⌬T tol.For the tolerated⌬T tol incon-
sistency values with regard to the T(p,s)equation,the
same statement held as given for the T(p,h)equation
under item(a).
(c)The saturation pressure calculated from the saturation-
temperature equation T s(p)was not allowed to deviate by
more than⌬p sϭϮ0.003percent from the p s value
determined from the saturation-pressure equation p s(T). The permissible numerical inconsistencies between the basic and backward equations,summarized under items(a)to(c),were extremely small,namely about one tenth of the uncertainties of the scientific standard IAPWS-95.
5.2Reference Constants.This section summarizes all ref-erence constants needed for evaluating the equations given in Section5.5.
The specific gas constant of ordinary water,
Rϭ0.461526kJ kgϪ1KϪ1,(1) results from the recommended values of the molar gas constant [11],
R mϭ8.31451J molϪ1KϪ1,(2) and from the molar mass of ordinary water,
Mϭ18.015257g molϪ1.(3)
Table1Most important property functions and their frequency of
use
Journal of Engineering for Gas Turbines and Power JANUARY2000,Vol.122/153
The value of M results from the molar masses obtained from isotopic molar masses in[12]and representative isotopic compo-sitions given in[13].
The values of the critical parameters
T cϭ647.096K,(4)
p cϭ22.064MPa,(5)
cϭ322kg mϪ3(6) are from the corresponding IAPWS release[14].The triple-point temperature is
T tϭ273.16K(7) according to the International Temperature Scale of1990(ITS-90) [6]and the triple-point pressure
p tϭ611.657Pa(8) was determined by Guildner et al.[15].According to the scientific standard of the thermodynamic properties of ordinary water,the IAPWS-95formulation[7,8],the temperature of the normal boiling point(at a pressure of0.101325MPa(1atm))amounts to
T bϭ373.1243K.(9) 5.3Auxiliary Equation for the Boundary between Regions 2and3.The boundary between regions2and3(see Fig.2)is defined by the following simple quadratic pressure-temperature relation,the B23-equation
ϭn1ϩn2ϩn32,(10) whereϭp/p*andϭT/T*with p*ϭ1MPa and T*ϭ1 K.The coefficients n1to n3of Eq.(10)are listed in Table A1of the appendix.Equation(10)describes roughly an isentropic line; the entropy values along this boundary line are between sϭ5.047 kJ kgϪ1KϪ1and sϭ5.261kJ kgϪ1KϪ1.
Alternatively Eq.(10)can be expressed explicitly in temperature as
ϭn4ϩ͓͑Ϫn5͒/n3͔0.5(11) withandas defined for Eq.(10)and the coefficients n3to n5 listed in Table A1.Equations(10)and(11)cover the range from 623.15K at16.5292MPa up to863.15K at100MPa.
5.4Functional Forms Adopted for Short Computing Times.Wide-range equations of state in reference quality are nowadays explicit in the Helmholtz free energy as function of density and temperature[7,8,16,17].As functional forms for such equations pure polynomials in density and temperature and particularly such polynomials combined with exponential func-tions in density have proved very successful.
Since,however,for IAPWS-IF97a short computing time was one of the most important criterions,the computing times of selected arithmetic operations were investigated[18].Compared with the two most important basic operations addition and multi-plication,all the other operations are slower by a factor of ten or more.After these tests it was clear that it was only possible to use polynomials in the form of series of additions and multiplications as basic functional forms for the new equations.Based on many tests for the general functional dependency
zϭz͑x,y͒,(12) where,for example,zϭf,xϭ,and yϭT,the following general functional expression has proved most effective[18]:
z͑x,y͒ϭi n iͩx aϩb ͪI iͩy cϩdͪJ i.(13)
This general expression forms the basis for the majority of the
equations of IAPWS-IF97.
Thefinal form of all equations(except for the saturation curve,
region4)of IAPWS-IF97was found by using the structure-
optimization method of Setzmann and Wagner[5]or a modified
version of Wagner’s method[19].These procedures require a
so-called bank of terms from which the best combination of an
optimum number of terms is determined.For the development of
the backward equations,these procedures were combined with
further optimization tools,see later.
5.5The Basic Equations for Regions1to5.For those
homogeneous regions of IAPWS-IF97for which it is thermody-
namically reasonable the corresponding equations of state were
established as function of the“technical”variables pressure p and
temperature T;this is the case for regions1,2,and5.For these
regions,the equations are formulated explicit in the specific Gibbs
free energy g which is,as a function of p and T,a fundamental
equation.Since region3contains the critical point,this region
cannot be reasonably covered by an equation with p and T as
independent variables.However,it can be represented by an equa-
tion as a function of densityand temperature T.Thus,for region
睢宁县李
集中学
3an equation explicit in the specific Helmholtz free energy as a邢钢客商function ofand T is used which is a fundamental equation,also.
One advantage of using fundamental equations(instead of equa-
tions of state in form of p(v,T)and v(p,T),respectively)is that
all thermodynamic properties can be calculated from derivatives of
the equations,no integrations with further information are needed.
If thefirst and second derivatives of g with respect to p and T and
of f with respect toand T,respectively,are correctly represented,
then any thermodynamic property,based on these derivatives
(which is the case for the vast majority of properties),can be
correctly calculated from such fundamental equations.
Proceeding from Eq.(13)with zϭg/(RT),xϭp,aϭp*,
yϭTϪ1,and cϭ(T*)Ϫ1one obtains the following general form
of a so-called combined polynomial for the g(p,T)equations:
g
ϭi n iͩpϩbͪI iͩT*ϩdͪJ i,(14)
where p*and T*are reducing parameters.
In the following sectionsfirst thefinal form of the corresponding
basic equation is given including all numerical information for its
use,then details of its development are summarized andfinally its
accuracy is discussed;all table numbers starting with an“A”are
listed in the appendix.
5.5.1The Gibbs Free Energy Equation for Region1.The
basic equation for this region is a fundamental equation for the
specific Gibbs free energy g.This equation is expressed in dimen-
sionless form,␥ϭg/(RT),and reads
g͑p,T͒
RT
ϭ␥͑,͒ϭiϭ134n i͑7.1Ϫ͒I i͑Ϫ1.222͒J i,(15)
whereϭp/p*andϭT*/T with p*ϭ16.53MPa and T*ϭ
1386K;R is given by Eq.(1).The coefficients n i and exponents
I i and J i of Eq.(15)are listed in Table A2.
All thermodynamic properties can be derived from Eq.(15)by
using the appropriate combinations of the dimensionless Gibbs
free energy␥and its derivatives.The relations of the relevant
thermodynamic properties to␥and its derivatives are summarized
in Table2.All required derivatives of the dimensionless Gibbs free
energy␥,Eq.(15),are explicitly given in Table3.
Since the5th International Conference on the Properties of
Steam in London in1956,the specific internal energy and the
specific entropy of the saturated liquid at the triple point have been
set equal to zero,as follows:
uЈtϭ0;sЈtϭ0.(16)
154/Vol.122,JANUARY2000Transactions of the ASME
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