ISO_TC_193___SC_1_N_344(natural gases).pdf

© ISO 2011 – All rights reserved

Document type: International Standard Document subtype: Document stage: (20) Preparatory Document language: E

ISO 193/SC 1 N 344 Date: 2011-01-19

ISO/CD 20765-2

ISO TC 193/SC 1/WG 13

Secretariat: NEN

Natural Gas — Calculation of thermodynamic properties — Part 2: Single-phase properties (gas, liquid, and dense fluid) for extended ranges of application

Élément introductif — Élément central — Partie 2: Titre de la partie

Warning

This document is not an ISO International Standard. It is distributed for review and comment. It is subject to change without notice and may not be referred to as an International Standard.

Recipients of this draft are invited to submit, with their comments, notification of any relevant patent rights of which they are aware and to provide supporting documentation.

ISO/CD 20765-2

© ISO 2010 - All rights reserved

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ISO/CD 20765-2


iv

Contents

Foreword ............................................................................................................................................................ vi

Introduction ...................................................................................................................................................... vii

1 Scope ..................................................................................................................................................... 1

2 Normative references ........................................................................................................................... 1

3 Terms and definitions........................................................................................................................... 2

4 Thermodynamic basis of the method ................................................................................................. 3 4.1 Principle ................................................................................................................................................. 3 4.2 The fundamental equation based on the Helmholtz free energy ..................................................... 4 4.2.1 Background ........................................................................................................................................... 4 4.2.2 The Helmholtz free energy ................................................................................................................... 4 4.2.3 The reduced Helmholtz free energy .................................................................................................... 4 4.2.4 The reduced Helmholtz free energy of the ideal gas ........................................................................ 5 4.2.5 The pure substance contribution to the residual part of the reduced Helmholtz free energy ..... 6 4.2.6 The departure function contribution to the residual part of the reduced Helmholtz free energy 6 4.2.7 Reducing functions .............................................................................................................................. 7 4.3 Thermodynamic properties derived from the Helmholtz free energy ............................................. 7 4.3.1 Background ........................................................................................................................................... 8 4.3.2 Relations for the calculation of thermodynamic properties in the homogeneous region ............ 8

5 Method of calculation ......................................................................................................................... 10 5.1 Input variables ..................................................................................................................................... 10 5.2 Conversion from pressure to reduced density ................................................................................ 10 5.3 Implementation ................................................................................................................................... 11

6 Ranges of application......................................................................................................................... 11 6.1 Pure gases ........................................................................................................................................... 12 6.2 Binary mixtures ................................................................................................................................... 13 6.3 Range of validity for natural gases ................................................................................................... 15

7 Uncertainty of the equation of state ................................................................................................. 17 7.1 Background ......................................................................................................................................... 17 7.2 Uncertainty for pure gases ................................................................................................................ 17 7.3 Uncertainty for binary mixtures ........................................................................................................ 20 7.4 Uncertainty for natural gases ............................................................................................................ 22 7.5 Uncertainties in other properties ...................................................................................................... 24 7.6 Impact of uncertainties of input variables........................................................................................ 24

8 Reporting of results ............................................................................................................................ 25

Annex A (normative) ........................................................................................................................................ 26

Annex B (normative) The reduced Helmholtz free energy of the ideal gas .............................................. 29 B.1 Calculation of the reduced Helmholtz free energy of the ideal gas .............................................. 29 B.2 Derivatives of the reduced Helmholtz free energy of the ideal gas ............................................... 30

Annex C (normative) Values of critical parameters and molar masses of the pure components .......... 35

Annex D (normative) The residual part of the reduced Helmholtz free energy ........................................ 36 D.1 Calculation of the residual part of the reduced Helmholtz free energy ........................................ 36 D.1.1 Derivatives of the residual part of the reduced Helmholtz free energy ........................................ 36 D.2 Calculation of the pure substance contribution to the residual part of the reduced Helmholtz

free energy ........................................................................................................................................... 37

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v

D.2.1 Derivatives of ),(r

o i wi ..................... 37

D.2.2 Coefficients and exponents of ),(ro i ......................................................................................... 39

D.3 Calculation of the departure function contribution to the residual part of the reduced Helmholtz free energy ......................................................................................................................... 44

D.3.1 Binary specific departure functions .................................................................................................. 44 D.3.2 Generalised departure functions ....................................................................................................... 44 D.3.3 No departure functions ....................................................................................................................... 44

D.3.4 Derivatives of ),(r ij with respect to the reduced mixture variables and . .......................... 45

D.3.5 Coefficients, exponents, and parameters for the departure functions .......................................... 46

Annex E (normative) The reducing functions for density and temperature .............................................. 50 E.1 Calculation of the reducing functions for density and temperature .............................................. 50 E.1.1 Binary parameters for mixtures with no or very poor experimental data ...................................... 50

Annex F (informative) Assignment of trace components ........................................................................... 57

Annex G (informative) Examples ................................................................................................................... 59

Bibliography ...................................................................................................................................................... 64

ISO/CD 20765-2


vi

Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.

International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.

The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote.

Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights.

ISO 20765-2 was prepared by Technical Committee ISO/TC 193, Natural Gas, Subcommittee SC 1, Analysis of Natural Gases.

ISO/CD 20765-2


Introduction

This International Standard specifies methods for the calculation of thermodynamic properties of natural gases, manufactured fuel gases, and similar mixtures. It comprises three parts:

Part 1: Gas phase properties for transmission and distribution applications

Part 2: Single-phase properties (gas, liquid, and dense fluid) for extended ranges of application (this document)

Part 3: Two-phase properties (vapor-liquid equilibria)

This part - Part 2 - has five normative annexes and two informative annexes.

ISO/CD 20765-2

© ISO 2011 – All rights reserved 1

Natural gas – Calculation of thermodynamic properties – Part 2: Single-phase properties (gas, liquid, and dense fluid) for extended ranges of application

1 Scope

This part of ISO 20765 specifies a method of calculation for the volumetric and caloric properties of natural gases, manufactured fuel gases, and similar mixtures, at conditions where the mixture may be in either the homogeneous (single-phase) gas state, the homogeneous liquid state, or the homogeneous supercritical (dense-fluid) state.

NOTE Although the primary application of this document is to natural gases, manufactured fuel gases, and similar

mixtures, the method presented is also applicable with high accuracy (i.e., to within experimental uncertainty) to each of the (pure) natural gas components and to numerous binary and multi-component mixtures related to or not related to

natural gas.

For mixtures in the gas phase and for both volumetric properties (compression factor and density) and caloric properties (for example, enthalpy, heat capacity, Joule-Thomson coefficient, and speed of sound), the method is at least equal in accuracy to the method described in Part 1 of this International Standard, over the full

ranges of pressure p, temperature T, and composition to which Part 1 applies. In some regions, the

performance is significantly better; for example, in the temperature range 250 K to 275 K. The method described here maintains an uncertainty of ≤ 0.1% for volumetric properties, and generally within 0.1% in speed of sound. Although the new equation accurately describes all volumetric and caloric properties in the homogeneous gas, liquid, and supercritical regions, and of vapor-liquid equilibrium states, its structure is more complex than that in Part 1.

NOTE All uncertainties in this document are expanded uncertainties given for a 95% confidence level (coverage factor k = 2).

The method described here is also applicable with no increase in uncertainty to wider ranges of temperature, pressure, and composition, for example, to natural gases with lower content of methane (down to 0.30 mole fraction), higher content of nitrogen (up to 0.55 mole fraction), carbon dioxide (up to 0.30 mole fraction), ethane (up to 0.25 mole fraction), and propane (up to 0.14 mole fraction), and to hydrogen-rich natural gases, to which the method of Part 1 is not applicable. The equations can be used for high CO2 mixtures found in carbon dioxide sequestration applications.

The mixture model presented here is valid by design over the entire fluid region. In the liquid and dense-fluid regions the paucity of high quality test data does not in general allow definitive statements of uncertainty for all sorts of multi-component natural gas mixtures. For saturated liquid densities of LNG-type fluids in the temperature range from 100 K to 140 K, the uncertainty is ≤ (0.1 – 0.3)%, which is in agreement with the estimated experimental uncertainty of available test data. The model represents experimental data for compressed liquid densities of various binary mixtures related to LNG to within deviations of ±(0.1 – 0.2)% at pressures up to 40 MPa, which is in agreement with the estimated experimental uncertainty as well. Due to the high accuracy of the equations developed for the binary subsystems, the mixture model can predict the thermodynamic properties for the liquid and dense-fluid regions with the best accuracy presently possible for multi-component natural gas fluids.

2 Normative references

The following referenced documents may be useful for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 31-3, Quantities and units – Part 3: Mechanics.

ISO/CD 20765-2

2 © ISO 2011 – All rights reserved

ISO 31-4, Quantities and units – Part 4: Heat.

ISO 7504, Gas Analysis – Vocabulary.

ISO 20765-1, Natural gas – Calculation of thermodynamic properties - Part 1: Gas phase properties for transmission and distribution applications.

ISO 14532, Natural gas – Vocabulary.

BIPM, IEC, IFCC, ISO, IUPAC, OIML, International vocabulary of basic and general terms in metrology (VIM), 2

nd edition, 1993.

Guide to the expression of uncertainty in measurement, International Organization for Standardization, 1st

edition, 1993, 100 p.

3 Terms and definitions

Except where given below, all terms and definitions relating to heat and thermodynamics are taken from ISO 31-4 and/or ISO 20765-1, all terms and definitions relating to gas analysis are taken from ISO 7504, and all terms and definitions relating to natural gas are taken from ISO 14532.

NOTE 1 See annex A for the list of symbols and units used in this part of ISO 20765.

NOTE 2 Figure 1 is a schematic representation of the phase behavior of a typical natural gas as a function of pressure and temperature. The positions of the bubble and dew lines depend upon the composition. This phase diagram may be

found useful in the interpretation of definitions presented below.

Figure 1 - Phase diagram for a typical natural gas

3.1 bubble point pressure pressure at which an infinitesimal amount of vapor is in equilibrium with a bulk liquid for a specified temperature

3.2 bubble point temperature temperature at which an infinitesimal amount of vapor is in equilibrium with a bulk liquid for a specified pressure

0

1

2

3

4

5

6

7

8

9

100 150 200 250 300

Temperature / K

Pre

ss

ure

/ M

Pa

bubble

cricondentherm

cricondenbar

GAS

PHASE

LIQUID PHASE

TWO-PHASEVAPOUR-LIQUID

line

dewline

critical point

SUPERCRITICAL

DENSE FLUIDSTATE

*

*

0

1

2

3

4

5

6

7

8

9

100 150 200 250 300

Temperature / K

Pre

ss

ure

/ M

Pa

bubble

cricondentherm

cricondenbar

GAS

PHASE

LIQUID PHASE

TWO-PHASEVAPOUR-LIQUID

line

dewline

critical point

SUPERCRITICAL

DENSE FLUIDSTATE

ISO/CD 20765-2


NOTE 1 The locus of bubble points is known as the bubble line.

NOTE 2 There may exist more than one bubble point temperature at a specific pressure. Moreover, more than one bubble point pressure may exist at a specified temperature.

3.3 cricondenbar maximum pressure at which two-phase separation can occur

3.4 cricondentherm maximum temperature at which two-phase separation can occur

3.5 critical point unique point on a two-phase vapor-liquid equilibrium line where the entire fluid has a single composition and a single density

NOTE 1 The critical point is the point at which the dew line and the bubble line meet.

NOTE 2 The pressure at the critical point is known as the critical pressure and the temperature as the critical temperature.

NOTE 3 A mixture of given composition may have one, more than one, or no critical points. Moreover, the phase behavior may be quite different than shown in Fig. 1 for mixtures (including natural gases) containing hydrogen or helium.

3.6 dew point pressure pressure at which an infinitesimal amount of liquid is in equilibrium with a bulk vapor for a specified temperature

NOTE 1 More than one dew point pressure may exist at the specified temperature.

3.7 dew point temperature temperature at which an infinitesimal amount of liquid is in equilibrium with a bulk vapor for a specified pressure

NOTE 1 More than one dew point temperature may exist at the specified pressure.

NOTE 2 The locus of dew points is known as the dew line.

3.8 super critical state dense phase region above the critical point (often taken as states above the critical temperature and pressure) within which no two-phase separation can occur

4 Thermodynamic basis of the method

4.1 Principle

The method is based on the concept that natural gas or any other type of mixture can be completely characterized in the calculation of its thermodynamic properties by component analysis. Such an analysis, together with the state variables of temperature and density, provides the necessary input data for the calculation of properties. In practice, the state variables available as input data are generally temperature and pressure, and it is thus necessary to first determine the density iteratively using the equations provided here.

The equation presented here expresses the Helmholtz free energy of the mixture as a function of density, temperature, and composition, from which all other thermodynamic properties in the homogeneous (single-phase) gas, liquid, and supercritical (dense-fluid) regions may be obtained in terms of the Helmholtz free

ISO/CD 20765-2


energy and its derivatives with respect to temperature and density.

NOTE The equation selected is also applicable to the calculation of two-phase properties (vapor-liquid equilibria).

However, derivatives in addition to those required for single-phase calculations are needed. The additional derivatives are presented in Part 3 of this International Standard.

The method uses a detailed molar composition analysis in which all components present in amounts exceeding 0.000 05 mole fraction are specified. For a typical natural gas, this might include alkane hydrocarbons up to about C7 or C8 together with nitrogen, carbon dioxide, and helium. Typically, isomers for alkanes above C5 may be lumped together by molecular weight and treated collectively as the normal isomer.

For some fluids, additional components such as C9, C10, water, and hydrogen sulfide may be present and need to be taken into consideration. For manufactured gases, hydrogen and carbon monoxide may also be present in the mixture.

More precisely, the method uses a 21-component analysis in which all of the major and minor components of natural gas are included (see section 6). Any trace component present but not identified as one of the 21 specified components may be assigned appropriately to one of these 21 components (see annex F).

4.2 The fundamental equation based on the Helmholtz free energy

4.2.1 Background

The GERG-2008 equation [1] was published by the Lehrstuhl für Thermodynamik at the Ruhr-Universität Bochum in Germany as a new wide-range equation of state for the volumetric and caloric properties of natural gases and other mixtures. It was originally published in 2004 [2] and later updated in 2008 [1]. The new equation improves upon the performance of the AGA-8 equation [3] for gas phase properties and in addition is applicable to the properties of the liquid phase, to the dense-fluid phase, to the vapor-liquid phase boundary, and to properties for two-phase states. The ranges of temperature, pressure, and composition to which the GERG-2008 equation of state applies are much wider than the AGA-8 equation and cover an extended range of application. The Groupe Européen de Recherches Gazières supported the development of the GERG-2008 equation of state over several years.

The GERG-2008 equation is explicit in the Helmholtz free energy, a formulation that enables all thermodynamic properties to be expressed analytically as functions of the free energy and of its derivatives with respect to the state conditions of temperature and density. There is generally no need for numerical differentiation or integration within any computer program that implements the method.

4.2.2 The Helmholtz free energy

The Helmholtz free energy a of a fluid mixture at a given mixture density , temperature T, and molar

composition xx can be expressed as the sum of ao describing the ideal gas behavior and a

r describing the

residual or real-gas behavior, as follows

),,(),,(),,( ro xTaxTaxTa . (1)

4.2.3 The reduced Helmholtz free energy

Usually, the Helmholtz free energy is used in its dimensionless form =a/RT as

),,(),,(),,( ro xxTx . (2)

In this equation, the reduced (dimensionless) mixture density is given by

r( )x, (3)

ISO/CD 20765-2


and the inverse reduced (dimensionless) mixture temperature is given by

T x

T

r( ) , (4)

where r and r are reducing functions for the mixture density and mixture temperature (see 4.2.7)

depending on the molar composition of the mixture only.

The residual part r of the reduced Helmholtz free energy is given by

),,(),,(),,( rro

r xxx . (5)

In this equation, the first term on the right-hand side ro describes the contribution of the residual parts of the

reduced Helmholtz free energy of the pure substance equations of state, which are multiplied by the mole

fraction of the corresponding substance and linearly combined using the reduced mixture variables and

(see equation (8)). The second term r is the departure function, which is the summation over all binary

specific and generalized departure functions developed for the respective binary mixtures (see equation (10)).

4.2.4 The reduced Helmholtz free energy of the ideal gas

The reduced Helmholtz free energy o represents the properties of the ideal-gas mixture at a given mixture

density , temperature , and molar composition xx according to

N

i

iii xTxxT

1

oo

o ln),(),,( . (6)

In this equation, the term ii xx ln is the contribution from the entropy of mixing, and ooi(, T ) is the

dimensionless form of the Helmholtz free energy in the ideal-gas state of component i as given by

),(oo Ti =

6,4

,co,o

o,o

,co3,o

,co2,o

o1,o

,c

sinhlnlnln

k

ikiki

ii

iii

i T

Tn

T

Tn

T

Tnn

R

R

7,5

,co

,o

o

,o coshlnk

i

kikiT

Tn , (7)

where

c,i and c,i are the critical parameters of the pure components (see Annex C). The values of the coefficients

nooi,k and the parameters

ooi,k for all 21 components are given in Annex B.

NOTE 1 The method prescribed is taken without change from the method prescribed in Part 1 of this International

Standard. The user should however be aware of significant differences that result inevitably from the change in definition

of inverse reduced temperature between Part 1 and Part 2.

NOTE 2 R = 8.314 472 J·mol-1

·K-1

is the current, internationally accepted standard for the molar gas constant [4].

Equation (7) results from the integration of the equations for the ideal-gas heat capacities taken from [5], where a different molar gas constant was used than the one adopted in the mixture model presented here. The ratio R

*/R with R

*=8.314 51

J·mol-1

·K-1

takes into account this difference and therefore leads to the exact solution of the original equations for the ideal-gas heat capacity.

ISO/CD 20765-2


4.2.5 The pure substance contribution to the residual part of the reduced Helmholtz free energy

The contribution of the residual parts of the reduced Helmholtz free energy of the pure substance equations of

state ro to the residual part of the reduced Helmholtz free energy of the mixture is

N

i

iixx

1

ro

ro ),(),,( , (8)

where

roi( , ) is the residual part of the reduced Helmholtz free energy of component i (i.e., the residual part of the

respective pure substance equation of state listed in table 2) and is given by

ii

i

kikiki

ikiki

KK

Kk

ctdki

K

k

tdi,ki enn

,Exp,Pol

,Pol

,o,o,o

,Pol,o,o

1

,o

1

oro ),( . (9)

The equations for roi use the same basic structure as further detailed in Annex D.2. The values of the

coefficients noi,k and the exponents doi,k, toi,k and coi,k for all 21 components are given in Annex D.2.2.

4.2.6 The departure function contribution to the residual part of the reduced Helmholtz free energy

The purpose of the departure function is to further improve the accuracy of the mixture model in the description of thermodynamic properties in addition to fitting the parameters of the reducing functions (see 4.2.7) when sufficiently accurate experimental data are available to characterize the properties of the mixture.

The departure function r of the multi-component mixture is the double summation over all binary specific

and generalized departure functions developed for the binary subsystems and is given by

1

1 1

rr ),,(),,(N

i

N

ij

ij xx (10)

with

),(),,( rr ijijjiij Fxxx . (11)

In this equation, the function rij( , ) is the part of the departure function r

ij( , , xx ) that depends only on

the reduced mixture variables and as given by

ijij

ij

kijkijkijkijkijkij

ij

kijkij

KK

Kk

td

kij

K

k

td

kijij

en

n

,Exp,Pol

,Pol

,,

2

,,,,

,Pol

,,

1

,

1

,

r ),(

,

(12)

Where

rij( , ) was developed either for a specific binary mixture (a binary specific departure function with binary

specific coefficients, exponents, and parameters) or for a group of binary mixtures (generalized departure function with a uniform structure for the group of binary mixtures).

ISO/CD 20765-2


(a) Binary specific departure functions

Binary specific departure functions were developed for the binary mixtures of methane with nitrogen, carbon

dioxide, ethane, propane, and hydrogen, and of nitrogen with carbon dioxide and ethane. For a binary

specific departure function, the adjustable factor Fij in equation (11) equals unity.

(b) Generalized departure functions

A generalized departure function was developed for the binary mixtures of methane with n-butane and

isobutane, of ethane with propane, n-butane, and isobutane, of propane with n-butane and isobutane, and of

n-butane with isobutane. For each mixture in the group of generalized binary mixtures, the parameter Fij is

fitted to the corresponding binary specific data (except for the binary system methane–n-butane, where Fij

equals unity).

(c) No departure functions

For all of the remaining binary mixtures, no departure function was developed, and Fij equals zero, i.e.,

rij( , , xx ) equals zero. For most of these mixtures, however, the parameters of the reducing functions for

density and temperature were fitted to selected experimental data (see 4.2.7 and 6.2).

The values of the coefficients nij,k, the exponents dij,k and tij,k, and the parameters ij,k, ij,k, ij,k, and ij,k for all

binary specific and generalized departure functions considered in the mixture model described here are given

in annex D.3, Table D.4. The non-zero Fij parameters are listed in Table D.5.

NOTE Compared to the reducing functions for density and temperature, the departure function is in general of minor

importance for the residual behavior of the mixture since it only describes an additional small residual deviation to the real

mixture behavior. The development of such a function was, however, necessary to fulfill the high demands on the

accuracy of the mixture model presented here in the description of the thermodynamic properties of natural gases and

other mixtures.

4.2.7 Reducing functions

The reduced mixture variables and are calculated from equations (3) and (4) by means of the

composition-dependent reducing functions for the mixture density and temperature

N

i

N

j jijiijv

jiijvijvji

xx

xxxx

x1 1

3

3/1,c

3/1,c

2,

,,r

11

8

1

)(

1

(13)

N

i

N

j

ji

jiijT

jiijTijTji TT

xx

xxxxxT

1 1

,c,c2,

,,r

5.0

)(

. (14)

These functions are based on quadratic mixing rules and are reasonably connected to physically well-founded

mixing rules. The binary parameters v,ij and v,ij in equation (13) and ,ij and ,ij in equation (14) are fitted

to data for binary mixtures. The values of the binary parameters for all binary mixtures are listed in table E.1

of Annex E. The critical parameters c,i and c,i of the pure components are given in Annex C. In equations

(13) and (14), ij=1/ji and ij=ji.

NOTE The binary parameters of equations (13) and (14) were fitted based on the deviations between the behavior of

the real mixture and the one resulting from ideal combining rules for the critical parameters of the pure components. In

those cases where sufficient experimental data are not available, the parameters of equations (13) and (14) are either set

to unity or modified (calculated) in such a manner that the critical parameters of the pure components are combined in a

different way, which proved to be more suitable for certain binary subsystems (see also Annex E.1).

4.3 Thermodynamic properties derived from the Helmholtz free energy

ISO/CD 20765-2


4.3.1 Background

The thermodynamic properties in the homogeneous gas, liquid, and supercritical regions of a mixture are

related to derivatives of the Helmholtz free energy with respect to the reduced mixture variables and , as

summarized in the following section (see table 1). All of the thermodynamic properties may be written

explicitly in terms of the reduced Helmholtz free energy and various derivatives thereof. The required

derivatives , , , , and are defined as follows:

x,

x,2

2

x,

x,2

2

xx ,,

(15)

Each derivative is the sum of an ideal-gas part (see Annex B) and a residual part (see Annex D). The

following substitutions help to simplify the appearance of the relevant relationships:

r2r2

,

2

1 212)(

ix

(16)

rr

,

2

2 1

ix

(17)

Detailed expressions for , , , , , , and can be found in Annexes B and D.

NOTE In addition to the derivatives of with respect to the reduced mixture variables and , composition

derivatives of and of the reducing functions for density and temperature are required for the calculation of vapor-liquid

equilibrium (VLE) properties as described in Part 3 of this International Standard.

4.3.2 Relations for the calculation of thermodynamic properties in the homogeneous region

The relations between common thermodynamic properties and the reduced Helmholtz free energy and its derivatives are summarized in table 1. The first column of this table defines the thermodynamic properties.

The second column gives their relation to the reduced Helmholtz free energy of the mixture. In equations

(23), (25), (26), (28), (29), and (31), the basic expressions for the properties s, h, cp, w, JT, and have been

additionally transformed, such that values of properties already derived can be used to simplify the subsequent calculations. This approach is useful for applications where several or all of the thermodynamic properties are to be determined.

In equations (22) to (27), the relations for the thermodynamic properties represent the molar quantities (i.e., quantity per mole, lower case symbols). Specific quantities (i.e., quantity per kilogram, represented normally

by upper case symbols) are obtained by dividing the molar variables (e.g., v, u, s, h, g, cv, and cp) by the molar

mass M.

The molar mass M of the mixture is derived from the composition xi and the molar masses Mi of the pure

substances as follows

N

i

ii MxxM

1

)( (18)

The mass-based density D is given by

MD (19)

NOTE 1 Values of the molar masses Mi of the pure substances are given in Annex C and are taken from [6]; these

values are not identical with those given in ISO 20765-1 and ISO 6976:1995 [7]. However, they are identical with the most

ISO/CD 20765-2


recent values adopted by the international community of metrologists. In these equations, R is the molar gas constant;

consequently R/M is the specific gas constant.

NOTE 2 See Annex B.1 for information on reference states for enthalpy and entropy.

Table 1 — Definitions of common thermodynamic properties and their relation to the reduced

Helmholtz free energy .

Property and definition Relation to and its derivatives

Pressure

xT

vaxTp,

),,( r1

),,(δ

RT

xp

(20)

Compression factor

RTpxTZ /),,(

r1),,( δxZ (21)

Internal energy

TsaxTu ),,(

RT

xu ),,(

(22)

Entropy

xv

TaxTs,

),,(

R

xs ),,(

RT

u

(23)

Isochoric heat capacity

xvv

TuxTc,

),,(

2),,(

R

xcv

(24)

Enthalpy

pvuxpTh ),,(

r1),,(

RT

xhZ

RT

u

(25)

Isobaric heat capacity

xpp

ThxpTc,

),,( 1

22),,(

2

R

xcp

1

22

RT

vc

(26)

Gibbs free energy

TshxpTg ),,(

r1

),,(

RT

xg

(27)

Speed of sound

xs

pMxpTw,

1),,(

2

2 22

1),,(

RT

MxwZ

(28)

Joule-thomson coefficient

xh

pTxpT,JT

),,(

122

21JT

2

R

1

1

2

pc

R

(29)

Isothermal throttling coefficient

xTT

phxpT,

),,( 1

21

T

(30)

Isentropic exponent

xs

vppvxpT,

),,(

12

22

r1 1

v

p

c

c

Z1

(31)

Second virial coefficient

xT

ZxTB,

0

lim),(

r

0rlim)(

B (32)

Third virial coefficient

xT

ZxTC,

22

0

lim2

1),(

r2

r0

lim)(

C (33)

ISO/CD 20765-2


5 Method of calculation

5.1 Input variables

The method presented in this standard internally uses reduced density, inverse reduced temperature, and

molar composition as the input variables. The actual allowable input variables are either

(a) Molar density, temperature , and molar composition x , or

(b) Absolute pressure p, temperature , and molar composition x .

If the mass-based density D is available as input, then is obtained directly as =D/M, where )(xM is the

molar mass given by equation (18). For given values of the molar density, temperature , and molar

composition x , the reduced mixture variables and can be calculated from equations (3) and (4) using the

reducing functions for density and temperature given by equations (13) and (14).

More often, however, absolute pressure, temperature, and molar composition are available as the input

variables. In consequence, it is usually necessary to first evaluate the reduced density and the inverse

reduced temperature from the available inputs. The conversion from temperature to inverse reduced temperature is given by equation (4). Section 5.2 explains how to obtain the reduced density given pressure and temperature.

The composition in mole fractions is required for the following 21 components: methane, nitrogen, carbon

dioxide, ethane, propane, n-butane, isobutane (2-methylpropane), n-pentane, isopentane (2-methylbutane), n-

hexane, n-heptane, n-octane, n-nonane, n-decane, hydrogen, oxygen, carbon monoxide, water, hydrogen

sulfide, helium, and argon. For natural gases and similar (multi-component) mixtures, the allowable ranges of

mole fraction are defined in 6.3. The sum of all mole fractions shall be unity.

NOTE 1 If the sum of all mole fractions is not unity within the limit of analytical resolution, then the composition is either faulty or incomplete. The user should not proceed until the source of this problem has been identified and eliminated.

NOTE 2 If the mole fractions of heptanes, octanes, nonanes, and decanes are unknown, then the use of a composite C6+ fraction may be acceptable. For VLE calculations (including dew points), this simplification is not acceptable since

even small amounts of heptanes, octanes, nonanes, decanes, and higher hydrocarbons have a significant influence on the phase behavior of the mixture. The user should carry out a sensitivity analysis in order to test whether a particular

approximation of this type degrades the result.

NOTE 3 Composition given in volume or mass fractions will need to be converted to mole fractions using the method

given in ISO 14912 [8].

5.2 Conversion from pressure to reduced density

Combination of the relations for the reduced mixture variables and (equations (3) and (4)) and equation (21) results in the following expression

p

x RT xZ x x

r r

r

( ) ( )( , , ) ( , , ) 1 , (34)

where

1

1 1

r

1

ro

,

rr ),(),(

),,(N

i

N

ij

ijijji

N

i

ii

x

Fxxxx

. (35)

If the input variables are available as pressure, inverse reduced temperature, and molar composition, equation

(34) may be solved for the reduced molar density . The derivatives of ori( , ) with respect toand the

ISO/CD 20765-2


coefficients noi,k and the exponents doi,k, toi,k, and coi,k involved (see equation (9)) are given in Annex D.2. The

derivatives of ),(r ij with respect to and the coefficients nij,k, the exponents dij,k and tij,k, and the

parameters Fij, ij,k, ij,k, ij,k, and ij,k involved (see equations (11) and (12)) are given in Annex D.3. Information on the reducing functions is given in annex E.

The solution of equation (34) may be obtained by any suitable numerical method but, in practice, a standard form of equation-of-state density-search algorithm may be the most convenient and satisfactory. Such algorithms usually use an initial estimate of the density (e.g., the ideal-gas approximation for low density

gaseous states) and proceed to calculate the pressure p. In an iterative procedure the value of is changed

in order to find the final reduced density which reproduces the known value of p to within a pre-established

level of agreement. A suitable criterion in the present case is that the pressure calculated from the iteratively

determined reduced density shall reproduce the input value of p at least to within 1 part in 106.

5.3 Implementation

Once the independent variables reduced density inverse reduced temperature , and molar composition x

of the mixture are known, the reduced Helmholtz free energy and the other thermodynamic properties (see

table 1) can be calculated. Equation (2) formulates the reduced Helmholtz free energy as =o+

r. The

relations for the ideal-gas part o are given in equations (6) and (7). The relations for the residual part

r,

which is formulated as a function of the reduced density the inverse reduced temperature , and the molar

composition x , are specified in equations (5) and (8) to (12) so as to give the following expression for :

N

i k

i

kiki

i

i

i

ii

i

iT

Tn

T

Tn

T

Tnn

R

Rxx

1 6,4

,co

,o

o

,o

,co

3,o

,co

2,o

o

1,o

,c

sinhlnlnln),,(

7,5

,co

,o

o

,o coshlnk

i

kikiT

Tn

N

i

ii xx1

ln

N

i

KK

Kk

ctd

ki

K

k

td

oi,ki

ii

i

kikiki

i

kiki ennx1 1

,o

1

,Exp,Pol

,Pol

,o,o,o

,Pol

,o,o (36)

1

1 1 1

,

1

,

,Exp,Pol

,Pol

,,2

,,,,

,Pol

,,

N

i

N

ij

KK

Kk

td

kij

K

k

td

kijijji

ijij

ij

kijkijkijkijkijkij

ij

kijkij ennFxx

For all 21 components, the values of the coefficients nooi,k and the parameters o

oi,k of the ideal-gas part of the reduced Helmholtz free energy are given in Annex B. The values of the coefficients noi,k and exponents doi,k,

toi,k, and coi,k in the contribution to the residual parts of the pure substance equations of state are listed in

Annex D. The values of the coefficients nij,k, the exponents dij,k and tij,k, and the parameters Fij, ij,k, ij,k, ij,k,

and ij,k in the departure functions for all relevant binary mixtures are given in Annex D as well.

Derivatives of with respect to the reduced mixture variables and that are needed for the calculation of the various thermodynamic properties may be obtained from Annexes B and D. Annex B lists the derivatives of the ideal-gas part of the reduced Helmholtz free energy, ao, with respect to the reduced density and inverse reduced temperature of the mixture. Derivatives of the contribution of the residual parts of the pure substance equations of state to the reduced residual Helmholtz free energy of the mixture, ar, with respect to the reduced

mixture variables and may be obtained from Annex D. Derivatives of the contribution of the departure

functions for binary mixtures to ar with respect to the reduced mixture variables and are given in Annex D

as well.

6 Ranges of application

ISO/CD 20765-2


6.1 Pure gases

The temperature and pressure ranges of validity for the pure fluid equations of state are listed in table 2. For these ranges, the equations have been tested with experimental data. The lower temperatures correspond to the triple point temperatures of the substances. For the main components, the equations have been tested for temperatures up to at least 600 K and pressures up to 300 MPa. For the secondary alkanes, the equations have been tested for temperatures up to 500 K and pressures up to at least 35 MPa. For the other secondary components the temperatures range up to at least 400 K (water up to 1273 K) and pressures range up to at least 100 MPa. The extrapolation to temperatures and pressures far beyond the listed (tested) ranges of validity yields reasonable results.

Table 2 — Validity range and references for the 21 components in the mixture modela

Pure substance Reference Tested range of validity Number

of terms Temperature Pressure

T/K pmax/MPa

Main components

Methane Klimeck (2000) [9] 90 – 623 300 24

Nitrogen Klimeck (2000) [9] 63 – 700 300 24

Carbon dioxide Klimeck (2000) [9] 90b – 900 300 22

Ethane Klimeck (2000) [9] 90 – 623 300 24

Secondary alkanes

Propane Span & Wagner (2003) [10] 85 – 623 100 12

n-Butane Span & Wagner (2003) [10] 134 – 693 70 12

Isobutane Span & Wagner (2003) [10] 113 – 573 35 12

n-Pentane Span & Wagner (2003) [10] 143 – 573 70 12

Isopentane Lemmon & Span (2006) [11] 112 – 500 35 12

n-Hexane Span & Wagner (2003) [10] 177 – 548 100 12

n-Heptane Span & Wagner (2003) [10] 182 – 523 100 12

n-Octane Span & Wagner (2003) [10] 216 – 548 100 12

n-Nonane Lemmon & Span (2006) [11] 219 – 600 800 12

n-Decane Lemmon & Span (2006) [11] 243 – 675 800 12

Other secondary components

Hydrogenc Kunz et al. (2007) [1] 14 – 700 300 14

Oxygen Span & Wagner (2003) [10] 54 – 500e 100 12

Carbon monoxide Lemmon & Span (2006) [11] 68 – 400 100 12

Water Kunz et al. (2007) [1] 273 – 1273 100 16

Heliumd Kunz et al. (2007) [1] 2.2 – 573 100 12

Argon Span & Wagner (2003) [10] 83 – 520 100 12

Hydrogen Sulfide Lemmon & Span (2006) [11] 187 – 760 170 12 a

The tabulated references correspond to the equations for the residual part of the Helmholtz free energy of the

pure substances. The equations for the isobaric heat capacity in the ideal-gas state from [5] were used to

derive the Helmholtz free energy of the ideal gas for all components. b

The equation can be extrapolated from the triple-point temperature 216 K down to 90 K. c Represents equilibrium hydrogen.

d Represents helium-4. The lower temperature limit of the equation of state is the helium I to helium II transition

point. e

The upper limit of oxygen has been increased to 500 K based on recent validation of the equation of state.

ISO/CD 20765-2


6.2 Binary mixtures

Table 3 summarizes the available data for volumetric and caloric properties of all binary mixtures considered

in this method. Almost half of the total number of data points are from those 15 mixtures for which binary

specific or generalized departure functions have been developed. The majority of the data (approximately

65%) describe the pT relation, approximately 25% are vapor-liquid equilibrium, and about 9% are caloric

properties.

(a) Specific departure functions

Table 4 lists the binary mixtures for which either binary specific or generalized departure functions have been

developed because sufficiently accurate and extensive data are available. Binary specific departure functions

are given for the binary mixtures of methane with nitrogen, carbon dioxide, ethane, propane, and hydrogen,

and of nitrogen with carbon dioxide and ethane (see also figure 2).

The temperature range of the experimental data used to test the equations developed for the binary mixtures

covers approximately 80 K to 700 K at pressures up to 70 MPa and more (e.g., 100 MPa for methane–carbon

dioxide and 750 MPa for methane–nitrogen). The composition ranges from nearly 0 to 1 for all mixtures listed

above; see [2] for further details.

(b) Generalized departure functions

A generalized departure function was developed for eight binary mixtures of secondary alkanes (see table 4).

The generalized departure function is used for the binary mixtures of methane with n-butane and isobutane,

ethane with propane, n-butane, and isobutane, propane with n-butane and isobutane, and n-butane with

isobutane.

Aside from experimental data for these binary mixtures, selected volumetric and caloric properties for the well-

measured binary alkane mixtures methane–ethane and methane–propane were also used for the

development of the generalized departure function for secondary binary alkane mixtures. The final structure

of the generalized departure function is almost completely based on accurate and comprehensive data sets

for the three binary systems methane–ethane, methane–propane, and methane–n-butane, with more than

6,200 selected data points used (about 1,000 data points for the methane–n-butane binary system). The

temperature range of the experimental data used to test the equations developed for the secondary binary

alkane mixtures covers roughly 95 K to 600 K at pressures up to 35 MPa (70 MPa for methane–n-butane).

The composition ranges from nearly 0 to 1 for all mixtures listed above; see [2] for further details.

(c) No departure functions

For all of the remaining binary mixtures, no departure function was developed (see figure 2). The

corresponding binary mixtures are either characterized by limited data, which do not permit the development

of binary specific or generalized departure functions, or are of minor importance for the description of the

thermodynamic properties of (multi-component) natural gases due to the small mole fractions of the respective

components. Fitting the parameters of the reducing functions for density and temperature to selected

experimental data is sufficient and yields accurate results for most of the binary mixtures. In those cases

where sufficient experimental data are not available, the parameters of the reducing functions for density and

temperature are either set to unity or modified (calculated) in such a manner that the critical parameters of the

pure components are combined in a different way, which proved to be more suitable for certain binary

subsystems (see also Annex E.1).

For the temperature, pressure, and composition ranges covered by the experimental data used to test the

equations developed for the binary mixtures without a departure function, i.e., by adjusting the reducing

functions for density and temperature only, see [2].

ISO/CD 20765-2


Table 3 — Summary of the available data for volumetric and caloric properties of binary mixtures

Data type Number of data

points

Temperature

ranges

Pressure ranges Composition

rangesc,d

totala used

b T/K p/MPa x

Density 51442 30252 66.9 – 800 0.00 – 1027 0.00 – 1.00

Isochoric heat capacity 1236 625 101 – 345 67.1ρ

– 902ρ 0.01 – 0.84

Speed of sound 2819 1805 157 – 450 0.00 – 1971 0.01 – 0.96

Isobaric heat capacity 1072 490 100 – 424 0.00 – 52.9 0.09 – 0.93

Enthalpy differences 1804 198 107 – 525 0.00 – 18.4 0.05 – 0.90

Excess molar enthalpy 177 20 221 – 373 0.8 – 15.0 0.01 – 0.98

Saturated liquid densitye 460 119 95.0 – 394 0.03 – 22.1 0.00 – 1.00

VLE data 20161 6350 15.5 – 700 0.00 – 422 0.00 – 1.00

Total 79171 39859 15.5 – 800 0.00 – 1971 0.00 – 1.00

a Number of available data points used in testing the GERG-2008 equation of state.

b Number of data points selected for the development of the GERG-2008 equation of state.

c Mole fractions of the second component of the binary mixture, i.e., component B of mixture A–B.

Values of 0.00 and 1.00 result from a mixture composition close to a pure component.

d The composition range for VLE data corresponds to bubble point compositions.

e Listed separately due to a different data format. Saturated liquid (and vapour) densities may also be

tabulated as ordinary pρT or VLE data. ρ

Density in kg∙m−3

instead of pressure.

Table 4 — List of the binary mixtures for which binary specific and generalized departure functions

were developed

Binary mixture Type of departure function Number of terms Type of termsa

Methane–Nitrogen binary specific 9 P (2), E (7)

Methane–Carbon dioxide binary specific 6 P (3), E (3)

Methane–Ethane binary specific 12 P (2), E (10)

Methane–Propane binary specific 9 P (5), E (4)

Methane–n-Butane generalized 10 P

Methane–Isobutane generalized 10 P

Methane–Hydrogen binary specific 4 P

Nitrogen–Carbon dioxide binary specific 6 P (2), E (4)

Nitrogen–Ethane binary specific 6 P (3), E (3)

Ethane–Propane generalized 10 P

Ethane–n-Butane generalized 10 P

Ethane–Isobutane generalized 10 P

Propane–n-Butane generalized 10 P

Propane–Isobutane generalized 10 P

n-Butane–Isobutane generalized 10 P

a ‖P‖ indicates polynomial terms, and ―E‖ indicates the exponential terms composed of a polynomial and

exponential expression according to equation (12). The numbers in parentheses indicate the respective

number of terms Kpol and Kexp .

ISO/CD 20765-2


Figure 2 Overview of the 210 binary combinations that result from the 21 natural gas components for the development of the GERG-2008 equation of state. The diagram shows the different formulations developed for the binary mixtures.

6.3 Range of validity for natural gases

The method described in this part of ISO 20765 applies to the calculation of thermodynamic properties of

natural gases and similar mixtures in the homogeneous (single-phase) region (gas, liquid, and dense fluid) for

normal and extended ranges of application. The ranges of validity in this part is defined as follows.

Pressure and temperature ranges

The relevant ranges of pressure and temperature are given in table 5. The method described in this part of

the standard applies strictly to mixtures in a homogeneous state (gas, liquid, and dense fluid).

Table 5 — Normal ranges of application

Pressure (absolute) 0 < p / MPa 35

Temperature 90 T / K 450

Composition ranges

Pipeline quality natural gas is taken as a natural (or similar) gas with mole fractions of the various components

that fall within the ranges given in the third column of table 6. The method described in ISO 20765-1 applies

only to pipeline quality natural gases for those ranges of pressure and temperature within which transmission

and distribution operations normally take place. The method given in this standard also applies to these

conditions as well as an expanded composition range of natural gas as given in the fourth column of table 6.

Accurate and extensive experimental data sets are available within the composition ranges listed in the table

below and have been used to validate the quality of the method presented here (see 7.4). Possible trace

components of natural gases, and details of how to deal with these, are discussed in Annex F. This method is

7.1

0 D

evelop

men

t of th

e Differen

t Bin

ary E

qu

atio

ns o

f Sta

te... 1

67

ISO/CD 20765-2


not applicable where the total of all trace components exceeds 0.000 5 mole fraction.

Beyond the expanded quality range is the extended composition range that covers the full composition range

for all mixture components, i.e., from 0 to 1. Due to the limited data situation, results obtained for multi-

component mixtures in the extended range of validity should be carefully assessed. The method can be used

for temperatures above 700 K and pressures above 70 MPa. However, due to the limited data situation it is

difficult to estimate the uncertainty of the method (see 7).

Table 6 — Mole fraction ranges for components of expanded and pipeline quality natural gas

Mole fraction for

i Component Pipeline quality range Expanded quality range

1 Methane 0.7 xCH4 1.00 0.3 xCH4 1.00

2 Nitrogen 0 xN2 0.20 0 xN2 0.55

3 Carbon dioxide 0 xCO2 0.20 0 xCO2 0.30

4 Ethane 0 xC2H6 0.10 0 xC2H6 0.25

5 Propane 0 xC3H8 0.035 0 xC3H8 0.14

6 + 7 n-Butane + Isobutanea 0 xC4H10 0.015 0 xC4H10 0.06

8 + 9 n-Pentane + Isopentanea 0 xC5H12 0.005 0 xC5H12 0.005

10 n-Hexane 0 xC6H14 0.001 0 xC6H14 0.002

11 n-Heptane 0 xC7H16 0.000 5 0 xC7H16 0.001

12 + 13 + 14 Octane+Nonane+Decanea 0 xC8+ 0.000 5 0 xC8+ 0.000 5

15 Hydrogen 0 xH2 0.10 0 xH2 0.40

16 Oxygen 0 xO2 0.000 2 0 xO2 0.02

17 Carbon monoxide 0 xCO 0.03 0 xCO 0.13

18 Water 0 xH2O 0.000 15 0 xH2O 0.000 2

19 Hydrogen sulfide 0 xH2S 0.000 2 0 xH2S 0.27

20 Helium 0 xHe 0.005 0 xHe 0.005

21 Argon 0 xAr 0.000 2 0 xAr 0.000 5

a Indicates the sum of the mole fractions of the components may not exceed the specified value.

NOTE 1 The method described in this part of the standard is applicable even for the individual pure components with

high accuracy (i.e., to within experimental uncertainty). For this application, the tested ranges of validity for temperature

and pressure is given in table 2.

NOTE 2 The accurate description of the thermodynamic properties of multi-component mixtures by the GERG-2008

equation of state is based on the accurate and wide-ranging equations of the binary subsystems, which were developed

with experimental data that generally cover the entire composition range. Therefore, it can be expected that even multi-

component natural gases of very unusual composition will be accurately described.

ISO/CD 20765-2


7 Uncertainty of the equation of state

7.1 Background

When ranges of uncertainties are given, the upper uncertainty value should be used unless further comparisons are made with the information given in the technical report to verify that the lower uncertainty value is valid for a particular application. The uncertainty in this document has a 95% confidence level (coverage factor k = 2).

7.2 Uncertainty for pure gases

7.2.1 Natural gas main components

The estimated uncertainties in calculated density and speed of sound for the natural gas main components methane, nitrogen, carbon dioxide, and ethane are summarized in table 8. For methane, more details are given in figures 3 and 4; details for nitrogen, carbon dioxide, and ethane are given in [1]. The estimated uncertainties in gas phase density and speed of sound range from 0.03% to 0.05% over wide ranges of temperature (e.g., up to 450 K) and at pressures up to 30 MPa. In the liquid phase at pressures up to 30 MPa, the estimated uncertainties in density range from 0.05% to 0.1%. At higher temperatures or pressures, the estimated uncertainties in calculated speed of sound are generally higher than in calculated density because of less accurate data.

7.2.2 Secondary alkanes

The estimated uncertainties in calculated density, speed of sound, and isobaric heat capacity for the secondary alkanes propane, n-butane, isobutane, n-pentane, isopentane, n-hexane, n-heptane, n-octane, n-nonane, and n-decane are summarized in table 9. For calculated densities, uncertainties of 0.2% were estimated, whereas calculations of speed of sound and isobaric heat capacity have estimated uncertainties between 1% and 2%.

ISO/CD 20765-2


Table 8 — Uncertainties of the equations of state for the natural gas main components methane,

nitrogen, carbon dioxide, and ethane with regard to different thermodynamic properties.

Density Speed of sound

Gas phasea

p ≤ 30 MPa

p > 30 MPac

0.03% – 0.05%

0.1% – 0.5%

0.03% – 0.05%b

0.5%d

Liquid phase

p ≤ 30 MPa

p > 30 MPac

0.05% – 0.1%

0.5%f

0.5% – 1.5%e

1.5%g

Saturated liquid

0.05%h

Saturated vapor 0.05%

a for temperatures up to 450 K; uncertainties at higher temperatures are given in figures 3 and 4 for methane and in [2] for

nitrogen, carbon dioxide, and ethane. b

This uncertainty range is not valid over the entire temperature and pressure ranges specified in the table; further details

are given in figure 4 for methane and in [2] for nitrogen, carbon dioxide, and ethane. c

States at pressures p > 100 MPa are not included in the table due to their limited technical relevance, but can be

obtained from figures 3 and 4 for methane and from [2] for nitrogen, carbon dioxide, and ethane. d

for methane; (1 – 2)% for nitrogen, (0.5 – 1)% for carbon dioxide, and ≥ 2% for ethane; see [2]. e

for methane; 1% (p ≤ 20 MPa) for nitrogen, (0.5 – 1)% for carbon dioxide, and (1 – 2)% for ethane; see [2].

f for methane, nitrogen, and ethane; 1% for carbon dioxide.

g for methane; ≥ 2% (p > 20 MPa) for nitrogen, (0.5 – 1)% for carbon dioxide, and ≥ 2% for ethane; see [2].

h for methane, nitrogen, and carbon dixoide; 0.1% for ethane.

Table 9 — Uncertainties of the equations of state for propane, n-butane, isobutane, n-pentane,

isopentane, n-hexane, n-heptane, n-octane, n-nonane, n-decane, hydrogen sulfide, oxygen,

and argon for various thermodynamic properties

Pressure range Uncertainty in

( , )T p w T p( , ) c T pp( , ) )(s

Tp ( )T ( )T

p ≤ 30 MPaa

0.2%b

(1 – 2)%c (1 – 2)%

c 0.2%

d 0.2% 0.4%

d,e

p > 30 MPaf 0.5% 2% 2% – – –

a Larger uncertainties exist in the extended critical region.

b In the extended critical region, p p/ is used instead of / .

c 1% at gaseous and gas-like supercritical states, 2% at liquid and liquid-like supercritical states.

d Larger relative uncertainties may result for small vapor pressures and their corresponding saturated vapor

densities. e

Combination of the uncertainties of the gas densities and vapor pressures; experimental data of this accuracy

are available for only a few substances. f

States at pressures p > 100 MPa are not included due to their limited technical relevance.

ISO/CD 20765-2


Figure 3 Uncertainty diagram for densities of methane calculated from the GERG-2008 equation of

state.

Figure 4 Uncertainty diagram for speeds of sound of methane calculated from the GERG-2008

equation of state.

ISO/CD 20765-2


7.2.3 Other secondary components

For oxygen and argon, the estimated uncertainties in calculated density, speed of sound, and isobaric heat

capacity are as stated in table 9 for the secondary alkanes.

The estimated uncertainties in calculated density for the other secondary components, namely hydrogen,

carbon monoxide, hydrogen sulfide, and helium, at supercritical temperatures and for pressures up to 30 MPa

are less than 0.2% and at higher pressures less than 0.5%. In general, higher uncertainties may occur in the

liquid phase and for other thermodynamic properties.

For hydrogen at temperatures above 270 K and pressures up to 30 MPa, the uncertainty in calculated density

is less than 0.1%. At pressures above 30 MPa, the uncertainty in density is slightly higher, approximately

(0.2 – 0.3)%. The equations for hydrogen and helium are designed to be valid at low (absolute) temperatures,

which occur in their sub- and supercritical regions. This is of particular importance for the mixture model

presented here since for pure substances the reduced temperature range 1.2 ≤ T/Tc ≤ 1.8 corresponds to the

region where the highest accuracy in the description of thermodynamic properties of typical natural gases is

demanded.

For water, calculated liquid densities and vapor pressures have an estimated uncertainty of 0.2%. Other

properties have higher uncertainties as detailed in the technical report.

7.3 Uncertainty for binary mixtures

The most accurate binary mixture data were used for the development of the GERG-2008 equation of state. However, in regions where these data are not available, less accurate data were also taken into account for the development and assessment of the equation. Experimental data for multi-component mixtures were used for the validation of the quality of the equation only. The total uncertainties of the most accurate experimental binary and multi-component mixture data with respect to selected thermodynamic properties are listed in table 10. The tabulated values represent the lowest uncertainties possible that can be achieved by the mixture model presented here. The corresponding experimental results are based on modern measurement techniques, which fulfill present quality standards. They are characterized by uncertainties equal to or below the lowest values listed in table 10. In contrast to the experimental uncertainties given for pure fluid properties measured using state-of-the-art techniques, the experimental uncertainties estimated for the properties of mixtures measured with the same apparatuses are, in general, higher due to the significant contribution of the uncertainty in the mixture composition.

Over wide ranges of temperature, pressure, and composition, the uncertainties tabulated below are mostly valid for those binary systems where binary specific departure functions were developed, see 6.2 and table 4. Due to limited experimental data (e.g., accurate speed of sound measurements are available for only a few binary systems), the uncertainties are partly valid for the remaining binary mixtures, including those binary systems for which a generalized departure function was developed, see 6.2 and [1].

General estimates of the uncertainties of the GERG-2008 equation of state in the description of selected thermodynamic properties are given in table 11. The different binary mixtures are distinguished by adjusted reducing functions and a binary specific departure function, adjusted reducing functions and a generalized departure function, or only adjusted reducing functions (without a departure function) to describe the mixtures. Uncertainty values are given for different pressures, temperatures, and (approximate) reduced temperature ranges.

ISO/CD 20765-2


Table 10 — Relative experimental uncertainties of the most accurate binary and multi-component mixture data

Data type Property Relative uncertainty

Density (gas phase) (0.05 – 0.1)%

Density (liquid phase) (0.1 – 0.3)%

Isochoric heat capacity vvcc (1 – 2)%

Speed of sound (gas phase) w w (0.05 – 0.1)%

Isobaric heat capacity c cp p (1 – 2)%

Enthalpy differences (gas phase) h h (0.2 – 0.5)%

Saturated liquid density (0.1 – 0.2)%

VLE data p ps s (1 – 3)%

NOTE h indicates a difference between two state points, h(T2,p2)-h(T1,p1).

From table 11 it is evident that binary systems with a binary specific departure function generally have the

lowest uncertainty for the different properties as compared to the other binary systems with either a

generalized departure function or only modified reducing parameters. Gas phase densities and speeds of

sound have uncertainties of ≤ 0.1% for binary mixtures with a binary specific departure function. The relative

uncertainty in isobaric and isochoric heat capacity is estimated to be less than (1 – 2)% in the homogeneous

gas, liquid, and supercritical regions independent of the type of developed binary equation.

Table 11 — Uncertainty of the GERG-2008 equation of state in the description of selected volumetric

and caloric properties of different binary mixturesa

Mixture regionb Adjusted reducing functions with a Only adjusted reducing

functions (no departure

function)

binary specific

departure function

generalized

departure function

Gas phase 0 – 30 MPa

1.2 T/Tr 1.4

0.1%

(0.1 – 0.2)%

(0.5 – 1)%


1.4 T/Tr 2.2

0.1%

0.1%

(0.3 – 0.5)%


1.2 T/Tr 1.4

w

w 0.1%

w

w 0.5%

w

w 1%


1.4 T/Tr 2.2

w

w 0.1%

w

w 0.3%

w

w 0.5%

Saturated liquid state

100 K ≤ T ≤ 140 K

(0.1 – 0.2)%

(0.2 – 0.5)%

(0.5 – 1)%

Liquid phase 0 – 40 MPa

T/Tr 0.7

(0.1 – 0.3)%

(0.2 – 0.5)%

(0.5 – 1)%

a The relative uncertainty in isobaric and isochoric heat capacity is estimated to be less than (1 – 2)%

in the homogeneous gas, liquid, and supercritical regions independent of the type of binary equation. b For a typical natural gas, temperatures of 250 K, 300 K, and 350 K correspond to reduced

temperatures T T xr( ) 1 of about 1.3, 1.5, and 1.8, respectively.

ISO/CD 20765-2


7.4 Uncertainty for natural gases

The GERG-2008 wide-range equation of state for natural gases and other (multi-component and binary) mixtures, consisting of the components listed in table 6, is valid in the gas phase, in the liquid phase, in the supercritical region, and for vapor-liquid equilibrium states. For natural gases and similar mixtures, a normal range of validity and an extended range of validity were defined. The extrapolation to temperatures and pressures even far beyond the extended range of validity yields reasonable results. The estimated uncertainties for the different ranges of validity, as described below, are based on the representation of the available experimental data for various thermodynamic properties of natural gases and other multi-component mixtures by the GERG-2008 equation of state as summarized in table 12.

In general, there are no restrictions in the composition range of binary and multi-component mixtures. But, since the estimated uncertainty of the GERG-2008 equation of state is based on the experimental data used for the development and evaluation of the equation, the uncertainty is mostly unknown for composition ranges not covered by experimental data. The data situation allows for well-founded uncertainty estimates only for selected properties and parts of the fluid surface.

Most of the available experimental data for multi-component mixtures describe the pT relation of natural

gases and similar mixtures in the gas phase. The majority of these data cover the temperature range 270 K ≤

T ≤ 350 K at pressures up to 30 MPa [1,2] and were measured for pipeline quality natural gas. There are a

number of additional experimental data available that define the composition range of wider quality natural gas, e.g., measurements on rich natural gases with comparatively high content of carbon dioxide, ethane, propane, and n-butane; see table 6 for the composition ranges defined for pipeline quality and expanded quality natural gases. As mentioned in 6.3, pipeline quality natural gases are a subset of the expanded quality natural gases.

Table 12 — Summary of the available data for volumetric and caloric properties of natural gases and

other multi-component mixtures

Data typea

Number of

data points

Covered ranges Maximum

number of

components

Temperature Pressure

T/K p/MPa

Density 21769 91.0 – 573 0.03 – 99.9 18

Speed of sound 1337 213 – 414 0.00 – 70.0 13

Isobaric heat capacity 325 105 – 350 0.5 – 30.0 8

Enthalpy differences 1166 105 – 422 0.2 – 16.5 10

Saturated liquid density 124 105 – 251 0.04 – 3.2 8

VLE data 2284 77.8 – 450 0.1 – 27.6 4

Total 27005 77.8 – 573 0.00 – 99.9 18 a

Further data not included in table 12 were used to validate the quality of the GERG-2008 equation of state, e.g., recent dew-point measurements for a number of different natural gases and other multi-component mixtures, see [2].

7.4.1 Uncertainty in the normal and expanded ranges of validity of natural gas

The normal range of validity of natural gas covers the temperature range 90 K ≤ T ≤ 450 K for pressures up to

35 MPa, see 6.3, table 5. This range corresponds to the use of the equation in both standard and advanced

technical applications using natural gases and similar mixtures, e.g., pipeline transport, natural gas storage,

and improved processes with liquefied natural gas. Estimated uncertainties for the composition subsets

―pipeline quality natural gas‖ and ―expanded quality natural gas‖ are summarized in table 13.

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Table 13 — Uncertainty of the GERG-2008 equation of state in the description of selected volumetric

and caloric properties of pipeline quality and expanded quality natural gases

Pipeline quality natural gas

Temperature region Pressure region Uncertainty

Density, gas phase 250 K T 450 K p 35 MPa

0.1%

Density, liquid phase 100 K T 140 K p 40 MPa

(0.1 – 0.5)%

Saturated liquid density 100 K T 140 K

(0.1 – 0.3)%

Speed of sound, gas phase 250 K T 270 K p 12 MPa w

w 0.1%

270 K T 450 K p 20 MPa w

w 0.1%

250 K T 270 K

250 K T 450 K

12 p 20 MPa

20 p 30 MPa

w

w (0.2 – 0.3)%

Enthalpy differences, gas

phase

250 K T 350 K p 20 MPa

h

h)( (0.2 – 0.5)%

Enthalpy differences, liquid

phase

h

h)( (0.5 – 1.0)%

Isobaric/isochoric heat capacity, gas and liquid phases

p

p

c

cor

v

v

c

c(1 – 2)%

Expanded rangea

Molar mass

Pressure region

Uncertainty

Density, gas phase M ≤ 26 kg∙kmol−1

p 30 MPa

0.1%

M > 26 kg∙kmol−1

p 30 MPa

(0.1 – 0.3)%

b

a For rich natural gases, i.e., for natural gas mixtures that contain comparatively large

amounts of carbon dioxide, ethane, propane, and further secondary alkanes, the

tested temperature range is as follows: 280 K T 350 K. b For mixtures with molar masses M > 30 kg∙kmol

−1 and compositions within the limits

stated in table 6, the upper uncertainty value in density is estimated to be 0.5%.

Pipeline quality natural gas Density data in the gas phase for pipeline quality natural gases are described by the equation with an

uncertainty of / ≤ 0.1% (over the temperature range 250 K ≤ T ≤ 450 K and for pressures up to 35 MPa).

The uncertainty in speed of sound is likewise less than 0.1%. However, due to limited experimental data, this uncertainty is restricted to pressures below 20 MPa; at temperatures below 270 K it is restricted to pressures below 12 MPa. The most accurate liquid or saturated liquid density data are described within 0.1% to 0.3%, which is in agreement with the estimated experimental uncertainty of the measurements.

Expanded quality natural gas This quality range of natural gases comprises a wider composition range than given by the pipeline quality natural gases. The wider composition range is almost identical to the composition range covered by the available experimental natural gas and similar multi-component mixture data, including several data sets for natural gases containing synthetic mixtures, ternary mixtures of natural gas main components, and rich

ISO/CD 20765-2


natural gases. Rich natural gases contain large amounts of carbon dioxide (up to 0.20 mole fraction), ethane (up to 0.18), propane (up to 0.14), n-butane (up to 0.06), n-pentane (0.005), and n-hexane (0.002).

For mixtures that fall within the wider composition range defined in table 6, the estimated uncertainty in gas

phase density is ≤ 0.1% for molar masses M ≤ 26 kg∙kmol−1

; see equation (18) for the calculation of the molar

mass from the given mixture composition. For mixtures with molar masses M > 26 kg∙kmol−1

, the uncertainty

in gas phase density is 0.1% to 0.3%. For other thermodynamic properties, well-founded estimates of uncertainty cannot be given due to the limited data situation.

NOTE 1 For rich natural gases, the lower temperature limit is increased because dew point temperatures are considerably higher for these types of mixtures, which contain comparatively large amounts of carbon dioxide, ethane,

propane, and the further secondary alkanes.

NOTE 2 Within the mole fraction limits defined for pipeline quality natural gas, the molar mass of any mixture will

always be lower than 26 kg∙kmol−1

.

Uncertainty in the extended range of validity, and calculation of properties beyond this range

The extended range of validity covers temperatures of 60 K ≤ T ≤ 700 K and pressures up to 70 MPa. The

uncertainty of the equation in gas phase density at temperatures and pressures outside the normal range of

validity is roughly estimated to be (0.2 – 0.5)%. For certain mixtures, the extended range of validity covers

temperatures of T > 700 K and pressures of p > 70 MPa. For example, the equation accurately describes gas

phase density data of air to within ±(0.1 – 0.2)% at temperatures up to 900 K and pressures up to 90 MPa.

For other thermodynamic properties, well-founded estimates of uncertainty cannot be given due to the limited

data situation outside the normal range of validity. However, the estimates given for the pure substance

equations and for the different binary systems give some clue as to what may be expected with respect to the

description of other thermodynamic properties in the extended range of validity.

When larger uncertainties are acceptable, tests have shown that the equation can be reasonably used outside

the extended range of validity. For example, density data (frequently of questionable and low accuracy

outside the extended range of validity) for certain binary mixtures are described to within ±(0.5 – 1)% at

pressures up to 100 MPa and more.

7.5 Uncertainties in other properties

An estimate of the uncertainty z of the equation of state in any property z other than the density and not

explicitly listed above, such as Joule-Thomson coefficient or isentropic exponent, can be obtained by

calculating

z z T x1 1 ( , , ) (37)

and

z z T x2 2 ( , , ) (38)

with

2 1 , (39)

where is the absolute uncertainty in density for given values of , T, and x . Then, the absolute

uncertainty in the property z corresponds to the difference

z z z 2 1 . (40)

This value most likely under estimates the uncertainty in the property.

7.6 Impact of uncertainties of input variables

The user should recognize that uncertainties in the input variables, usually pressure, temperature and composition by mole fractions, will have additional effects upon the uncertainty of any calculated result. The uncertainties given so far for calculated results in 7 assume the input data are exact. In any particular

ISO/CD 20765-2


application where the additional uncertainty may be of importance the user should carry out sensitivity tests to determine its magnitude. Varying the input variables may do this.

8 Reporting of results

When reported in accordance with the units given in Annex A, results for the thermodynamic properties shall be quoted with rounding to the number of digits after the decimal point as given in table 14. The report shall identify the temperature, pressure (or density), and detailed composition to which the results refer. The method of calculation used shall be identified by reference to ISO 20765-2 Natural gas – Calculation of thermodynamic properties.

For the validation of calculations and for subsequent calculations based on thermodynamic properties obtained using this standard it may be appropriate to carry extra digits (see example calculations in Annex G).

Table 14 — Reporting of results

Symbol Property Units Decimal places

Z Compression factor - 4

ρ Molar density kmol/m3 3

D Mass-based density kg/m3 2

p Pressure MPa 3

u Molar internal energy kJ/kmol 0

U Specific internal energy kJ/kg 1

h Molar enthalpy kJ/kmol 0

H Specific enthalpy kJ/kg 1

s Molar entropy kJ/(kmol·K) 2

S Specific entropy kJ/(kg·K) 3

g Molar Gibbs free energy kJ/kmol 0

G Specific Gibbs free energy kJ/kg 1

cv Molar isochoric heat capacity kJ/(kmol·K) 2

Cv Specific isochoric heat capacity kJ/(kg·K) 3

cp Molar isobaric heat capacity kJ/(kmol·K) 2

Cp Specific isobaric heat capacity kJ/(kg·K) 3

µ Joule-Thomson coefficient K/MPa 2

δT Isothermal throttling coefficient 10−3

·m3/kmol 2

Isentropic exponent - 2

w Speed of sound m/s 1

B Second virial coefficient 10−3

·m3/kmol 4

C Third virial coefficient 10−6

·m6/kmol

2 4

ISO/CD 20765-2


Annex A (normative)

Symbols and units

Symbol Meaning Source of values Units

a Molar Helmholtz free energy equation (1) kJ/kmol

B Second virial coefficient equation (32) m3/kmol

c Density exponent equation (9) ---

cp Molar isobaric heat capacity equation (26) kJ/(kmol·K)

cv Molar isochoric heat capacity equation (24) kJ/(kmol·K)

C Third virial coefficient equation (33) m6/kmol

2

Cv Specific isochoric heat capacity table 14 kJ/(kg·K)

Cp Specific isobaric heat capacity table 14 kJ/(kg·K)

d Density exponent equation (9) ---

D Mass-based density equation (19) kg/m3

F Mixture parameter equation (11) ---

g Molar Gibbs free energy equation (27) kJ/kmol

h Molar enthalpy equation (25) kJ/kmol

H Specific enthalpy table 14 kJ/kg

i Serial number equation (6) ---

j Serial number equation (10) ---

k Coverage factor Scope ---

k Serial number equation (7) ---

K Number of terms equation (9) ---

M Molar mass equation (18) kg/kmol

n Coefficient equation (7) ---

N Number of components in the mixture equation (6) ---

p Pressure equation (20) MPa

R Molar gas constant, R = 8.314 472, [4] equation (7) kJ/(kmol∙K)

R* Molar gas constant, R*= 8.314 51, used in [5] equation (7) kJ/(kmol∙K)

s Molar entropy equation (23) kJ/(kmol∙K)

S Specific entropy table 14 kJ/(kg∙K)

t Temperature exponent equation (9) ---

T Temperature in K (ITS-90) equation (1) K

u Molar internal energy equation (22) kJ/kmol

U Specific internal energy table 14 kJ/kg

ISO/CD 20765-2


v Molar volume equation (20) m3/kmol

w Speed of sound equation (28) m/s

x Mole fraction equation (6) ---

xx Molar composition (vector of mole fractions) equation (1) ---

z Any property equation (37) ---

Z Compression factor equation (21) ---

Greek Symbols

Reduced molar Helmholtz free energy, a/(RT) equation (2) ---

Parameter equation (12) ---

T Parameter equation (14) ---

V Parameter equation (13) ---


T Parameter equation (14) ---

V Parameter equation (13) ---

Reduced density, /r equation (2) ---

Isothermal throttling coefficient equation (30) m³/kmol


Partial derivative equation (15) ---

Departure function for the reduced molar Helmholtz

free energy equation (5) ---



Isentropic exponent equation (31) ---

JT Joule-Thomson coefficient equation (29) K/MPa

Molar density equation (1) kmol/m3

Inverse reduced temperature, r equation (2) ---

Inferior indices

c At the critical point equation (7) ---

Exp Exponential term equation (9) ---

i Serial number equation (6) ---

j Serial number equation (10) ---

k Serial number equation (7) ---

o Property of the pure substance equation (5) ---

Pol Polynomial term equation (9) ---

r Reducing property equation (3) ---

s At saturation (phase equilibrium) table 9 ---

Partial derivative with respect to equation (15) ---

Partial derivative with respect to equation (15) ---

ISO/CD 20765-2


0 Reference state, T0 = 298.15 K, p0 = 0.101325 MPa equation (B.4) ---

Superior indices

o Ideal-gas state equation (1) ---

r Residual part equation (1) ---

' Saturated liquid state table 9 ---

" Saturated vapor state table 9 ---

ISO/CD 20765-2


Annex B (normative)

The reduced Helmholtz free energy of the ideal gas

B.1 Calculation of the reduced Helmholtz free energy of the ideal gas

For a pure component, the Helmholtz free energy of the ideal gas is given by

),()(),( TTsRTThTa ooo . (B.1)

For the ideal gas, the enthalpy ho is a function of temperature only, whereas the entropy s

o depends on

temperature and density. Both properties can be expressed in terms of the ideal-gas isobaric heat capacity cop

as follows:

o

0

oo

0

d)( hTcThT

Tp , (B.2)

o

0o

0

o

o lnd),(0

sRTT

RcTs

T

T

p

. (B.3)

When the above expressions for ho(T) and s

o(,T) are inserted into equation (B.1), one obtains

o

0o

0

o

o

0

oo lndd),(00

sRTT

RcTRThTcTa

T

T

pT

Tp

, (B.4)

where all variables with the subscript ―0‖ refer to an arbitrary reference state. The reference state of zero

enthalpy and zero entropy is here adopted at T0 = 298.15 K and p0 = 0.101 325 MPa for the ideal gas. The

integration constants ho0 and s

o0 are then determined so as to conform to this definition. The reference density

o0 is given by

o0=p0/(RT0).

In order to obtain an explicit equation for ao(,T), an equation for the ideal-gas isobaric heat capacity c

op is

needed. The ideal-gas isobaric heat capacity may be written as follows [2]

7,5

2

co

co

o

6,4

2

co

co

oo

3

o

coshsinh

1k

k

k

k

k

k

k

k

p

T

TT

T

n

T

TT

T

nnR

c

. (B.5)

The values of the coefficients n0k and parameters

0k of equation (B.5) are given in table B.1.

NOTE The equation for the ideal-gas isobaric heat capacity is taken from [5] and given as a function of the temperature

T. The values resulting from the parameter product ok Tc (for k = 4 to 7) are in agreement with the original

ISO/CD 20765-2


parameters DO, F

O, H

O, and J

O, respectively, and the coefficients n

ok (for k = 4 to 7) have identical values as the

coefficients CO

, EO

, GO, and I

O, respectively. The coefficient n

o

3=Bo–1.

For a mixture at a given mixture density , temperature T, and molar composition x , the reduced Helmholtz

free energy of the ideal gas is given as follows:

N

i

iii xTxxT

1

oo

o ln),(),,( . (B.6)

In this equation, ooi(, T ) is the dimensionless form of the Helmholtz free energy in the ideal-gas state of

component i as given by

),(oo Ti

6,4

,

,,

,

3,

,

2,1,

,

sinhlnlnlnk

i

kiki

i

i

i

ii

i T

Tn

T

Tn

T

Tnn

R

R coo

oo

coo

coo

oo

c

7,5

,co,o

o,o coshln

k

i

kiki T

Tn , (B.7)

and the term ii xx ln accounts for the entropy of mixing of the ideal-gas mixture.

NOTE To indicate a pure substance as a component in a mixture, the inferior indices ―o‖ (referring to pure substance)

and ―i‖ (referring to the considered component) are introduced.

It is important to observe that in equation (B.6) o is a function of the molar density , the temperature T, and

the molar composition x , where is the molar density of the real mixture (i.e., not the molar density of the

ideal gas). If the input variables are the absolute pressure p, the temperature T and the molar composition x ,

may be calculated from the input variables as described in 5.

B.2 Derivatives of the reduced Helmholtz free energy of the ideal gas

For some of the thermodynamic properties, the calculation requires first and second partial derivatives of the

reduced Helmholtz free energy of the ideal gas o= o (, T , x ) (equation (B.6)) with respect to the reduced

mixture density or inverse reduced mixture temperature . The relevant mathematical expressions are as

follows:

1

1

o

,c

r

,

oo

*

r

,/

o

N

iT

i

i

x

R

R

ic

ix (B.8)

2

*2

r

,

1

12

o22

,c

r

,

2

o2o

/

o

R

RN

iT

i

i

x ic

ix

(B.9)

0

//1

o2

r

,c

,c

r

o2o

,,

o

N

i

i

i

i

x

TTT

Tx

icic

i

(B.10)

ISO/CD 20765-2


N

i

i

i

x

TTT

Tx

ic

i

1

o

r

,c

,

oo

/,

o

(B.11)

7,5

,co

,o

o

,o

o

,o

6,4 ,co

,o

o

,oo

,o

,c

o

3,o

o

2,o

o

tanh

tanh/,

o

k

i

kikiki

k i

ki

ki

ki

i

ii T

Tn

T

Tn

T

Tnn

R

R

TT ic

i

(B.12)

N

i

i

i

xTTT

Tx

ic

i

12

o22

r

,c

,

2

o2o

/,

o

(B.13)

7,52

,co

,o

2o

,oo

,o

6,42

,co

,o

2o

,oo

,o

2

,c

o

3,o2

o2

coshsinh

/,

o

k

i

ki

ki

ki

k

i

ki

ki

ki

i

i

T

T

n

T

T

nT

Tn

R

R

TT ic

i

(B.14)

ISO/CD 20765-2


Table B.1 — Values of the coefficients and parameters in equation (B.7) for the 21 componentsa

k n i koo

, ooi k, k n i ko

o, o

oi k,

Methane

1 19.597508817 – 5 0.00460 0.936220902

2 −83.959667892 – 6 8.74432 5.577233895

3 3.00088 – 7 −4.46921 5.722644361

4 0.76315 4.306474465

Nitrogen

1 11.083407489 – 5 −0.14660 −5.393067706

2 −22.202102428 – 6 0.90066 13.788988208

3 2.50031 – 7 – –

4 0.13732 5.251822620

Carbon dioxide

1 11.925152758 – 5 −1.06044 −2.844425476

2 −16.118762264 – 6 2.03366 1.589964364

3 2.50002 – 7 0.01393 1.121596090

4 2.04452 3.022758166

Ethane

1 24.675437527 – 5 1.23722 0.731306621

2 −77.425313760 – 6 13.1974 3.378007481

3 3.00263 – 7 −6.01989 3.508721939

4 4.33939 1.831882406

Propane

1 31.602908195 – 5 3.19700 0.543210978

2 −84.463284382 – 6 19.1921 2.583146083

3 3.02939 – 7 −8.37267 2.777773271

4 6.60569 1.297521801

n-Butane

1 20.884168790 – 5 6.89406 0.431957660

2 −91.638478026 – 6 24.4618 4.502440459

3 3.33944 – 7 14.7824 2.124516319

4 9.44893 1.101487798

ISO/CD 20765-2


Isobutane

1 20.413726078 – 5 5.25156 0.485556021

2 −94.467620036 – 6 25.1423 4.671261865

3 3.06714 – 7 16.1388 2.191583480

4 8.97575 1.074673199

n-Pentane

1 14.536611217 – 5 21.8360 1.789520971

2 −89.919548319 – 6 33.4032 3.777411113

3 3.0 – 7 – –

4 8.95043 0.380391739

Isopentane

1 15.449907693 – 5 20.1101 1.977271641

2 −101.298172792 – 6 33.1688 4.169371131

3 3 – 7 – –

4 11.7618 0.635392636

n-Hexane

1 14.345969349 – 5 26.8142 1.691951873

2 −96.165722367 – 6 38.6164 3.596924107

3 3.0 – 7 – –

4 11.6977 0.359036667

n-Heptane

1 15.063786601 – 5 30.4707 1.548136560

2 −97.345252349 – 6 43.5561 3.259326458

3 3.0 – 7 – –

4 13.7266 0.314348398

n-Octane

1 15.864687161 – 5 33.8029 1.431644769

2 −97.370667555 – 6 48.1731 2.973845992

3 3.0 – 7 – –

4 15.6865 0.279143540

n-Nonane

1 16.313913248 – 5 38.12350 1.370586158

2 −102.160247463 – 6 53.34150 2.848860483

3 3.0 – 7 – –

4 18.02410 0.263819696

ISO/CD 20765-2


n-Decane

1 15.870791919 – 5 43.49310 1.353835195

2 −108.858547525 – 6 58.36570 2.833479035

3 3.0 – 7 – –

4 21.00690 0.267034159

Hydrogen

1 13.796443393 – 5 0.45444 9.847634830

2 −175.864487294 – 6 1.56039 49.765290750

3 1.47906 – 7 −1.37560 50.367279301

4 0.95806 6.891654113

Oxygen

1 10.001843586 – 5 1.01334 7.223325463

2 −14.996095135 – 6 – –

3 2.50146 – 7 – –

4 1.07558 14.461722565

Carbon monoxide

1 10.813340744 – 5 0.00493 5.305158133

2 −19.834733959 – 6 – –

3 2.50055 – 7 – –

4 1.02865 11.675075301

Hydrogen Sulfide

1 9.336197742 – 5 1.00243 2.270653980

2 −16.266508995 – 6 – –

3 3.0 – 7 – –

4 3.11942 4.914580541

Water

1 8.203520690 – 5 0.98763 1.763895929

2 −11.996306443 – 6 3.06904 3.874803739

3 3.00392 – 7 – –

4 0.01059 0.415386589

Helium

1 13.628409737 – 3 1.5 –

2 −143.470759602 –

Argon

1 8.316631500 – 3 1.5 –

2 −4.946502600 – a

The values of the coefficients and parameters are also valid for equation (7).

ISO/CD 20765-2


Annex C (normative)

Values of critical parameters and molar masses of the pure components

Table C.1 — Critical parameters and molar masses of the 21 components

Component Formula c,i /(mol∙dm−3) T ic, /K Mi /(g∙mol−1)a

Methane CH4 10.139342719 190.564 16.04246

Nitrogen N2 11.1839 126.192 28.0134

Carbon dioxide CO2 10.624978698 304.1282 44.0095

Ethane C2H6 6.870854540 305.322 30.06904

Propane C3H8 5.000043088 369.825 44.09562

n-Butane n-C4H10 3.920016792 425.125 58.1222

Isobutane i-C4H10 3.860142940 407.817 58.1222

n-Pentane n-C5H12 3.215577588 469.7 72.14878

Isopentane i-C5H12 3.271 460.35 72.14878

n-Hexane n-C6H14 2.705877875 507.82 86.17536

n-Heptane n-C7H16 2.315324434 540.13 100.20194

n-Octane n-C8H18 2.056404127 569.32 114.22852

n-Nonane n-C9H20 1.81 594.55 128.2551

n-Decane n-C10H22 1.64 617.7 142.28168

Hydrogen H2 14.94 33.19 2.01588

Oxygen O2 13.63 154.595 31.9988

Carbon monoxide CO 10.85 132.86 28.0101

Water H2O 17.873716090 647.096 18.01528

Hydrogen sulfide H2S 10.19 373.1 34.08088

Helium He 17.399 5.1953 4.002602

Argon Ar 13.407429659 150.687 39.948 a According to IUPAC Technical Report (2006) [6].

ISO/CD 20765-2


Annex D (normative)

The residual part of the reduced Helmholtz free energy

D.1 Calculation of the residual part of the reduced Helmholtz free energy

The residual part r of the reduced Helmholtz free energy of the mixture is given by

),,(),,(),,( xxx rr

o

r , (D.1)

where

N

i

iixx1

),(),,( r

o

r

o (D.2)

and

1

1 1

rr ),(),,(

N

i

N

ij

ijijji Fxxx . (D.3)

In equation (D.1), the first term on the right-hand side, ro( , , x ), describes the contribution of the residual

parts of the reduced Helmholtz free energy of the pure substance equations of state, which are multiplied by

the mole fraction of the corresponding substance and linearly combined using the reduced variables and

of the mixture, to the residual part of the reduced Helmholtz free energy of the mixture. The second term is the

departure function, r ( , , x ), which is the summation over all binary specific and generalized departure

functions. The variables and are the reduced mixture density and inverse reduced mixture temperature,

respectively, as given by

r( )x (D.4)

and

T x

T

r( ) , (D.5)

where r and Tr are reducing functions for the mixture density and mixture temperature depending on the

molar composition of the mixture only (see 4.2.7 and Annex E.1).

D.1.1 Derivatives of the residual part of the reduced Helmholtz free energy

The derivatives of the residual part r( , , x ) of the reduced Helmholtz free energy of the mixture with

respect to the reduced mixture variables and are as follows:

ISO/CD 20765-2


1

1 1

r

1

r

o

,

rr

N

i

N

ij

ij

ijji

N

i

i

i

x

Fxxx

(D.6)

1

1 12

r2

12

r

o

2

,

2

r2r

N

i

N

ij

ij

ijji

N

i

i

i

x

Fxxx

(D.7)

1

1 1

r2

1

r

o

2r2

rN

i

N

ij

ij

ijji

N

i

i

i

x

Fxxx

(D.8)

1

1 1

r

1

r

o

,

rr

N

i

N

ij

ij

ijji

N

i

i

i

x

Fxxx

(D.9)

1

1 12

r2

12

r

o

2

,

2

r2r

N

i

N

ij

ij

ijji

N

i

i

i

x

Fxxx

(D.10)

D.2 Calculation of the pure substance contribution to the residual part of the reduced Helmholtz free energy

The contribution of the residual parts of the reduced Helmholtz free energy of the pure substance equations of

state ro to the residual part of the reduced Helmholtz free energy of the mixture is given by equation (D.2),

where the residual part of the reduced Helmholtz free energy of component i, roi( , ), (i.e., the residual part

of the respective pure substance equation of state) is given by

ii

i

kikiki

i

kiki

KK

Kk

ctd

ki

K

k

td

i,ki enn,,

,

,,,

,

,,

1

,

1

),(ExpPol

Pol

ooo

Pol

oo

oo

r

o

. (D.11)

For each of the pure substances, the equations for roi use the same basic structure, however, the number of

terms differ. For the main components, the equations consist of 24 terms for methane, nitrogen, and ethane and of 22 terms for carbon dioxide. For the secondary alkanes and the other secondary components (oxygen, carbon monoxide, helium, hydrogen sulfide, and argon), 12 terms are used, while the equations for hydrogen

and water consist of 14 and 16 terms, respectively. The values of the coefficients noi,k and exponents doi,k, toi,k,

and coi,k for the 21 components are given in Annex D.2.2.

D.2.1 Derivatives of ),(r

o i with respect to the reduced mixture variables

The derivatives of the residual part of the reduced Helmholtz free energy of component i, roi( , ), (equation

(D.11)) with respect to the reduced mixture variables and are as follows:

ISO/CD 20765-2


ii

i

kikikiki

ikiki

KK

Kk

ctc

kiki

d

ki

K

k

td

kiki

i

cdn

dn

,Exp,Pol

,Pol

,o,o,o,o

,Pol,o,o

1,o,o

1

,o

1

1

,o,o

r

o

exp

(D.12)

ii

i

kikikikikiki

ikiki

KK

Kk

ctc

ki

c

kiki

c

kiki

d

ki

K

k

td

kikiki

i

ccdcdn

ddn

,Exp,Pol

,Pol

,o,o,o,o,o,o

,Pol,o,o

1

2

,o,o,o,o,o

2

,o

1

2

,o,o,o2

r

o

2

exp1

1

(D.13)

ii

i

kikikiki

ikiki

KK

Kk

ctc

kiki

d

kiki

K

k

td

kikiki

i

cdtn

tdn

,Exp,Pol

,Pol

,o,o,o,o

,Pol,o,o

1

1

,o,o

1

,o,o

1

11

,o,o,o

r

o

2

exp

(D.14)

ii

i

kikiki

ikiki

KK

Kk

ctd

kiki

K

k

td

kiki

i

tn

tn

,Exp,Pol

,Pol

,o,o,o

,Pol,o,o

1

1

,o,o

1

1

,o,o

r

o

exp

(D.15)

ii

i

kikiki

ikiki

KK

Kk

ctd

kikiki

K

k

td

kikiki

i

ttn

ttn

,Exp,Pol

,Pol

,o,o,o

,Pol,o,o

1

2

,o,o,o

1

2

,o,o,o2

r

o

2

exp1

1

(D.16)

ISO/CD 20765-2


D.2.2 Coefficients and exponents of ),(ro i

Table D.1 — Values of the coefficients and exponents in equation (D.11) for methane, nitrogen, and ethane a,b

.

k n i ko , n i ko , n i ko ,

Methane Nitrogen Ethane

1 0.57335704239162 0.59889711801201 0.63596780450714

2 −0.16760687523730×101

−0.16941557480731×101

−0.17377981785459×101

3 0.23405291834916 0.24579736191718 0.28914060926272

4 −0.21947376343441 −0.23722456755175 −0.33714276845694

5 0.16369201404128×10−1

0.17954918715141×10−1

0.22405964699561×10−1

6 0.15004406389280×10−1

0.14592875720215×10−1

0.15715424886913×10−1

7 0.98990489492918×10−1

0.10008065936206 0.11450634253745

8 0.58382770929055 0.73157115385532 0.10612049379745×101

9 −0.74786867560390 −0.88372272336366 −0.12855224439423×101

10 0.30033302857974 0.31887660246708 0.39414630777652

11 0.20985543806568 0.20766491728799 0.31390924682041

12 −0.18590151133061×10−1

−0.19379315454158×10−1

−0.21592277117247×10−1

13 −0.15782558339049 −0.16936641554983 −0.21723666564905

14 0.12716735220791 0.13546846041701 −0.28999574439489

15 −0.32019743894346×10−1

−0.33066712095307×10−1

0.42321173025732

16 −0.68049729364536×10−1

−0.60690817018557×10−1

0.46434100259260×10−1

17 0.24291412853736×10−1

0.12797548292871×10−1

−0.13138398329741

18 0.51440451639444×10−2

0.58743664107299×10−2

0.11492850364368×10−1

19 −0.19084949733532×10−1

−0.18451951971969×10−1

−0.33387688429909×10−1

20 0.55229677241291×10−2

0.47226622042472×10−2

0.15183171583644×10−1

21 −0.44197392976085×10−2

−0.52024079680599×10−2

−0.47610805647657×10−2

22 0.40061416708429×10−1

0.43563505956635×10−1

0.46917166277885×10−1

23 −0.33752085907575×10−1

−0.36251690750939×10−1

−0.39401755804649×10−1

24 −0.25127658213357×10−2

−0.28974026866543×10−2

−0.32569956247611×10−2

k c i ko , d i ko , t i ko , k c i ko , d i ko , t i ko ,

1 – 1 0.125 13 2 2 4.5

2 – 1 1.125 14 2 3 4.75

3 – 2 0.375 15 2 3 5

4 – 2 1.125 16 2 4 4

5 – 4 0.625 17 2 4 4.5

6 – 4 1.5 18 3 2 7.5

7 1 1 0.625 19 3 3 14

8 1 1 2.625 20 3 4 11.5

9 1 1 2.75 21 6 5 26

10 1 2 2.125 22 6 6 28

11 1 3 2 23 6 6 30

12 1 6 1.75 24 6 7 16 a The values of the coefficients and exponents are also valid for equation (9).

b KPol,i = 6, KExp,i = 18.

ISO/CD 20765-2


Table D.2 — Values of the coefficients and exponents in equation (D.11) for propane, n-butane,

isobutane, n-pentane, isopentane, n-hexane, n-heptane, n-octane, n-nonane, n-

decane, oxygen, carbon monoxide, hydrogen sulfide, and argona-c

.

k n i ko , n i ko , n i ko ,

Propane n-Butane Isobutane

1 0.10403973107358×101

0.10626277411455×101

0.10429331589100×101

2 −0.28318404081403×101

−0.28620951828350×101

−0.28184272548892×101

3 0.84393809606294 0.88738233403777 0.86176232397850

4 −0.76559591850023×10−1

−0.12570581155345 −0.10613619452487

5 0.94697373057280×10−1

0.10286308708106 0.98615749302134×10−1

6 0.24796475497006×10−3

0.25358040602654×10−3

0.23948208682322×10−3

7 0.27743760422870 0.32325200233982 0.30330004856950

8 −0.43846000648377×10−1

−0.37950761057432×10−1

−0.41598156135099×10−1

9 −0.26991064784350 −0.32534802014452 −0.29991937470058

10 −0.69313413089860×10−1

−0.79050969051011×10−1

−0.80369342764109×10−1

11 −0.29632145981653×10−1

−0.20636720547775×10−1

−0.29761373251151×10−1

12 0.14040126751380×10−1

0.57053809334750×10−2

0.13059630303140×10−1

n-Pentane Isopentane n-Hexane

1 0.10968643098001×101

0.11017531966644×101

0.10553238013661×101

2 −0.29988888298061×101

−0.30082368531980×101

−0.26120615890629×101

3 0.99516886799212 0.99411904271336 0.76613882967260

4 −0.16170708558539 −0.14008636562629 −0.29770320622459

5 0.11334460072775 0.11193995351286 0.11879907733358

6 0.26760595150748×10−3

0.29548042541230×10−3

0.27922861062617×10−3

7 0.40979881986931 0.36370108598133 0.46347589844105

8 −0.40876423083075×10−1

−0.48236083488293×10−1

0.11433196980297×10−1

9 −0.38169482469447 −0.35100280270615 −0.48256968738131

10 −0.10931956843993 −0.10185043812047 −0.93750558924659×10−1

11 −0.32073223327990×10−1

−0.35242601785454×10−1

−0.67273247155994×10−2

12 0.16877016216975×10−1

0.19756797599888×10−1

−0.51141583585428×10−2

n-Heptane n-Octane n-Nonane

1 0.10543747645262×101

0.10722544875633×101

0.11151x101

2 −0.26500681506144×101

−0.24632951172003×101

−0.27020x101

3 0.81730047827543 0.65386674054928 0.83416

4 −0.30451391253428 −0.36324974085628 −0.38828

5 0.12253868710800 0.12713269626764 0.13760

6 0.27266472743928×10−3

0.30713572777930×10−3

0.28185x10−3

7 0.49865825681670 0.52656856987540 0.62037

8 −0.71432815084176×10−3

0.19362862857653×10−1

0.15847x10−1

9 −0.54236895525450 −0.58939426849155 −0.61726

10 −0.13801821610756 −0.14069963991934 −0.15043

11 −0.61595287380011×10−2

−0.78966330500036×10−2

−0.12982x10−1

12 0.48602510393022×10−3

0.33036597968109×10−2

0.44325x10−2

ISO/CD 20765-2


n-Decane Oxygen Carbon monoxide

1 0.10461×101 0.88878286369701 0.92310041400851

2 −0.24807×101 −0.24879433312148×10

1 −0.24885845205800×10

1

3 0.74372 0.59750190775886 0.58095213783396

4 −0.52579 0.96501817061881×10−2

0.28859164394654×10−1

5 0.15315 0.71970428712770×10−1

0.70256257276544×10−1

6 0.32865×10−3 0.22337443000195×10

−3 0.21687043269488×10

−3

7 0.84178 0.18558686391474 0.13758331015182

8 0.55424×10−1 −0.38129368035760×10

−1 −0.51501116343466×10

−1

9 −0.73555 −0.15352245383006 −0.14865357483379

10 −0.18507 −0.26726814910919×10−1

−0.38857100886810×10−1

11 −0.20775×10−1 −0.25675298677127×10

−1 −0.29100433948943×10

−1

12 0.12335×10−1 0.95714302123668×10

−2 0.14155684466279×10

−1

Hydrogen Sulfide Argon

1 0.87641 0.85095714803969

2 −0.20367×101 −0.24003222943480×10

1

3 0.21634 0.54127841476466

4 −0.50199×10−1 0.16919770692538×10

−1

5 0.66994×10−1 0.68825965019035×10

−1

6 0.19076×10−3 0.21428032815338×10

−3

7 0.20227 0.17429895321992

8 −0.45348×10−2 −0.33654495604194×10

−1

9 −0.22230 −0.13526799857691

10 −0.34714×10−1 −0.16387350791552×10

−1

11 −0.14885×10−1 −0.24987666851475×10

−1

12 0.74154×10−2 0.88769204815709×10

−2

k c i ko , d i ko , t i ko , k c i ko , d i ko , t i ko ,

1 – 1 0.25 7 1 2 0.625

2 – 1 1.125 8 1 5 1.75

3 – 1 1.5 9 2 1 3.625

4 – 2 1.375 10 2 4 3.625

5 – 3 0.25 11 3 3 14.5

6 – 7 0.875 12 3 4 12 a The values of the coefficients and exponents are also valid for equation (9).


c For the simultaneously optimized equations of state of Span and Wagner (2003) [10], the old molar

gas constant R* was substituted with the recent one R without conversion. This has nearly no effect

on the quality of the equations of state (page 104 of [2]).

ISO/CD 20765-2


Table D.3 — Values of the coefficients and exponents in equation (D.11) for carbon dioxide,

hydrogen, water, and heliuma.

k c i ko , d i ko , t i ko , n i ko ,

Carbon dioxideb

1 – 1 0 0.52646564804653

2 – 1 1.25 −0.14995725042592×101

3 – 2 1.625 0.27329786733782

4 – 3 0.375 0.12949500022786

5 1 3 0.375 0.15404088341841

6 1 3 1.375 −0.58186950946814

7 1 4 1.125 −0.18022494838296

8 1 5 1.375 −0.95389904072812×10−1

9 1 6 0.125 −0.80486819317679×10−2

10 1 6 1.625 −0.35547751273090×10−1

11 2 1 3.75 −0.28079014882405

12 2 4 3.5 −0.82435890081677×10−1

13 3 1 7.5 0.10832427979006×10−1

14 3 1 8 −0.67073993161097×10−2

15 3 3 6 −0.46827907600524×10−2

16 3 3 16 −0.28359911832177×10−1

17 3 4 11 0.19500174744098×10−1

18 5 5 24 −0.21609137507166

19 5 5 26 0.43772794926972

20 5 5 28 −0.22130790113593

21 6 5 24 0.15190189957331×10−1

22 6 5 26 −0.15380948953300×10−1

Hydrogenc

1 – 1 0.5 0.53579928451252×101

2 – 1 0.625 −0.62050252530595×101

3 – 2 0.375 0.13830241327086

4 – 2 0.625 −0.71397954896129×10−1

5 – 4 1.125 0.15474053959733×10−1

6 1 1 2.625 −0.14976806405771

7 1 5 0 −0.26368723988451×10−1

8 1 5 0.25 0.56681303156066×10−1

9 1 5 1.375 −0.60063958030436×10−1

10 2 1 4 −0.45043942027132

11 2 1 4.25 0.42478840244500

12 3 2 5 −0.21997640827139×10−1

13 3 5 8 −0.10499521374530×10−1

14 5 1 8 −0.28955902866816×10−2

ISO/CD 20765-2


Table D.3 (continued)

k c i ko , d i ko , t i ko , n i ko ,

Waterd

1 – 1 0.5 0.82728408749586

2 – 1 1.25 −0.18602220416584×101

3 – 1 1.875 −0.11199009613744×101

4 – 2 0.125 0.15635753976056

5 – 2 1.5 0.87375844859025

6 – 3 1 −0.36674403715731

7 – 4 0.75 0.53987893432436×10−1

8 1 1 1.5 0.10957690214499×101

9 1 5 0.625 0.53213037828563×10−1

10 1 5 2.625 0.13050533930825×10−1

11 2 1 5 −0.41079520434476

12 2 2 4 0.14637443344120

13 2 4 4.5 −0.55726838623719×10−1

14 3 4 3 −0.11201774143800×10−1

15 5 1 4 −0.66062758068099×10−2

16 5 1 6 0.46918522004538×10−2

Heliume

1 – 1 0 −0.45579024006737

2 – 1 0.125 0.12516390754925×101

3 – 1 0.75 −0.15438231650621×101

4 – 4 1 0.20467489707221×10−1

5 1 1 0.75 −0.34476212380781

6 1 3 2.625 −0.20858459512787×10−1

7 1 5 0.125 0.16227414711778×10−1

8 1 5 1.25 −0.57471818200892×10−1

9 1 5 2 0.19462416430715×10−1

10 2 2 1 −0.33295680123020×10−1

11 3 1 4.5 −0.10863577372367×10−1

12 3 2 5 −0.22173365245954×10−1

a The values of the coefficients and exponents are also valid for equation (9).


c KPol,i = 5, KExp,i = 9.

d KPol,i = 7, KExp,i = 9.

e KPol,i = 4, KExp,i = 8.

ISO/CD 20765-2


D.3 Calculation of the departure function contribution to the residual part of the reduced Helmholtz free energy

The purpose of the departure function is to further improve the accuracy of the mixture model in the

description of thermodynamic properties in cases where fitting the parameters of the reducing functions for

density and temperature (see Annex E) to accurate experimental data is not sufficient. The departure function

r of the multicomponent mixture is the double summation over all binary specific and generalized departure

functions developed for the binary subsystems and is given by

1

1 1

rr ),,(),,(

N

i

N

ij

ij xx (D.17)

with

),(),,( rr ijijjiij Fxxx . (D.18)

In this equation, the function rij( , ) is the part of the departure function

rij( , , x ) that depends only on

the reduced mixture variables and , as given by

ijij

ij

kijkijkijkijkijkij

ij

kijkij

KK

Kk

tdkij

K

k

tdkijij enn

,Exp,Pol

,Pol

,,2

,,,,

,Pol

,,

1

,

1

,r ),(

, (D.19)

where rij( , ) was developed either for a specific binary mixture (a binary specific departure function

with binary specific coefficients, exponents, and parameters) or for a group of binary mixtures (generalized

departure function with a uniform structure for the group of binary mixtures).

The values of the coefficients nij,k, the exponents dij,k and tij,k, and the parameters ij,k, ij,k, ij,k, and ij,k in

equation (D.19) for all binary specific and generalized departure functions considered in the mixture model are

given in table D.4. The values of the non-zero Fij parameters in equation (D.18) are listed in table D.5.

D.3.1 Binary specific departure functions

Binary specific departure functions were developed for the binary mixtures of methane with nitrogen, carbon

dioxide, ethane, propane, and hydrogen, and of nitrogen with carbon dioxide and ethane. For a binary specific

departure function, the adjustable factor Fij in equation (D.18) equals unity.

D.3.2 Generalised departure functions

A generalized departure function was developed for the binary mixtures of methane with n-butane and

isobutane, of ethane with propane, n-butane, and isobutane, of propane with n-butane and isobutane, and of

n-butane with isobutane. For each mixture in the group of generalized binary mixtures, the parameter Fij is

fitted to the corresponding binary specific data (except for the binary system methane–n-butane, where Fij

equals unity).

D.3.3 No departure functions

For all of the remaining binary mixtures, no departure function was developed, and Fij equals zero, i.e.,

rij( , , x ) equals zero. For most of these mixtures, however, the parameters of the reducing functions for

density and temperature are fitted to selected experimental data (see Annex E).

ISO/CD 20765-2


D.3.4 Derivatives of ),(r ij with respect to the reduced mixture variables and .

The derivatives of the function rij( , ) (equation D.19) with respect to the reduced mixture variables and

are as follows:

ijij

ij

kijkij

ijkijkij

KK

Kkkijkijkij

kij

kijkijkijkij

td

kij

K

k

td

kijkij

ij

d

n

dn

,Exp,Pol

,Pol

,,

,Pol,,

1,,,

,

,,

2

,,,

1

1

,,

r

2exp

(D.20)

kij

kij

kijkijkij

kij

kijkijkijkij

dd

ijij

ij

kijkijij

kijkij

KK

Kk

td

kij

K

k

td

kijkijkij

ijnddn

,2

,

2

,,,

,

,,

2

,, 22exp

,Exp,Pol

,Pol

,,,Pol

,,

1,

1

2

,,,2

r2

1

(D.21)

kijkijkij

kij

kijkijkijkij

d

ijij

ij

kijkijij

kijkij

KK

Kk

td

kijkij

K

k

td

kijkijkij

ijtntdn

,,,

,

,,

2

,, 2exp

,Exp,Pol

,Pol

,,,Pol

,,

1

1

,,1

11

,,,

r2

(D.22)

ijij

ij

kijkij

ijkijkij

KK

Kkkijkijkijkij

td

kijkij

K

k

td

kijkij

ij

tn

tn

,Exp,Pol

,Pol

,,

,Pol,,

1,,

2

,,

1

,,

1

1

,,

r

exp

(D.23)

ijij

ij

kijkij

ijkijkij

KK

Kkkijkijkijkij

td

kijkijkij

K

k

td

kijkijkij

ij

ttn

ttn

,Exp,Pol

,Pol

,,

,Pol,,

1,,

2

,,

2

,,,

1

2

,,,2

r2

exp1

1

(D.24)

ISO/CD 20765-2


D.3.5 Coefficients, exponents, and parameters for the departure functions

Table D.4 — Values of the coefficients, exponents, and parameters in equation (D.19) for the

binary specific and generalized departure functionsa.

k dij k, tij k, nij k, ij k, ij k, kij ,

ij k,

Methane–Nitrogenb

1 1 0 −0.98038985517335×10−2

– – – –

2 4 1.85 0.42487270143005×10−3

– – – –

3 1 7.85 −0.34800214576142×10−1

1 0.5 1 0.5

4 2 5.4 −0.13333813013896 1 0.5 1 0.5

5 2 0 −0.11993694974627×10−1

0.25 0.5 2.5 0.5

6 2 0.75 0.69243379775168×10−1

0 0.5 3 0.5

7 2 2.8 −0.31022508148249 0 0.5 3 0.5

8 2 4.45 0.24495491753226 0 0.5 3 0.5

9 3 4.25 0.22369816716981 0 0.5 3 0.5

Methane–Carbon dioxidec

1 1 2.6 −0.10859387354942 – – – –

2 2 1.95 0.80228576727389×10−1

– – – –

3 3 0 −0.93303985115717×10−2

– – – –

4 1 3.95 0.40989274005848×10−1

1 0.5 1 0.5

5 2 7.95 −0.24338019772494 0.5 0.5 2 0.5

6 3 8 0.23855347281124 0 0.5 3 0.5

Methane–Ethaned

1 3 0.65 −0.80926050298746×10−3

– – – –

2 4 1.55 −0.75381925080059×10−3

– – – –

3 1 3.1 −0.41618768891219×10−1

1 0.5 1 0.5

4 2 5.9 −0.23452173681569 1 0.5 1 0.5

5 2 7.05 0.14003840584586 1 0.5 1 0.5

6 2 3.35 0.63281744807738×10−1

0.875 0.5 1.25 0.5

7 2 1.2 −0.34660425848809×10−1

0.75 0.5 1.5 0.5

8 2 5.8 −0.23918747334251 0.5 0.5 2 0.5

9 2 2.7 0.19855255066891×10−2

0 0.5 3 0.5

10 3 0.45 0.61777746171555×101

0 0.5 3 0.5

11 3 0.55 −0.69575358271105×101

0 0.5 3 0.5

12 3 1.95 0.10630185306388×101

0 0.5 3 0.5

ISO/CD 20765-2




ij k,

Methane–Propanee

1 3 1.85 0.13746429958576×10−1

– – – –

2 3 3.95 −0.74425012129552×10−2

– – – –

3 4 0 −0.45516600213685×10−2

– – – –

4 4 1.85 −0.54546603350237×10−2

– – – –

5 4 3.85 0.23682016824471×10−2

– – – –

6 1 5.25 0.18007763721438 0.25 0.5 0.75 0.5

7 1 3.85 −0.44773942932486 0.25 0.5 1 0.5

8 1 0.2 0.19327374888200×10−1

0 0.5 2 0.5

9 2 6.5 −0.30632197804624 0 0.5 3 0.5

Nitrogen–Carbon dioxidef

1 2 1.85 0.28661625028399 – – – –

2 3 1.4 −0.10919833861247 – – – –

3 1 3.2 −0.11374032082270×101

0.25 0.5 0.75 0.5

4 1 2.5 0.76580544237358 0.25 0.5 1 0.5

5 1 8 0.42638000926819×10−2

0 0.5 2 0.5

6 2 3.75 0.17673538204534 0 0.5 3 0.5

Nitrogen–Ethaneg

1 2 0 −0.47376518126608 – – – –

2 2 0.05 0.48961193461001 – – – –

3 3 0 −0.57011062090535×10−2

– – – –

4 1 3.65 −0.19966820041320 1 0.5 1 0.5

5 2 4.9 −0.69411103101723 1 0.5 1 0.5

6 2 4.45 0.69226192739021 0.875 0.5 1.25 0.5

Methane–Hydrogenh

1 1 2 −0.25157134971934 – – – –

2 3 −1 −0.62203841111983×10−2

– – – –

3 3 1.75 0.88850315184396×10−1

– – – –

4 4 1.4 −0.35592212573239×10−1

– – – –

ISO/CD 20765-2




ij k,

Methane–n-Butane, Methane–Isobutane, Ethane–Propane, Ethane–n-Butane,

Ethane–Isobutane, Propane–n-Butane, Propane–Isobutane, and n-Butane–Isobutanei

1 1 1 0.25574776844118×101

– – – –

2 1 1.55 −0.79846357136353×101

– – – –

3 1 1.7 0.47859131465806×101

– – – –

4 2 0.25 −0.73265392369587 – – – –

5 2 1.35 0.13805471345312×101

– – – –

6 3 0 0.28349603476365 – – – –

7 3 1.25 −0.49087385940425 – – – –

8 4 0 −0.10291888921447 – – – –

9 4 0.7 0.11836314681968 – – – –

10 4 5.4 0.55527385721943×10−4

– – – – a The values of the coefficients, exponents, and parameters are also valid for equation (12).

b KPol,ij = 2, KExp,ij = 7.

f KPol,ij = 2, KExp,ij = 4.

c KPol,ij = 3, KExp,ij = 3.

g KPol,ij = 3, KExp,ij = 3.

d KPol,ij = 2, KExp,ij = 10.

h KPol,ij = 4, KExp,ij = 0.

e KPol,ij = 5, KExp,ij = 4.

i KPol,ij = 10, KExp,ij = 0.

Table D.5 — Values of the non-zero Fij parameters in equation (D.18) for the binary specific and

generalized departure functionsa,b

.

Mixture i–j Fij

Methane–Nitrogen 1

Methane–Carbon dioxide 1

Methane–Ethane 1

Methane–Propane 1

Methane–n-Butane 1

Methane–Isobutane 0.771035405688

Methane–Hydrogen 1

Nitrogen–Carbon dioxide 1

Nitrogen–Ethane 1

Ethane–Propane 0.130424765150

Ethane–n-Butane 0.281570073085

Ethane–Isobutane 0.260632376098

Propane–n-Butane 0.312572600489×10−1

Propane–Isobutane −0.551609771024×10−1

n-Butane–Isobutane −0.551240293009×10−1

ISO/CD 20765-2


a The values of the Fij parameters are also valid for equation (11).

b The Fij parameters equal zero for all other binary combinations.

ISO/CD 20765-2


Annex E (normative)

The reducing functions for density and temperature

E.1 Calculation of the reducing functions for density and temperature

The composition-dependent reducing functions are used to calculate the reduced mixture variables and (i.e., dimensionless mixture density and temperature, respectively) according to

r( )x (E.1)

and

T x

T

r( ) . (E.2)

The reducing functions for density and temperature can be written as

N

i

N

j jijiijv

jiijvijvji

xx

xxxx

x1 1

3

3/1,c

3/1,c

2,

,,r

11

8

1

)(

1

(E.3)

and

N

i

N

j

ji

jiijT

jiijTijTji TT

xx

xxxxxT

1 1

,c,c2,

,,r

5.0

)(

, (E.4)

with v,ij = 1/v,ji, v,ij = v,jj, ,ij = 1/,ji, and ,ij = ,ji. These functions are based on quadratic mixing rules and

with that they are reasonably connected to physically well-founded mixing rules. The binary parameters of

equations (E.3) and (E.4) consider the deviation between the behavior of the real mixture and the one

resulting from the ideal combining rules for the critical parameters of the pure components.

The values of the binary parameters v,ij, v,ij, ,ij, and ,ij in equations (E.3) and (E.4) for all binary mixtures

are listed in table E.1. The critical parameters c,i and Tc,i of the pure components are given in Annex C.

E.1.1 Binary parameters for mixtures with no or very poor experimental data

The binary parameters v,ij and v,ij in equation (E.3) and ,ij and ,ij in equation (E.4) are fitted to

experimental data for binary mixtures. In those cases where sufficient experimental data are not available, the

parameters of equations (E.3) and (E.4) are either set to unity or modified (calculated) in such a manner that

the critical parameters of the pure components are combined in a different way, which proved to be more

suitable for certain binary subsystems. For example, for binary hydrocarbon mixtures for which no data or only

few or very poor data are available, v,ij and ,ij are set to one, and the binary parameters v,ij and ,ij are

calculated from the following conversions:

ISO/CD 20765-2


3

3/1,

3/1,

,,,

11

11

4

ji

jiijv

cc

cc

and

5.0

,,

,,,

2

1

ji

jiijT

TT

TT

cc

cc

. (E.5)

In this way, the original Lorentz and Berthelot combining rules for the critical parameters of the pure

components in equations (E.3) and (E.4) are substituted by the arithmetic mean of the critical parameters. The

use of different combining rules is superfluous when data are used to adjust the binary parameters.

Table E.1 — Values of the binary parameters in equations (E.3) and (E.4) for the reducing

functions for density and temperature.

Mixture i–j v ij, v ij, T ij, T ij,

CH4–N2 0.998721377 1.013950311 0.998098830 0.979273013

CH4–CO2 0.999518072 1.002806594 1.022624490 0.975665369

CH4–C2H6 0.997547866 1.006617867 0.996336508 1.049707697

CH4–C3H8 1.004827070 1.038470657 0.989680305 1.098655531

CH4–n-C4H10 0.979105972 1.045375122 0.994174910 1.171607691

CH4–i-C4H10 1.011240388 1.054319053 0.980315756 1.161117729

CH4–n-C5H12 0.948330120 1.124508039 0.992127525 1.249173968

CH4–i-C5H12 1.0 1.343685343 1.0 1.188899743

CH4–n-C6H14 0.958015294 1.052643846 0.981844797 1.330570181

CH4–n-C7H16 0.962050831 1.156655935 0.977431529 1.379850328

CH4–n-C8H18 0.994740603 1.116549372 0.957473785 1.449245409

CH4–n-C9H20 1.002852287 1.141895355 0.947716769 1.528532478

CH4–n-C10H22 1.033086292 1.146089637 0.937777823 1.568231489

CH4–H2 1.0 1.018702573 1.0 1.352643115

CH4–O2 1.0 1.0 1.0 0.950000000

CH4–CO 0.997340772 1.006102927 0.987411732 0.987473033

CH4–H2O 1.012783169 1.585018334 1.063333913 0.775810513

CH4–H2S 1.012599087 1.040161207 1.011090031 0.961155729

CH4–He 1.0 0.881405683 1.0 3.159776855

CH4–Ar 1.034630259 1.014678542 0.990954281 0.989843388

N2–CO2 0.977794634 1.047578256 1.005894529 1.107654104

N2–C2H6 0.978880168 1.042352891 1.007671428 1.098650964

N2–C3H8 0.974424681 1.081025408 1.002677329 1.201264026

N2–n-C4H10 0.996082610 1.146949309 0.994515234 1.304886838

N2–i-C4H10 0.986415830 1.100576129 0.992868130 1.284462634

N2–n-C5H12 1.0 1.078877166 1.0 1.419029041

N2–i-C5H12 1.0 1.154135439 1.0 1.381770770

N2–n-C6H14 1.0 1.195952177 1.0 1.472607971

N2–n-C7H16 1.0 1.404554090 1.0 1.520975334

N2–n-C8H18 1.0 1.186067025 1.0 1.733280051

ISO/CD 20765-2


N2–n-C9H20 1.0 1.100405929 0.956379450 1.749119996

N2–n-C10H22 1.0 1.0 0.957934447 1.822157123

N2–H2 0.972532065 0.970115357 0.946134337 1.175696583

N2–O2 0.999521770 0.997082328 0.997190589 0.995157044

N2–CO 1.0 1.008690943 1.0 0.993425388

N2–H2O 1.0 1.094749685 1.0 0.968808467

N2–H2S 0.910394249 1.256844157 1.004692366 0.960174200

N2–He 0.969501055 0.932629867 0.692868765 1.471831580

N2–Ar 1.004166412 1.002212182 0.999069843 0.990034831

CO2–C2H6 1.002525718 1.032876701 1.013871147 0.900949530

CO2–C3H8 0.996898004 1.047596298 1.033620538 0.908772477

CO2–n-C4H10 1.174760923 1.222437324 1.018171004 0.911498231

CO2–i-C4H10 1.076551882 1.081909003 1.023339824 0.929982936

CO2–n-C5H12 1.024311498 1.068406078 1.027000795 0.979217302

CO2–i-C5H12 1.060793104 1.116793198 1.019180957 0.961218039

CO2–n-C6H14 1.0 0.851343711 1.0 1.038675574

CO2–n-C7H16 1.205469976 1.164585914 1.011806317 1.046169823

CO2–n-C8H18 1.026169373 1.104043935 1.029690780 1.074455386

CO2–n-C9H20 1.0 0.973386152 1.007688620 1.140671202

CO2–n-C10H22 1.000151132 1.183394668 1.020028790 1.145512213

CO2–H2 0.904142159 1.152792550 0.942320195 1.782924792

CO2–O2 1.0 1.0 1.0 1.0

CO2–CO 1.0 1.0 1.0 1.0

CO2–H2O 0.949055959 1.542328793 0.997372205 0.775453996

CO2–H2S 0.906630564 1.024085837 1.016034583 0.926018880

CO2–He 0.846647561 0.864141549 0.768377630 3.207456948

CO2–Ar 1.008392428 1.029205465 0.996512863 1.050971635

C2H6–C3H8 0.997607277 1.003034720 0.996199694 1.014730190

C2H6–n-C4H10 0.999157205 1.006179146 0.999130554 1.034832749

C2H6–i-C4H10 1.0 1.006616886 1.0 1.033283811

C2H6–n-C5H12 0.993851009 1.026085655 0.998688946 1.066665676

C2H6–i-C5H12a

1.0 1.045439935 1.0 1.021150247

C2H6–n-C6H14 1.0 1.169701102 1.0 1.092177796

C2H6–n-C7H16 1.0 1.057666085 1.0 1.134532014

C2H6–n-C8H18 1.007469726 1.071917985 0.984068272 1.168636194

C2H6–n-C9H20a

1.0 1.143534730 1.0 1.056033030

C2H6–n-C10H22 0.995676258 1.098361281 0.970918061 1.237191558

C2H6–H2 0.925367171 1.106072040 0.932969831 1.902008495

C2H6–O2 1.0 1.0 1.0 1.0

C2H6–CO 1.0 1.201417898 1.0 1.069224728

C2H6–H2O 1.0 1.0 1.0 1.0

C2H6–H2S 1.010817909 1.030988277 0.990197354 0.902736660

C2H6–He 1.0 1.0 1.0 1.0

C2H6–Ar 1.0 1.0 1.0 1.0

ISO/CD 20765-2


C3H8–n-C4H10 0.999795868 1.003264179 1.000310289 1.007392782

C3H8–i-C4H10 0.999243146 1.001156119 0.998012298 1.005250774

C3H8–n-C5H12 1.044919431 1.019921513 0.996484021 1.008344412

C3H8–i-C5H12 1.040459289 0.999432118 0.994364425 1.003269500

C3H8–n-C6H14 1.0 1.057872566 1.0 1.025657518

C3H8–n-C7H16 1.0 1.079648053 1.0 1.050044169

C3H8–n-C8H18 1.0 1.102764612 1.0 1.063694129

C3H8–n-C9H20 1.0 1.199769134 1.0 1.109973833

C3H8–n-C10H22 0.984104227 1.053040574 0.985331233 1.140905252

C3H8–H2 1.0 1.074006110 1.0 2.308215191

C3H8–O2 1.0 1.0 1.0 1.0

C3H8–CO 1.0 1.108143673 1.0 1.197564208

C3H8–H2O 1.0 1.011759763 1.0 0.600340961

C3H8–H2S 0.936811219 1.010593999 0.992573556 0.905829247

C3H8–He 1.0 1.0 1.0 1.0

C3H8–Ar 1.0 1.0 1.0 1.0

n-C4H10–i-C4H10 1.000880464 1.000414440 1.000077547 1.001432824

n-C4H10–n-C5H12 1.0 1.018159650 1.0 1.002143640

n-C4H10–i-C5H12a

1.0 1.002728434 1.0 1.000792201

n-C4H10–n-C6H14 1.0 1.034995284 1.0 1.009157060

n-C4H10–n-C7H16 1.0 1.019174227 1.0 1.021283378

n-C4H10–n-C8H18 1.0 1.046905515 1.0 1.033180106

n-C4H10–n-C9H20a

1.0 1.049219137 1.0 1.014096448

n-C4H10–n-C10H22 0.976951968 1.027845529 0.993688386 1.076466918

n-C4H10–H2 1.0 1.232939523 1.0 2.509259945

n-C4H10–O2 1.0 1.0 1.0 1.0

n-C4H10–COa

1.0 1.084740904 1.0 1.173916162

n-C4H10–H2O 1.0 1.223638763 1.0 0.615512682

n-C4H10–H2S 0.908113163 1.033366041 0.985962886 0.926156602

n-C4H10–He 1.0 1.0 1.0 1.0

n-C4H10–Ar 1.0 1.214638734 1.0 1.245039498

i-C4H10–n-C5H12a

1.0 1.002779804 1.0 1.002495889

i-C4H10–i-C5H12a

1.0 1.002284353 1.0 1.001835788

i-C4H10–n-C6H14a

1.0 1.010493989 1.0 1.006018054

i-C4H10–n-C7H16a

1.0 1.021668316 1.0 1.009885760

i-C4H10–n-C8H18a

1.0 1.032807063 1.0 1.013945424

i-C4H10–n-C9H20a

1.0 1.047298475 1.0 1.017817492

i-C4H10–n-C10H22a

1.0 1.060243344 1.0 1.021624748

i-C4H10–H2a

1.0 1.147595688 1.0 1.895305393

i-C4H10–O2 1.0 1.0 1.0 1.0

i-C4H10–COa

1.0 1.087272232 1.0 1.161390082

i-C4H10–H2O 1.0 1.0 1.0 1.0

i-C4H10–H2S 1.012994431 0.988591117 0.974550548 0.937130844

ISO/CD 20765-2


i-C4H10–He 1.0 1.0 1.0 1.0

i-C4H10–Ar 1.0 1.0 1.0 1.0

n-C5H12–i-C5H12a

1.0 1.000024335 1.0 1.000050537

n-C5H12–n-C6H14a

1.0 1.002480637 1.0 1.000761237

n-C5H12–n-C7H16a

1.0 1.008972412 1.0 1.002441051

n-C5H12–n-C8H18 1.0 1.069223964 1.0 1.016422347

n-C5H12–n-C9H20 1.0 1.034910633 1.0 1.103421755

n-C5H12–n-C10H22 1.0 1.016370338 1.0 1.049035838

n-C5H12–H2a

1.0 1.188334783 1.0 2.013859174

n-C5H12–O2 1.0 1.0 1.0 1.0

n-C5H12–COa

1.0 1.119954454 1.0 1.206043295

n-C5H12–H2O 1.0 0.956677310 1.0 0.447666011

n-C5H12–H2S 0.984613203 1.076539234 0.962006651 0.959065662

n-C5H12–He 1.0 1.0 1.0 1.0

n-C5H12–Ar 1.0 1.0 1.0 1.0

i-C5H12–n-C6H14a

1.0 1.002995876 1.0 1.001204174

i-C5H12–n-C7H16a

1.0 1.009928206 1.0 1.003194615

i-C5H12–n-C8H18a

1.0 1.017880545 1.0 1.005647480

i-C5H12–n-C9H20a

1.0 1.028994325 1.0 1.008191499

i-C5H12–n-C10H22a

1.0 1.039372957 1.0 1.010825138

i-C5H12–H2a

1.0 1.184340443 1.0 1.996386669

i-C5H12–O2 1.0 1.0 1.0 1.0

i-C5H12–COa

1.0 1.116694577 1.0 1.199326059

i-C5H12–H2O 1.0 1.0 1.0 1.0

i-C5H12–H2S 1.0 0.835763343 1.0 0.982651529

i-C5H12–He 1.0 1.0 1.0 1.0

i-C5H12–Ar 1.0 1.0 1.0 1.0

n-C6H14–n-C7H16 1.0 1.001508227 1.0 0.999762786

n-C6H14–n-C8H18a

1.0 1.006268954 1.0 1.001633952

n-C6H14–n-C9H20 1.0 1.020761680 1.0 1.055369591

n-C6H14–n-C10H22 1.001516371 1.013511439 0.997641010 1.028939539

n-C6H14–H2 1.0 1.243461678 1.0 3.021197546

n-C6H14–O2 1.0 1.0 1.0 1.0

n-C6H14–COa

1.0 1.155145836 1.0 1.233272781

n-C6H14–H2O 1.0 1.170217596 1.0 0.569681333

n-C6H14–H2S 0.754473958 1.339283552 0.985891113 0.956075596

n-C6H14–He 1.0 1.0 1.0 1.0

n-C6H14–Ar 1.0 1.0 1.0 1.0

n-C7H16–n-C8H18 1.0 1.006767176 1.0 0.998793111

n-C7H16–n-C9H20 1.0 1.001370076 1.0 1.001150096

n-C7H16–n-C10H22 1.0 1.002972346 1.0 1.002229938

n-C7H16–H2 1.0 1.159131722 1.0 3.169143057

ISO/CD 20765-2


n-C7H16–O2 1.0 1.0 1.0 1.0

n-C7H16–COa

1.0 1.190354273 1.0 1.256123503

n-C7H16–H2O 1.0 1.0 1.0 1.0

n-C7H16–H2S 0.828967164 1.087956749 0.988937417 1.013453092

n-C7H16–He 1.0 1.0 1.0 1.0

n-C7H16–Ar 1.0 1.0 1.0 1.0

n-C8H18–n-C9H20a

1.0 1.001357085 1.0 1.000235044

n-C8H18–n-C10H22 1.0 1.002553544 1.0 1.007186267

n-C8H18–H2a

1.0 1.305249405 1.0 2.191555216

n-C8H18–O2 1.0 1.0 1.0 1.0

n-C8H18–COa

1.0 1.219206702 1.0 1.276565536

n-C8H18–H2O 1.0 0.599484191 1.0 0.662072469

n-C8H18–H2S 1.0 1.0 1.0 1.0

n-C8H18–He 1.0 1.0 1.0 1.0

n-C8H18–Ar 1.0 1.0 1.0 1.0

n-C9H20–n-C10H22a

1.0 1.000810520 1.0 1.000182392

n-C9H20–H2a

1.0 1.342647661 1.0 2.234354040

n-C9H20–O2 1.0 1.0 1.0 1.0

n-C9H20–COa

1.0 1.252151449 1.0 1.294070556

n-C9H20–H2O 1.0 1.0 1.0 1.0

n-C9H20–H2S 1.0 1.082905109 1.0 1.086557826

n-C9H20–He 1.0 1.0 1.0 1.0

n-C9H20–Ar 1.0 1.0 1.0 1.0

n-C10H22–H2 1.695358382 1.120233729 1.064818089 3.786003724

n-C10H22–O2 1.0 1.0 1.0 1.0

n-C10H22–CO 1.0 0.870184960 1.049594632 1.803567587

n-C10H22–H2O 1.0 0.551405318 0.897162268 0.740416402

n-C10H22–H2S 0.975187766 1.171714677 0.973091413 1.103693489

n-C10H22–He 1.0 1.0 1.0 1.0

n-C10H22–Ar 1.0 1.0 1.0 1.0

H2–O2 1.0 1.0 1.0 1.0

H2–CO 1.0 1.121416201 1.0 1.377504607

H2–H2O 1.0 1.0 1.0 1.0

H2–H2S 1.0 1.0 1.0 1.0

H2–He 1.0 1.0 1.0 1.0

H2–Ar 1.0 1.0 1.0 1.0

O2–CO 1.0 1.0 1.0 1.0

O2–H2O 1.0 1.143174289 1.0 0.964767932

O2–H2S 1.0 1.0 1.0 1.0

O2–He 1.0 1.0 1.0 1.0

O2–Ar 0.999746847 0.993907223 1.000023103 0.990430423

ISO/CD 20765-2


CO–H2O 1.0 1.0 1.0 1.0

CO–H2S 0.795660392 1.101731308 1.025536736 1.022749748

CO–He 1.0 1.0 1.0 1.0

CO–Ar 1.0 1.159720623 1.0 0.954215746

H2O–H2S 1.0 1.014832832 1.0 0.940587083

H2O–He 1.0 1.0 1.0 1.0

H2O–Ar 1.0 1.038993495 1.0 1.070941866

H2S–He 1.0 1.0 1.0 1.0

H2S–Ar 1.0 1.0 1.0 1.0

He–Ar 1.0 1.0 1.0 1.0 a

The values of the binary parameters v ij, and T ij, were calculated from equation (E.5).

ISO/CD 20765-2


Annex F (informative)

Assignment of trace components

In order to calculate, with the use of the method described in this part of ISO 20765, the thermodynamic properties of a natural gas or similar mixture that contains trace amounts of one or more components that do not appear in table 6, it is necessary to assign each trace component to one of the 21 major and minor components for which the GERG-2008 equation of state was developed. Recommendations for appropriate assignments are given in table F.1.

Each recommendation is based on an assessment of which substance is likely to give the best overall compromise of accuracy for the complete set of thermodynamic properties. The factors taken into account in this assessment include molar mass, critical temperature, and critical volume. Because, however, no single assignment is likely to be equally satisfactory for all properties, it is not unreasonable that the user may prefer an alternative assignment for a particular application in which, for example, only a single property is needed. For this reason the recommendations are not normative. Implementations of the method that include assignments for trace components need to be carefully documented in this respect.

NOTE The additional components given in table F.1 are the same as those included in ISO 6976 [7].

Table F.1 — Assignment of trace components

trace component formula recommended assignment

2,2-dimethylpropane (neo-pentane) C5H12 n-pentane

2-methylpentane C6H14 n-hexane

3-methylpentane C6H14 n-hexane

2,2-dimethylbutane C6H14 n-hexane

2,3-dimethylbutane C6H14 n-hexane

ethylene (ethene) C2H4 ethane

propylene (propene) C3H6 propane

1-butene C4H8 n-butane

cis-2-butene C4H8 n-butane

trans-2-butene C4H8 n-butane

2-methylpropene C4H8 n-butane

1-pentene C5H10 n-pentane

propadiene C3H4 propane

1,2-butadiene C4H6 n-butane

1,3-butadiene C4H6 n-butane

ISO/CD 20765-2


acetylene (ethyne) C2H2 ethane

cyclopentane C5H10 n-pentane

methylcyclopentane C6H12 n-hexane

ethylcyclopentane C7H14 n-heptane

cyclohexane C6H12 n-hexane

methylcyclohexane C7H14 n-heptane

ethylcyclohexane C8H16 n-octane

benzene C6H6 n-pentane

toluene (methylbenzene) C7H8 n-hexane

ethylbenzene C8H10 n-heptane

o-xylene C8H10 n-heptane

all other C6 hydrocarbons n-hexane

all other C7 hydrocarbons n-heptane

all other C8 hydrocarbons n-octane

all other C9 hydrocarbons n-nonane

all other C10 hydrocarbons n-decane

all higher hydrocarbons n-decane

methanol (methyl alcohol) CH3OH ethane

methanethiol (methyl mercaptan) CH3SH propane

ammonia NH3 methane

hydrogen cyanide HCN ethane

carbonyl sulfide (carbon oxysulfide) COS n-butane

carbon disulfide CS2 n-pentane

sulfur dioxide SO2 n-butane

nitrous oxide N2O carbon dioxide

neon Ne argon

krypton Kr argon

xenon Xe argon

ISO/CD 20765-2


Annex G (informative)

Examples

The following examples are provided for the purpose of software validation.

Table G.1 – Gas analysis by mole fraction

Component Gas 1 Gas 2 Gas 3 Gas 4 Gas 5 Gas 6

1 Methane 0.796 0.650 0.720 0.550

2 Nitrogen 0.100 0.065 0.010 0.25

3 Carbon dioxide 0.010 0.190 0.010 0.15

4 Ethane 0.057 0.010 0.150

5 Propane 0.020 0.010 0.010 0.100 0.10

6 n-Butane 0.005 0.010 0.010 0.10

7 iso-Butane 0.005 0.010 0.010 0.10

8 n-Pentane 0.002 0.010 0.010 0.10

9 iso-Pentane 0.002 0.010 0.010 0.10

10 n-Hexane 0.001 0.010 0.020 0.10

11 n-Heptane 0.001 0.020 0.010 0.10

12 n-Octane 0.001 0.010 0.10

13 n-Nonane 0.10

14 n-Decane 0.05

15 Hydrogen 0.200 0.10

16 Oxygen 0.010 0.049 999 0.10

17 Carbon monoxide 0.010 0.100 0.10

18 Water 0.000 001 0.05

19 Hydrogen sulfide 0.10

20 Helium 0.005 0.10

ISO/CD 20765-2


21 Argon 0.10

Total 1.000 1.000 1.000 1.000 1.000 1.000

ISO/CD 20765-2


Table G.2 – Calculations

Gas 1 M = 19.778 kg/kmol

T p D Z U H S Cv Cp w μ

K MPa kg/m³ kJ/kg kJ/kg kJ/(kg∙K) kJ/(kg∙K) kJ/(kg∙K) m/s K/MPa

180.00 10.000 389.18 0.33956 -587.21 -561.52 -3.8341 1.6233 3.3622 796.80 0.0080269

220.00 10.000 267.65 0.40397 -438.06 -400.70 -3.0329 1.6813 5.0560 427.36 1.8447

200.00 20.000 380.27 0.62552 -548.37 -495.77 -3.6247 1.6149 3.1519 826.44 -0.052377

250.00 20.000 283.04 0.67232 -399.28 -328.61 -2.8802 1.6215 3.4875 568.65 0.77140

305.00 3.0000 24.835 0.94211 -123.61 -2.8081 -1.0611 1.5553 2.1284 394.93 4.5112

350.00 10.000 74.667 0.91021 -88.582 45.346 -1.3808 1.7043 2.5065 430.98 2.6236




180.00 13.000 548.61 0.42502 -476.13 -452.44 -2.7551 1.3262 2.4027 945.53 -0.28327

220.00 11.000 452.06 0.35708 -372.89 -348.55 -2.2159 1.3850 2.8949 615.88 0.26965

250.00 20.000 425.35 0.60721 -318.05 -271.03 -1.9729 1.4023 2.7340 615.09 0.26645

350.00 20.000 226.57 0.81426 -77.691 10.583 -1.0241 1.5242 2.6216 414.60 1.6170

355.00 3.0000 28.743 0.94921 31.065 135.44 -0.15609 1.4383 1.8513 357.41 4.0492

400.00 10.000 88.041 0.91676 67.533 181.12 -0.38181 1.5813 2.1613 383.33 2.4644




150.00 10.000 504.68 0.38600 -625.94 -606.13 -3.5589 1.6072 2.6157 1267.9 -0.46108

200.00 10.000 431.60 0.33852 -492.03 -468.86 -2.7711 1.6072 2.9153 889.61 -0.17932

250.00 20.000 376.46 0.62095 -371.21 -318.08 -2.2087 1.6569 3.0291 731.32 0.072674

300.00 20.000 289.77 0.67228 -227.15 -158.13 -1.6264 1.7557 3.3290 523.24 0.83762

ISO/CD 20765-2


380.00 5.0000 41.741 0.92111 90.218 210.00 -0.16124 1.8869 2.4301 378.29 3.5788

400.00 10.000 82.020 0.89066 99.474 221.40 -0.34656 1.9839 2.6995 392.34 2.6674

Gas 4

M = 18.037 kg/kmol



180.00 25.000 378.94 0.79511 -587.57 -521.60 -3.9519 1.6680 3.1028 849.32 -0.15640

220.00 25.000 307.84 0.80081 -474.55 -393.33 -3.3091 1.6573 3.2831 660.80 0.27099

240.00 1.7000 16.192 0.94898 -240.28 -135.30 -1.2079 1.5070 2.0930 371.93 5.5554

260.00 25.000 242.33 0.86078 -365.93 -262.76 -2.7635 1.6780 3.1984 564.69 0.69309

300.00 10.000 80.581 0.89738 -193.05 -68.950 -1.7100 1.6729 2.5985 425.19 2.6008

400.00 10.000 54.907 0.98774 1.9610 184.09 -0.98174 1.8988 2.5469 495.53 1.3589

Gas 5

M = 27.610 kg/kmol



150.00 30.000 704.83 0.94229 -353.11 -310.55 -2.4550 0.89982 1.7205 713.43 -0.26388

200.00 25.000 502.27 0.82643 -269.63 -219.85 -1.8878 0.84706 1.8818 485.49 0.41958

250.00 20.000 299.16 0.88802 -188.56 -121.70 -1.3922 0.81223 1.6418 391.32 1.3328

300.00 15.000 172.80 0.96087 -123.76 -36.952 -1.0017 0.79210 1.3398 387.22 1.5610

350.00 10.000 95.443 0.99409 -69.179 35.595 -0.65788 0.78914 1.1995 401.08 1.4409

400.00 5.0000 41.380 1.0031 -19.179 101.65 -0.27250 0.79492 1.1359 416.75 1.2625

Gas 6

M = 81.365 kg/kmol



180.00 10.000 758.08 0.71716 -302.28 -289.09 -0.47202 1.4744 1.9300 1603.4 -0.54381

250.00 10.000 699.02 0.55998 -163.58 -149.27 0.18277 1.6214 2.0839 1283.4 -0.48029

ISO/CD 20765-2


300.00 25.000 671.21 1.2150 -63.863 -26.617 0.54977 1.7870 2.2353 1184.5 -0.43573

400.00 5.0000 549.38 0.22266 196.06 205.16 1.3054 2.1625 2.7652 645.44 -0.091974

450.00 0.50000 11.811 0.92057 557.91 600.24 2.3202 2.2505 2.4030 203.37 11.662

500.00 2.0000 52.175 0.75024 648.58 686.91 2.3827 2.5111 2.8942 176.26 11.031

ISO/CD 20765-2


Bibliography

[1] KUNZ, O. and WAGNER, W., The GERG-2008 Wide-Range Equation of State for Natural Gases and Other Mixtures, to be published.

[2] KUNZ, O., KLIMECK, R., WAGNER, W., and JAESCHKE, M., The GERG-2004 Wide-Range Equation of State for Natural Gases and Other Mixtures: GERG Technical Monograph 15 and Fortschr.-Ber. VDI, Reihe 6, Nr. 557, VDI Verlag, Düsseldorf, 2007.

[3] STARLING, K. E. and SAVIDGE, J. L., Compressibility Factors of Natural Gas and Other Related Hydrocarbon Gases. American Gas Association, Transmission Measurement Committee Report No. 8, Second Edition, Arlington, Virginia, 1992, and Errata No. 1, 1994.

[4] MOHR, P. J., TAYLOR, B. N., and NEWELL, D. B., CODATA recommended values of the fundamental physical constants: 2006, J. Phys. Chem. Ref. Data, 37, 1187-1284, 2008.

[5] JAESCHKE, M. and SCHLEY, P. , Ideal-gas thermodynamic properties for natural-gas applications. Int. J. Thermophysics, 16, 1381-1392, 1995.

[6] WIESER, M. E., Atomic weights of the elements 2005 (IUPAC Technical Report). Pure Appl. Chem., 78, 2051-2066, 2006.

[7] ISO 6976, Natural gas – Calculation of calorific values, density, relative density and Wobbe index from composition.

[8] ISO 14912, Gas analysis – Conversion of gas mixture composition data.

[9] KLIMECK, R., Entwicklung einer Fundamentalgleichung für Erdgase für das Gas- und Flüssigkeitsgebiet sowie das Phasengleichgewicht. Dissertation, Fakultät für Maschinenbau, Ruhr-Universität Bochum, 2000.

[10] SPAN, R. and WAGNER, W., Equations of state for technical applications. II. Results for nonpolar fluids, Int. J. Thermophysics, 24, 41-109, 2003.

[11] LEMMON, E. W. AND SPAN, R., Short fundamental equations of state for 20 industrial fluids. J. Chem. Eng. Data, 51, 785-850, 2006.

ISO/CD 20765-2


ISO_TC_193___SC_1_N_344(natural gases).pdf

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