SIST EN ISO 20765-2:2018

SLOVENSKI STANDARD
01-december-2018
=HPHOMVNLSOLQ,]UDþXQWHUPRGLQDPLþQLKODVWQRVWLGHO/DVWQRVWLHQRID]QLK
VLVWHPRYSOLQWHNRþLQDLQJRVWDWHNRþLQD]DUD]ãLUMHQREVHJXSRUDEH,62

Natural gas - Calculation of thermodynamic properties - Part 2: Single-phase properties
(gas, liquid, and dense fluid) for extended ranges of application (ISO 20765-2:2015)
Erdgas - Berechnung thermodynamischer Eigenschaften - Teil 2:
Einphaseneigenschaften (gasförmig, flüssig und dickflüssig) für den erweiterten
Anwendungsbereich (ISO 20765-2:2015)
Gaz naturel - Calcul des propriétés thermodynamiques -- Partie 2: Propriétés des phases
uniques (gaz, liquide, fluide dense) pour une gamme étendue d'applications (ISO 20765-
2:2015)
Ta slovenski standard je istoveten z: EN ISO 20765-2:2018
ICS:
71.040.40 Kemijska analiza Chemical analysis
75.060 Zemeljski plin Natural gas
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 20765-2
EUROPEAN STANDARD
NORME EUROPÉENNE
September 2018
EUROPÄISCHE NORM
ICS 75.060
English Version
Natural gas - Calculation of thermodynamic properties -
Part 2: Single-phase properties (gas, liquid, and dense
fluid) for extended ranges of application (ISO 20765-
2:2015)
Gaz naturel - Calcul des propriétés thermodynamiques Erdgas - Berechnung thermodynamischer
-- Partie 2: Propriétés des phases uniques (gaz, liquide, Eigenschaften - Teil 2: Einphaseneigenschaften
fluide dense) pour une gamme étendue d'applications (gasförmig, flüssig und dicht-flüssig) für den
(ISO 20765-2:2015) erweiterten Anwendungsbereich (ISO 20765-2:2015)
This European Standard was approved by CEN on 31 August 2018.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this
European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references
concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN
member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by
translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management
Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,
Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland,
Turkey and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels
© 2018 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 20765-2:2018 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
The text of ISO 20765-2:2015 has been prepared by Technical Committee ISO/TC 193 "Natural gas” of
the International Organization for Standardization (ISO) and has been taken over as EN ISO 20765-
2:2018 by Technical Committee CEN/TC 238 “Test gases, test pressures, appliance categories and gas
appliance types” the secretariat of which is held by AFNOR.
This European Standard shall be given the status of a national standard, either by publication of an
identical text or by endorsement, at the latest by March 2019, and conflicting national standards shall
be withdrawn at the latest by March 2019.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. CEN shall not be held responsible for identifying any or all such patent rights.
According to the CEN-CENELEC Internal Regulations, the national standards organizations of the
following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,
Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta,
Netherlands, Norway, Poland, Portugal, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland,
Turkey and the United Kingdom.
Endorsement notice
The text of ISO 20765-2:2015 has been approved by CEN as EN ISO 20765-2:2018 without any
modification.
INTERNATIONAL ISO
STANDARD 20765-2
First edition
2015-01-15
Natural gas — Calculation of
thermodynamic properties —
Part 2:
Single-phase properties (gas, liquid,
and dense fluid) for extended ranges
of application
Gaz naturel — Calcul des propriétés thermodynamiques —
Partie 2: Propriétés des phases uniques (gaz, liquide, fluide dense)
pour une gamme étendue d’applications
Reference number
ISO 20765-2:2015(E)
©
ISO 2015
ISO 20765-2:2015(E)
© ISO 2015
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form
or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior
written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of
the requester.
ISO copyright office
Case postale 56 • CH-1211 Geneva 20
Tel. + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2015 – All rights reserved

ISO 20765-2:2015(E)
Contents Page
Foreword .v
1 Scope . 1
2 Normative references . 2
3 Terms and definitions . 2
4 Thermodynamic basis of the method . 4
4.1 Principle . 4
4.2 The fundamental equation based on the Helmholtz free energy . 4
4.2.1 Background. 4
4.2.2 The Helmholtz free energy . 5
4.2.3 The reduced Helmholtz free energy . 5
4.2.4 The reduced Helmholtz free energy of the ideal gas . 6
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 . 7
4.2.7 Reducing functions . 8
4.3 Thermodynamic properties derived from the Helmholtz free energy . 8
4.3.1 Background. 8
4.3.2 Relations for the calculation of thermodynamic properties in the
homogeneous region . 9
5 Method of calculation .11
5.1 Input variables .11
5.2 Conversion from pressure to reduced density .11
5.3 Implementation .12
6 Ranges of application .13
6.1 Pure gases .13
6.2 Binary mixtures .14
6.3 Natural gases .17
7 Uncertainty of the equation of state .18
7.1 Background .18
7.2 Uncertainty for pure gases . .18
7.2.1 Natural gas main components.18
7.2.2 Secondary alkanes .19
7.2.3 Other secondary components .21
7.3 Uncertainty for binary mixtures .21
7.4 Uncertainty for natural gases .23
7.4.1 Uncertainty in the normal and intermediate ranges of applicability of
natural gas .24
7.4.2 Uncertainty in the full range of applicability, and calculation of properties
beyond this range .25
7.5 Uncertainties in other properties .25
7.6 Impact of uncertainties of input variables .25
8 Reporting of results .25
Annex A (normative) Symbols and units .27
Annex B (normative) The reduced Helmholtz free energy of the ideal gas .29
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
Annex E (normative) The reducing functions for density and temperature .48
Annex F (informative) Assignment of trace components .55
ISO 20765-2:2015(E)
Annex G (informative) Examples .57
Bibliography .60
iv © ISO 2015 – All rights reserved

ISO 20765-2:2015(E)
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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
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. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 193, Natural Gas, Subcommittee SC 1, Analysis
of Natural Gas.
ISO 20765 consists of the following parts, under the general title Natural gas — Calculation of
thermodynamic properties:
— 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
— Part 3: Two-phase properties (vapour-liquid equilibria)
INTERNATIONAL STANDARD ISO 20765-2:2015(E)
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 to calculate 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 1 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 (–10 °F to 35 °F). The method described here maintains an uncertainty of ≤ 0,1 % for volumetric
properties, and generally within 0,1 % for speed of sound. It accurately describes volumetric and
caloric properties of homogeneous gas, liquid, and supercritical fluids as well as those in vapour-liquid
equilibrium. Therefore its structure is more complex than that in Part 1.
NOTE 2 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 which the method of Part 1 is not applicable. For example, it
is applicable 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. A practical usage is
the calculation of properties of highly concentrated CO 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 (–280 °F to –208 °F), 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
to within ±(0,1 – 0,2) % at pressures up to 40 MPa (5800 psia), which is also in agreement with the
estimated experimental uncertainty. 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.
ISO 20765-2:2015(E)
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 7504, Gas Analysis — Vocabulary
ISO 14532, Natural gas — Vocabulary
ISO 20765-1, Natural gas — Calculation of thermodynamic properties — Part 1: Gas phase properties for
transmission and distribution applications
ISO 80000-5:2007, Quantities and units — Part 5: Thermodynamics
3 Terms and definitions
For the purposes of this document, the terms and definitions in ISO 80000-5:2007 and/or ISO 20765-1,
ISO 7504, ISO 14532, and the following apply.
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 behaviour 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 useful in understanding the definitions below.
SUPERCRITICAL
cricondenbar
DENSE FLUID
STATE
critical point dew
line
LIQUID PHASE
TWO-PHASE
cricondentherm
bubble
VAPOUR-
LIQUID
line
GAS
PHASE
100 150 200 250 300 350 400
Figure 1 — Phase diagram for a typical natural gas
3.1
bubble pressure
pressure at which an infinitesimal amount of vapour is in equilibrium with a bulk liquid for a
specified temperature
2 © ISO 2015 – All rights reserved
Pressure/MPa
ISO 20765-2:2015(E)
3.2
bubble temperature
temperature at which an infinitesimal amount of vapour is in equilibrium with a bulk liquid for a
specified pressure
Note 1 to entry: The locus of bubble points is known as the bubble line.
Note 2 to entry: More than one bubble temperature may exist at a specific pressure. Moreover, more than one
bubble pressure may exist at a specified temperature, as explained in the example given in 3.6.
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 saturation point along the two-phase vapour-liquid equilibrium boundary where both the vapour
and liquid phases have the same composition and density
Note 1 to entry: The critical point is the point at which the dew line and the bubble line meet.
Note 2 to entry: The pressure at the critical point is known as the critical pressure and the temperature as the
critical temperature.
Note 3 to entry: A mixture of given composition may have one, more than one, or no critical points. In addition,
the phase behaviour may be quite different from that shown in Fig. 1 for mixtures (including natural gases)
containing, e.g., hydrogen or helium.
3.6
dew pressure
pressure at which an infinitesimal amount of liquid is in equilibrium with a bulk vapour for a
specified temperature
Note 1 to entry: More than one dew pressure may exist at the specified temperature. For example, isothermal
compression at 300 K with a gas similar to that shown in Figure 1: At low pressure the mixture is a gas. At just
above 2 MPa (the dew pressure), a liquid phase initially forms. As pressure increases more liquid forms in the
two-phase region, but a further increase in pressure reduces the amount of liquid (retrograde condensation) until
at about 8 MPa where the liquid phase disappears at the upper dew pressure, and the mixture is in the dense gas
phase. In the two-phase region, the overall composition is as specified, however the coexisting vapour and liquid
will have different compositions.
3.7
dew temperature
temperature at which an infinitesimal amount of liquid is in equilibrium with a bulk vapour for a
specified pressure
Note 1 to entry: More than one dew temperature may exist at a specified pressure, similar to the example given in 3.6.
Note 2 to entry: The locus of dew points is known as the dew line.
3.8
supercritical state
dense phase region above the critical point (often considered to be a state above the critical temperature
and pressure) within which no two-phase separation can occur
ISO 20765-2:2015(E)
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 iteratively determine the density using the
equations provided here.
These equations express 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 energy
and its derivatives with respect to temperature and density. For example, pressure is proportional to
the first derivative of the Helmholtz energy with respect to density (at constant temperature).
NOTE These equations are also applicable in the calculation of two-phase properties (vapour-liquid
equilibria). Additional composition-dependent derivatives are required and 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 (50 ppm) are specified. For a typical natural gas, this might include
alkane hydrocarbons up to about C or C together with nitrogen, carbon dioxide, and helium. Typically,
7 8
isomers for alkanes C and higher may be lumped together by molar mass and treated collectively as the
normal isomer.
For some fluids, additional components such as C , C , water, and hydrogen sulfide may be present and
9 10
need to be taken into consideration. For manufactured gases, hydrogen, carbon monoxide, and oxygen
may also be present in the mixture.
More precisely, the method uses a 21-component analysis in which all of the major and most of the minor
components of natural gas are included (see Clause 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
[1]
The GERG-2008 equation 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
[2] [1]
natural gases and other mixtures. It was originally published in 2007 and later updated in 2008.
[3]
The new equation improves upon the performance of the AGA-8 equation for gas phase properties and
in addition is applicable to the properties of the liquid phase, to the dense-fluid phase, to the vapour-
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
(GERG) supported the development of this 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 © ISO 2015 – All rights reserved

ISO 20765-2:2015(E)
4.2.2 The Helmholtz free energy
The Helmholtz free energy a of a fluid mixture at a given mixture density ρ, temperature T, and molar
o r
composition x can be expressed as the sum of a describing the ideal gas behaviour and a describing
the residual or real-gas contribution, as follows:
or
aT(,ρρ,)xa=+(,Tx,) aT(,ρ ,)x (1)
4.2.3 The reduced Helmholtz free energy
The Helmholtz free energy is often used in its dimensionless form α=a/(RT) as
or
αδ(,τα,)xT=+(,ρα,)xx(,δτ,) (2)
In this equation, the reduced (dimensionless) mixture density δ is given by
ρ
δ = (3)
ρ ()x
r
and the inverse reduced (dimensionless) mixture temperature τ is given by
Tx()
r
τ = (4)
T
where
ρ and Τ are reducing functions for the mixture density and mixture temperature (see 4.2.7) depending
r r
on the molar composition of the mixture only.
r
The residual part α of the reduced Helmholtz free energy is given by
r rr
αδ(,τα,)xx=+(,δτ,) Δαδ(,τ,)x (5)
o
r
In this equation, the first term on the right-hand side α describes the contribution of the residual parts
o
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 calculated at the reduced mixture variables δ and
r
τ (see equation (8)). The second term Δα is the departure function, which is the double summation over
all binary specific and generalized departure functions developed for the respective binary mixtures
(see equation (10)).
ISO 20765-2:2015(E)
4.2.4 The reduced Helmholtz free energy of the ideal gas
o
The reduced Helmholtz free energy α represents the properties of the ideal-gas mixture at a given
mixture density ρ, temperature Τ, and molar composition x according to
N
o o
αρ(,Tx,)=+xT[(αρ,) lnx ] (6)
∑ iio i
i=1
o
In this equation, the term ∑x lnx is the contribution from the entropy of mixing, and αρ(,T) is the
i i
oi
dimensionless form of the Helmholtz free energy in the ideal-gas state of component i, as given by
∗

 
T T   T 
ρ R
o o o c,i o c,,i o o c,i

αρ(,T)l= nl++nn +n n + n ln sinh ϑ
 
   
oi oii,,12o oi,3 ∑ oik,,oik
 
ρ R T T T

c,i    
 
k=46,

(7)

 T 
o o c,i

− n lnncosh ϑ
 
∑ oik, oik,
T

 
k=57,

where
ρ and Τ are the critical parameters of the pure components (see Annex C).
c,i c,i
o o
The values of the coefficients n and the parameters ϑ for all 21 components are given in Annex B.
oik, oik,
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 the inverse reduced temperature τ between Part 1 and Part 2.
-1 -1 [4]
NOTE 2 R = 8,314 472 J·mol ·K was the internationally accepted standard for the molar gas constant at the
time of development of the equation of state. 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
-1 -1
the mixture model presented here. The ratio R*/R with R*=8,314 51 J·mol ·K takes into account this difference
and therefore leads to the exact solution of the original equations for the ideal-gas heat capacity.
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
r
equations of state α to the residual part of the reduced Helmholtz free energy of the mixture is
o
N
r r
αδ(,τα,)xx= (,δτ) (8)
o ∑ iio
i=1
6 © ISO 2015 – All rights reserved

ISO 20765-2:2015(E)
where
r
αδ(,τ) is the residual part of the reduced Helmholtz free energy of component i (i.e., the residual part
oi
of the respective pure substance equation of state listed in Table 2) and is given by
K KK+
PolE,,iixp
Pol,i
cc
oik,
dt dt
r −δ
ooik,,ik ooik,,ik
αδ(,τδ)=+nnτδ τ e (9)
oii∑ oo,k ∑ ik,
k=1 kK=+1
Pol,i
r
The equations for α use the same basic structure as further detailed in Annex D.2. The values of the
oi
coefficients n and the exponents d , t and c for all 21 components are given in Annex D.2.2.
oi,k oi,k oi,k oi,k
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
r
mixture. The departure function Δα 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
N−1 N
rr
ΔΔαδ(,τα,)xx= (,δτ,) (10)
ij
∑ ∑
i=1 ji=+1
with
rr
Δαδ(,τα,)xx= xF (,δτ) (11)
ij ij ij ij
r r
In this equation, the function α (δ,τ) is the part of the departure function Δα (δ,τ,x ) that depends only
ij ij
on the reduced mixture variables δ and τ, as given by
K
Pol,ij
dt
r
ij,,kijk
αδ(,τδ)= n τ
ij ∑ ij,k
k=1
(12)
KK+
PolE,,ij xp ij
dt −ηηδ()−−εβ ()δγ−
ij,,kijk ij,,kijk ij,,kijk
+ neδτ
∑ ij,k
kK=+1
Pol,ij
where
r
αδ(,τ) was developed either for a specific binary mixture (a binary specific departure function with
ij
binary specific coefficients and exponents) or for a group of binary mixtures (generalized departure
function with a uniform structure for the group of binary mixtures).
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 F in equation (11) equals unity.
ij
b) Generalized departure function
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
ISO 20765-2:2015(E)
parameter F is fitted to the corresponding binary specific data (except for the binary system methane–
ij
n-butane, where F equals unity).
ij
c) No departure function
For all of the remaining binary mixtures, no departure function was developed, and F equals zero, i.e.,
ij
r
Δαδ(,τ ,)x equals zero. For most of these mixtures, however, the parameters of the reducing functions
ij
for density and temperature were fitted to selected experimental data (see 4.2.7 and 6.2).
The values of the coefficients n , the exponents d and t , and the parameters η , ε , β , and γ
ij,k ij,k ij,k ij,k ij,k ij,k 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 number of digits given in these tables is as presented in
the source publication; the effect of truncation is not obvious and all of the digits shall be used in all
calculations. The non-zero F parameters are listed in Table D.5.
ij
NOTE Compared to the reducing functions for density and temperature, the departure function is in general
of minor importance for the residual behaviour of the mixture since it only describes an additional small residual
deviation to the real mixture behaviour. The development of such a function was, however, necessary to fulfil
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 δ and τ are calculated from equations (3) and (4) by means of the
composition-dependent reducing functions for the mixture density and temperature
N N  
xx+
11 11
ij  
 
= xx βγ + (13)
ij vi,,jv ij
∑∑  
21/3 13/
 
ρ ()x 8
β xx+   ρρ
r
i=1 j=1 vi, ji j
cc,i ,jj
 
NN N
xx+ 05.
ij
Tx()= xx βγ ()TT⋅ (14)
rc∑∑ ij Ti,,jT ij ,,ijc
β xx+
Ti, ji j
i=1 j=1
These functions are based on quadratic mixing rules and are reasonably connected to physically
well-founded mixing rules. The binary parameters β and γ in equation (13) and β and γ in
v,ij v,ij T,ij T,ij
equation (14) are fitted to data for binary mixtures subject to the conditions β =1/β and γ =γ . The
ij ji ij ji
values of the binary parameters for all binary mixtures are listed in Table E.1 of Annex E. The critical
parameters ρ and Τ of the pure components are given in Annex C.
c,i c,i
NOTE The binary parameters of equations (13) and (14) were fitted based on the deviations between the
behaviour of the real mixture (determined by experimental data) and the one resulting from ideal combining
rules (with β and γ set to 1) 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
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 δ
and τ, as summarized in the following section (see Table 1). All of the thermodynamic properties may
8 © ISO 2015 – All rights reserved

ISO 20765-2:2015(E)
be written explicitly in terms of the reduced Helmholtz free energy α and its various derivatives. The
required derivatives α , α , α , α , and α are defined as follows:
τ ττ δ δδ δτ
2 2
     
∂α  ∂ α ∂α  ∂ α ∂ ∂α 
α = α =  α = α =  α =  (15)
τ   ττ δ   δδ δτ  
2 2
     
∂τ ∂δ ∂τ ∂δ
  ∂τ   ∂δ  
δ ,x τ ,x τ ,x
     
δ ,x τ ,x δ ,x
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:
∂()δα2
 
δ 22rr
α = =+21δα δα =+2δα +δα (16)
1   δδδδ δδ
∂δ
 
τ,x
i
 
∂ δα 
2 rr
δ
ατ=− =−δα δταδ=+1 αδ− τα (17)
 
2   δδτδ δτ
∂τ τ
 
 
δ,x
i
Detailed expressions for α , α , α , α , α , α , and α can be found in Annexes B and D.
τ ττ δ δδ δτ 1 2
NOTE In addition to the derivatives of α with respect to the reduced mixture variables δ and τ, composition
derivatives of α and of the reducing functions for density and temperature are required for the calculation of
vapour-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 α
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 α of the
mixture. In equations (26), (28), (29), (30), and (31), the basic expressions for the properties c, w, μ , ϕ,
p JT
and κ 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, c , and c )
v p
by the molar mass M.
The molar mass M of the mixture is derived from the composition x and the molar masses M of the pure
i i
substances, as follows
N
Mx()=⋅xM (18)
∑ ii
i=1
The mass-based density D is given by
DM=ρ (19)
NOTE 1 Values of the molar masses M of the pure substances are given in Annex C and are taken from [6]; these
i
[7]
values are not identical with those given in ISO 20765-1 and ISO 6976:1995. However, they are identical with the
most 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.
ISO 20765-2:2015(E)
Table 1 — Definitions of common thermodynamic properties and their relation to the reduced
Helmholtz free energy α
Property and definition Relation to α and its derivatives
Pressure (20)
p
r
=+1 δα
pa=−(/∂∂v)
δ
Tx,
ρRT
Compression factor (21)
r
Z=+1 δα
δ
Zp= /(ρRT)
Internal energy (22)
u
=τα
ua=+Ts
τ
RT
Entropy (23)
s
=−τα α
sa=−(/∂∂T)
τ
vx,
R
Isochoric heat capacity (24)
c
v 2
=−τα
cu=∂(/∂T)
ττ
vv,x
R
Enthalpy (25)
h
r
=+1 δα +τα
hu=+pv
δτ
RT
Isobaric heat capacity (26)
c
α
p
ch=∂(/∂T)
=−τα +
pp,x
ττ
R α
Gibbs free energy (27)
g
r
=+1 δα +α
gh=−Ts
δ
RT
Speed of sound (28)
2 2
c
wM α
p
2 2
=−α ==Zκα
wM=∂(/1 )( p/)∂ρ
1 1
sx,
RT c
τα
v
ττ
Joule-Thomson coefficient (29)
 
αα− R α
21 2
μ=∂(/Tp∂ )
μρR = =−1
hx,  
c α
ατ− αα
p  1 
2 ττ 1
Isothermal throttling coefficient (30)
α
φρ=−1
φ =∂(/hp∂ )
Tx,
α
Isentropic exponent (31)
 
c
α α α
p
12 1
κ =−(/vp)(∂∂pv/)
sx,   κ = 1− =
r 2
 
Z c
1+δα τα α
v
δτ τ 1 
Second virial coefficient (32)
r
Bρα= lim( )
r δ
BZ=∂lim( /)∂ρ
Tx, δ→0
ρ→0
Third virial coefficient (33)
2 r
Cρα= lim( )
r δδ
CZ=∂lim( /)∂ρ /2 δ→0
Tx,
ρ→0
10 © ISO 2015 – All rights reserved

ISO 20765-2:2015(E)
5 Method of calculation
5.1 Input variables
The method presented in this standard uses reduced density, inverse reduced temperature, and
...