SIST EN ISO 18453:2006

SLOVENSKI STANDARD
SIST EN ISO 18453:2006
01-januar-2006
=HPHOMVNLSOLQ±.RUHODFLMDPHGGHOHåHPYRGHLQYRGQLPURVLãþHP,62

Natural gas - Correlation between water content and water dew point (ISO 18453:2004)
Erdgas - Beziehung zwischen Wassergehalt und Taupunkt (ISO 18453:2004)
Gaz naturel - Corrélation entre la teneur en eau et le point de rosée eau (ISO
18453:2004)
Ta slovenski standard je istoveten z: EN ISO 18453:2005
ICS:
75.060 Zemeljski plin Natural gas
SIST EN ISO 18453:2006 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------

SIST EN ISO 18453:2006

---------------------- Page: 2 ----------------------

SIST EN ISO 18453:2006
EUROPEAN STANDARD
EN ISO 18453
NORME EUROPÉENNE
EUROPÄISCHE NORM
November 2005
ICS 75.060

English Version
Natural gas - Correlation between water content and water dew
point (ISO 18453:2004)
Gaz naturel - Corrélation entre la teneur en eau et le point Erdgas - Beziehung zwischen Wassergehalt und Taupunkt
de rosée eau (ISO 18453:2004) (ISO 18453:2004)
This European Standard was approved by CEN on 7 October 2005.
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 Central Secretariat 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 Central Secretariat has the same status as the official
versions.
CEN members are the national standards bodies of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France,
Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Slovakia,
Slovenia, Spain, Sweden, Switzerland and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36  B-1050 Brussels
© 2005 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 18453:2005: E
worldwide for CEN national Members.

---------------------- Page: 3 ----------------------

SIST EN ISO 18453:2006
EN ISO 18453:2005 (E)






Foreword



The text of ISO 18453:2004 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 18453:2005 by CMC.

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 May 2006, and conflicting national standards
shall be withdrawn at the latest by May 2006.

According to the CEN/CENELEC Internal Regulations, the national standards organizations of
the following countries are bound to implement this European Standard: Austria, Belgium,
Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary,
Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland,
Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.



Endorsement notice

The text of ISO 18453:2004 has been approved by CEN as EN ISO 18453:2005 without any
modifications.

2

---------------------- Page: 4 ----------------------

SIST EN ISO 18453:2006


INTERNATIONAL ISO
STANDARD 18453
First edition
2004-07-01


Natural gas — Correlation between water
content and water dew point
Gaz naturel — Corrélation entre la teneur en eau et le point de rosée
de l'eau




Reference number
ISO 18453:2004(E)
©
ISO 2004

---------------------- Page: 5 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but
shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In
downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat
accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation
parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In
the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.


©  ISO 2004
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing 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 2004 – All rights reserved

---------------------- Page: 6 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope. 1
2 Terms and definitions. 1
3 Development of the correlation . 2
4 Range of application and uncertainty of the correlation . 3
5 Correlation . 4
Annex A (normative) Thermodynamic principles . 8
Annex B (informative) Traceability . 15
Annex C (informative) Examples of calculations . 17
Annex D (informative) Subscripts, symbols, units, conversion factors and abbreviations . 19
Bibliography . 21

© ISO 2004 – All rights reserved iii

---------------------- Page: 7 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(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.
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 18453 was prepared by Technical Committee ISO/TC 193, Natural gas, Subcommittee SC 1, Analysis of
natural gas.
iv © ISO 2004 – All rights reserved

---------------------- Page: 8 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
Introduction
ISO/TC 193, Natural gas, was established in May 1989, with the task of creating new standards, and updating
existing standards relevant to natural gas. This includes gas analysis, direct measurement of properties,
quality designation and traceability.
This document provides a reliable mathematical relationship between water content and water dew point in
natural gas. The calculation method was developed by GERG; it is applicable in both ways, i.e. either to
calculate the water content or to calculate the water dew point. Information relating to the thermodynamic
principles is given in Annex A; information relating to the traceability, applications and uncertainties associated
with this work is given in Annex B.
Some of the operational problems in the natural gas industry can be traced back to water content in natural
gases. Even with low water vapour content in the gas, changing operating pressure and temperature
conditions can cause water to condense and thus lead to corrosion problems, hydrates or ice formation. To
avoid these problems, expensive dehydration units have been installed by natural gas companies. The design
and cost of these installations depend on the exact knowledge of the water content at the dew point and the
(contractually) required water content.
The instruments resulting from the improvements of moisture measurement equipment during the last
decades focus on the determination of water content rather than on water dew point. Therefore, if the water
content is measured, a correlation is needed for the expression of water dew point.
1)
The GERG Group identified a need to build a comprehensive and accurate database of measured water
content and corresponding water dew point values for a number of representative natural gases in the range
of interest before validating the existing correlations between water content and water dew point.
It was subsequently shown that the uncertainty range of the existing correlations could be improved.
Therefore, as a result, a more accurate, composition-dependent correlation was successfully developed on
the basis of the new database.
The aim of this International Standard is to standardize the calculation procedure developed by GERG
concerning the relationship between water content and water dew point (and vice versa) in the field of natural
gas typically for custody transfer.


1) GERG is an abbreviation of Groupe Européen de Recherche Gazière.
© ISO 2004 – All rights reserved v

---------------------- Page: 9 ----------------------

SIST EN ISO 18453:2006

---------------------- Page: 10 ----------------------

SIST EN ISO 18453:2006
INTERNATIONAL STANDARD ISO 18453:2004(E)

Natural gas — Correlation between water content and water
dew point
1 Scope
This International Standard specifies a method to provide users with a reliable mathematical relationship
between water content and water dew point in natural gas when one of the two is known. The calculation
method, developed by GERG; is applicable to both the calculation of the water content and the water dew
point.
This International Standard gives the uncertainty for the correlation but makes no attempt to quantify the
measurement uncertainties.
2 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
2.1
correlation
relationship between two or several random variables within a distribution of two or more random variables
[ISO 3534-1]
NOTE The indication of the range of temperature, pressure and composition for which the correlation was validated
is given in Clause 3.
2.2
working range
range of parameters for which the correlation has been validated
2.3
extended working range
range of parameters for which the correlation has been developed, but outside the range for which the
correlation has been validated
2.4
uncertainty of the correlation
absolute deviation of calculated value from the experimental database
NOTE This does not include any measurement uncertainty in the field.
2.5
acentric factor
parameter to characterize the acentricity or non-sphericity of a molecule
NOTE This definition was taken from reference [1] in the Bibliography.
© ISO 2004 – All rights reserved 1

---------------------- Page: 11 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
2.6
normal reference conditions
reference conditions of pressure, temperature and humidity (state of saturation) equal to 101,325 kPa and
273,15 K for the real, dry gas
[ISO 14532:2001]
2.6
traceability
property of the result of a measurement or the value of a standard whereby it can be related to stated
references, usually national or international standards, through an unbroken chain of comparisons all having
stated uncertainties
[ISO 14532:2001]
3 Development of the correlation
In the past, GERG has identified the necessity for an accurate conversion between the water content and the
water dew point for natural gases with sales gas characteristics. To achieve this goal, the GERG defined a
research program. In the first phase of the project, reliable data on water content together with data on water
dew point were collected for several natural gases for the dew-point temperature range of interest: −15 °C to
+5 °C and for the (absolute) pressure range of interest: 0,5 MPa to 10 MPa. In addition to the measurements
on the seven representative natural gases, measurements were also carried out on the key binary system
methane/water. The procedure used for gathering the measured data was the saturation method.
Taking the determined values for the repeatability and reproducibility of the Karl Fischer instrument as
consistency criteria for all measured water contents, only a few inconsistent values were detected, which were
mainly situated in the range of low water content (high pressure, low temperature range). Values which failed
the consistency check were either rejected or, in a few cases, weighted much lower in the data pool. In most
cases, these values were replaced by repeated measurements carried out at the same pressure and
temperature conditions.
Detailed information on the experimental procedure and the composition of the natural gases used during the
[2]
experiments can be found in the GERG Monograph .
The developed relationship is validated for dew-point temperatures ranging from −15 °C to +5 °C and
(absolute) pressures ranging from 0,5 MPa to 10 MPa.
The representative natural gases used for validating the correlation were sampled technically free of glycol,
3
methanol, liquid hydrocarbon and with a maximum content of H S of 5 mg/m (in normal conditions). No
2
attempt was made to investigate the impact of the uncertainties resulting from the inclusion of such
contaminants.
The thermodynamic background of the developed relationship makes it possible to extend the range of
applicability outside the working range to temperatures of −50 °C to +40 °C and (absolute) pressures from
0,1 MPa to 30 MPa with unknown uncertainties.
It is intended that the correlation be interpreted as reciprocal between the water content and the water dew
point. Note that this relationship was derived under laboratory conditions using several compositions of natural
gas sampled in the field. Under practical field operational conditions, significant additional uncertainties are
generated.
Besides the uncertainty in the conversion of the measurement itself, the uncertainties of the measured values
should also be considered.
Unless explicitly otherwise stated, the volume is stated under normal reference conditions (2.6).
2 © ISO 2004 – All rights reserved

---------------------- Page: 12 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
4 Range of application and uncertainty of the correlation
4.1 Working range
The working range is within the ranges defined above, and the associated uncertainties are as follows.
a) Range of pressure: 0,5 MPa u p u 10 MPa
b) Range of dew-point temperature: −15 °C u t u +5 °C
c) Range of composition: the correlation accepts water and the components given in Table 1 as input
parameters. The calculation method is applicable to natural gases that meet the limitations listed in
Table 1. Examples of the influence of composition are given in Annex C.
Table 1 — Range of composition for percentage molar composition
Compound Percentage molar composition
Methane (CH) W 40,0 %
4
Nitrogen (N ) u 55,0 %
2
Carbon dioxide (CO ) u 30,0 %
2
Ethane (C H ) u 20,0 %
2 6
Propane (C H ) u 4,5 %
3 8
2-Methyl propane (C H ) u 1,5 %
4 10
n-Butane (C H ) u 1,5 %
4 10
2,2-Dimethyl propane (C H ) u 1,5 %
5 12
2-Methyl butane (C H ) u 1,5 %
5 12
n-Pentane (C H ) u 1,5 %
5 12
C (sum of hexane + higher hydrocarbons) (C H ) u 1,5 %
6+ 6 14
NOTE C is treated as n-hexane.
6+
Within the range above the uncertainty are the following:
 for the water dew point calculated from the water content: ± 2 °C
 for the water content calculated from the water dew point:
3 3
1) β < 580 mg/m : 0,14 + 0,021 × β ± 20 (mg/m );
w w
3 3
2) β W 580 mg/m : −18,84 + 0,053 7 × β ± 20 (mg/m ).
w w
For the application of these formulae, refer to Annex B and the examples given in Annex C.
NOTE The conversion between normal reference conditions and standard reference conditions is given in ISO 13443.
4.2 Extended working range
Extension of the application range may be extrapolated within the following ranges, but the associated
uncertainties are unknown.
a) Range of pressure: extended range of (absolute) pressure is 0,1 MPa u p < 0,5 MPa and
10 MPa < p u 30 MPa;
© ISO 2004 – All rights reserved 3

---------------------- Page: 13 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
b) Range of temperature: extended range of dew-point temperature is −50 °C u t < −15 °C and
+5 °C < t u +40 °C;
c) Range of composition: range of components is the same as in 4.1.
5 Correlation
5.1 General
The correlational method is based on the Peng and Robinson equation of state (see Annex A for detailed
information).
In order to ensure an accurate calculation of water vapour pressure above ice and liquid, it was decided to
divide the new α-function into two parts:
 the temperature range of 223,15 K to 273,16 K, i.e. vapour pressure data above ice;
 the temperature range of 273,16 K to 313,15 K, i.e. vapour pressure data over liquid water.
2
24

1/ 2 1/ 2 1/ 2
αTA=+11−T +A1−T +A1−T
()
R1R )2 R ) 3 R )
( ( (

where T is the reduced temperature as follows:
R
T
T =
R
T
crit
The coefficients of the new α-function are listed as follows.
a) For the range: 223,15 K u T < 273,16 K
1) A = 0,106 025
1
2) A = 2,683 845
2
3) A = −4,756 38
3
b) For the range: 273,16 K u T < 313,15 K
1) A = 0,905 436
1
2) A = −0,213 781
2
3) A = 0,260 05
3
A reliable estimate for the parameter was obtained from an appropriate set of vapour-liquid equilibria data.
The optimum parameters for binary parameters k are found by satisfying a specified statistical criterion
ij
(minimization of an objective function through a least squares fit algorithm). For the binary systems, carbon
dioxide/water, methane/water and ethane/water, it was necessary to introduce temperature-dependent
interaction parameters to obtain a satisfactory description of the vapour-liquid equilibrium. The temperature
dependence is given as:
T

kT=+k k −1
()
ij ij,0 ij,1

273,15

4 © ISO 2004 – All rights reserved

---------------------- Page: 14 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
This definition of k (T) has the advantage that k equals k when the temperature equals 0 °C. The
ij ij ij,0
parameters of the binary water system are optimized for the extended working range of this correlation
(−50 °C up to 40 °C). Extrapolation of the data beyond the extended working range is not allowed.
Pure component data are listed in Table 2 and an overview over the complete binary interaction parameters is
given in Table 3.
Table 2 — Pure component data (compound properties used in the calculation)
Component ω p T Source
c c
[12]
Water (HO) 0,344 37 220,64 647,14 Knapp (1982)
2
[12]
Nitrogen (N) 0,035 93 33,99 126,26 Knapp (1982)
2
[12]
Carbon dioxide (CO) 0,223 94 73,86 304,21 Knapp (1982)
2
[12]
Methane (CH) 0,011 4 45,99 190,55 Knapp (1982)
4
[12]
Ethane (C H) 0,099 09 48,72 305,33 Knapp (1982)
2 6
[12]
Propane (C H) 0,156 11 42,46 369,85 Knapp (1982)
3 8
[12]
2–Methyl propane (i-C H) 0,184 65 36,4 407,85 Knapp (1982)
4 10
[12]
n–Butane (n-C H) 0,197 77 37,84 425,14 Knapp (1982)
4 10
[12]
2,2–Dimethyl propane (neo-C H) 0,195 28 31,96 433,75 Knapp (1982)
5 12
[12]
2–Methyl butane (i-C H) 0,226 06 33,7 460,39 Knapp (1982)
5 12
[12]
n–Pentane (n-C H) 0,249 83 33,64 469,69 Knapp (1982)
5 12
[12]
n–Hexane (C H) 0,296 30,2 507,85 Knapp (1982)
6 14
ω is the acentric factor
p is the critical pressure, expressed in bar
c
T is the critical temperature, expressed in kelvins
c
© ISO 2004 – All rights reserved 5

---------------------- Page: 15 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
Table 3 — Binary interaction parameters
Component i Component j k k Source
ij,0 ij,1
[2]
Water Nitrogen 0,480 0 0 GERG
[2]
Water Carbon dioxide 0,184 0,236 GERG
[2]
Water Methane 0,651 −1,385 GERG
[2]
Water Ethane 0,635 −0,93 GERG
[2]
Water Propane 0,53 0 GERG
[2]
Water n-Butane 0,69 0 GERG
[2]
Water n-Pentane 0,5 0 GERG
[2]
Water n-Hexane 0,5 0 GERG
[2]
Water 2-Methyl propane 0,69 0 GERG
[2]
Water 2,2-Dimethyl propane 0,5 0 GERG
[2]
Water 2-Methyl butane 0,5 0 GERG
[9]
Nitrogen Carbon dioxide −0,017 0 0 Knapp (1982)
[9]
Nitrogen Methane 0,031 1 0 Knapp (1982)
[9]
Nitrogen Ethane 0,051 5 0 Knapp (1982)
[9]
Nitrogen Propane 0,085 2 0 Knapp (1982)
[9]
Nitrogen n-Butane 0,080 0 0 Knapp (1982)
[9]
Nitrogen n-Pentane 0,100 0 0 Knapp (1982)
[9]
Nitrogen n-Hexane 0,149 6 0 Knapp (1982)
[9]
Nitrogen 2-Methyl propane 0,103 3 0 Knapp (1982)
[8]
Nitrogen 2,2-Dimethyl propane 0,093 0 0 Avlonitis (1994)
[9]
Nitrogen 2-Methyl butane 0,092 2 0 Knapp (1982)
[9]
Carbon dioxide Methane 0,091 9 0 Knapp (1982)
[9]
Carbon dioxide Ethane 0,132 2 0 Knapp (1982)
[9]
Carbon dioxide Propane 0,124 1 0 Knapp (1982)
[9]
Carbon dioxide n-Butane 0,133 3 0 Knapp (1982)
[9]
Carbon dioxide n-Pentane 0,122 2 0 Knapp (1982)
[9]
Carbon dioxide n-Hexane 0,110 0 0 Knapp (1982)
[9]
Carbon dioxide 2-Methyl propane 0,120 0 0 Knapp (1982)
[10]
Carbon dioxide 2,2-Dimethyl propane 0,126 0 0 Kordas (1994)
[9]
Carbon dioxide 2-Methyl butane 0,121 9 0 Knapp (1982)
[9]
Methane Ethane −0,002 6 0 Knapp (1982)
[9]
Methane Propane 0,014 0 0 Knapp (1982)
[9]
Methane n-Butane 0,013 3 0 Knapp (1982)
[9]
Methane n-Pentane 0,023 0 0 Knapp (1982)
[9]
Methane n-Hexane 0,042 2 0 Knapp (1982)
[9]
Methane 2-Methyl propane 0,025 6 0 Knapp (1982)
[11]
Methane 2,2-Dimethyl propane 0,018 0 0 Kordas (1995)
[9]
Methane 2-Methyl butane −0,005 6 0 Knapp (1982)
[9]
Ethane Propane 0,001 1 0 Knapp (1982)

6 © ISO 2004 – All rights reserved

---------------------- Page: 16 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
Table 3 (continued)
Component i Component j k k Source
ij,0 ij,1
[9]
Ethane n-Butane 0,009 6 0 Knapp (1982)
[9]
Ethane n-Pentane 0,007 8 0 Knapp (1982)
[9]
Ethane n-Hexane −0,010 0 0 Knapp (1982)
[9]
Ethane 2-Methyl propane −0,006 7 0 Knapp (1982)
[7]
Ethane 2,2-Dimethyl propane 0,023 0 0 Nishiumi (1988)
[7]
Ethane 2-Methyl butane 0,016 0 0 Nishiumi (1988)
[9]
Propane n-Butane 0,003 3 0 Knapp (1982)
[9]
Propane n-Pentane 0,026 7 0 Knapp (1982)
[2]
Propane n-Hexane 0,000 7 0 GERG
[9]
Propane 2-Methyl propane −0,007 8 0 Knapp (1982)
Propane 2,2-Dimethyl propane 0 0
[9]
Propane 2-Methyl butane 0,011 1 0 Knapp (1982)
[2]
2-Methyl propane n-Butane −0,000 4 0 GERG
2-Methyl propane n-Pentane 0 0
2-Methyl propane n-Hexane 0 0
2-Methyl propane 2,2-Dimethyl propane 0 0
2-Methyl propane 2-Methyl butane 0 0
[9]
n-Butane n-Pentane 0,017 4 0 Knapp (1982)
[2] [12]
n-Butane n-Hexane −0,005 6 0 K-BP
n-Butane 2,2-Dimethyl propane 0 0
n-Butane 2-Methyl butane 0 0
2,2-Dimethyl propane 2-Methyl butane 0 0
2,2-Dimethyl propane n-Pentane 0 0
2,2-Dimethyl propane n-Hexane 0 0
[9]
2-Methyl butane n-Pentane 0,060 0 Knapp (1982)
2-Methyl butane n-Hexane 0 0
n-Pentane n-Hexane 0 0
5.2 Input and output
5.2.1 Input
The input parameters for the water content/water dew point correlation are:
a) dry gas composition (mol %),
b) absolute pressure (bar),
3
c) water content (mg/m ) or water dew point (°C).
5.2.2 Output
3
The correlation calculates either the water dew point (°C) or the water content (mg/m ).
© ISO 2004 – All rights reserved 7

---------------------- Page: 17 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
Annex A
(normative)

Thermodynamic principles
NOTE This annex provides more details to Clause 5 and reference [6] describes a PC program for calculating the
water content or water dew point.
A.1 Phase equilibrium thermodynamics
A.1.1 General
The second principle of thermodynamics defines the state of thermodynamic equilibrium of a closed system
as the state of maximum entropy. The entropy, S, of an isolated system can only increase, therefore the initial
state of equilibrium of an isolated system is stable.
When a system is perturbed in the region of equilibrium it returns to equilibrium as soon as the perturbation
has ceased. In some cases, the return to equilibrium can take an infinite time; this is referred to as asymptotic
stability. Thus the condition of equilibrium is written as follows:
∆ This perturbation can be developed for all orders:
11 1
23 4
∆SS=δδ+++S δS δS+ . (A.2)
2! 3! 4!
where δ denotes the differential forms.
When the terms for all orders greater than 1 are negative:
23 4
δSS=<0 andδδδ0S< 0S< 0 . (A.3)
the equilibrium is said to be stable.
When the second order term is positive:
2
δδSS=>0 and 0 (A.4)
the equilibrium is said to be unstable.
When some of the higher order terms are positive:
23 4
∆=SS0 andδδ< 0 butS> 0δS> 0 (A.5)
the equilibrium is said to be metastable.
The limit of metastability is defined by the following:
2
δ S = 0 (A.6)
8 © ISO 2004 – All rights reserved

---------------------- Page: 18 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
The study of the thermodynamic state of equilibrium of a system is conducted on the basis of thermodynamic
potentials; in terms of Gibbs' free energy, G, at fixed T and p, the state of equilibrium is defined by a minimum:
2
dGG=>0 and d 0 (A.7)
In the same way, at fixed T and V the state of equilibrium will be defined by the minimum of Helmholtz free
energy A. For a system including “nc” constituents distributed over “nΦ” phases, the free energy to be
minimized is:
nnΦ c
Gn= µ (A.8)
∑∑ ij ij
ji==11
with the following constraints
nΦ
nn= (A.9)
ij i
∑
j=1
where
n is the total number of moles of constituent i;
i
n is the number of moles of constituent i in phase j;
ij
µ is the chemical potential of constituent i in phase j.
ij
The direct search for a state of equilibrium is thus a minimizing of a function of (nΦ,nc) variables under nc
constraints. The simplicity of this statement of the problem masks the practical difficulties of such a search.
Applying the first law of thermodynamics for a homogeneous closed system, more specifically pure or
constant composition fluids, the following equation is obtained.
ddUW=+dQ (A.10)
In a reversible process, from the Second Law of Thermodynamics
ddWp=− V (A.11)
and
ddQT= S (A.12)
Combining Equations (A.10), (A.11) and (A.12) gives
ddUT=−S pdV (A.13)
A heterogeneous closed system contains two or more phases. In such a system, each phase may be viewed
as a homogeneous open system, because any component in the system may move across the phase
boundaries from one phase to another.
The internal energy, U, of a homogeneous open system is dependent on additional variables: the quantity of
each component in the system. The quantity of the components may be represented by the number of moles
of the components.
For an open system containing N components,
UU= (S,V,n ,n ,.n ) (A.14)
12 N
© ISO 2004 – All rights reserved 9

---------------------- Page: 19 ----------------------

SIST EN ISO 18453:2006
ISO 18453:2004(E)
Applying the chain rule to Equation (A.14) to obtain total differential gives
N

δδUU δU
ddUS=+ dV+ dn (A.15)

  ∑ i
δδSV δn
 
i
Vn,,Sn
i SV,,n
j
where j represents all components other than i.
Since
δδUU
...