Natural gas - Calculation of thermodynamic properties - Part 1: Gas phase properties for transmission and distribution applications (ISO 20765-1:2005)

This part of ISO 20765 specifies a method of calculation for the volumetric and caloric properties of natural gases, natural gases containing synthetic admixture and similar mixtures, at conditions where the mixture can exist only as a gas.
The method is applicable to pipeline-quality gases within the ranges of pressure and temperature at which transmission and distribution operations normally take place. For volumetric properties (compression factor and density), the uncertainty of calculation is about ± 0,1 % (95 % confidence interval). For caloric properties (for example enthalpy, heat capacity, Joule-Thomson coefficient, speed of sound), the uncertainty of calculation is usually greater.

Erdgas - Berechnung thermodynamischer Eigenschaften - Teil 1: Eigenschaften der Gasphase für Zwecke des Transports und der Verteilung (ISO 20765-1:2005)

Gaz naturel - Calcul des propriétés thermodynamiques - Partie 1: Propriétés de la phase gazeuse pour des applications de transport et de distribution (ISO 20765-1:2005)

L'ISO 20765-1:2005 spécifie une méthode de calcul des propriétés volumétriques et calorifiques des gaz naturels, de gaz naturels contenant un adjuvant synthétique et de mélanges similaires, dans des conditions telles que le mélange ne peut exister que sous forme gazeuse.
La méthode est applicable aux gaz de qualité réseau dans les plages de pression et de température auxquelles s'effectuent normalement les opérations de transport et de distribution. Pour les propriétés volumétriques (facteur de compression et masse volumique), l'incertitude de calcul avoisine ± 0,1 % (intervalle de confiance de 95 %). En revanche, l'incertitude de calcul est généralement plus élevée pour les propriétés calorifiques (par exemple l'enthalpie, la capacité thermique, le coefficient de Joule-Thomson, la vitesse du son).

Zemeljski plin - Izračun termodinamičnih lastnosti - 1. del: Lastnosti plinaste faze za prenos in distribucijo (ISO 20765-1:2005)

Ta del standarda ISO 20765 določa metodo za izračun volumetričnih in kaloričnih lastnosti zemeljskih plinov, zemeljskih plinov, ki vsebujejo sintetične dodatke, in podobnih mešanic pri pogojih, v katerih lahko mešanica obstaja samo v plinastem stanju.
Metoda se uporablja za pline iz cevovoda v razponu tlaka in temperature, v katerem običajno potekajo dejavnosti prenosa in distribucije. Negotovost izračuna za volumetrične lastnosti (kompresijski faktor in gostota) znaša približno ±0,1 % (95-odstotni interval zaupanja). Negotovost izračuna za kalorične lastnosti (na primer entalpija, toplotna kapaciteta, Joul-Thomsonov koeficient, hitrost zvoka) je običajno večja.

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Published
Publication Date
25-Sep-2018
Current Stage
6060 - Definitive text made available (DAV) - Publishing
Due Date
26-Sep-2018
Completion Date
26-Sep-2018

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2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.Erdgas - Berechnung thermodynamischer Eigenschaften - Teil 1: Eigenschaften der Gasphase für Zwecke des Transports und der Verteilung (ISO 20765-1:2005)Gaz naturel - Calcul des propriétés thermodynamiques - Partie 1: Propriétés de la phase gazeuse pour des applications de transport et de distribution (ISO 20765-1:2005)Natural gas - Calculation of thermodynamic properties - Part 1: Gas phase properties for transmission and distribution applications (ISO 20765-1:2005)75.060Zemeljski plinNatural gasICS:Ta slovenski standard je istoveten z:EN ISO 20765-1:2018SIST EN ISO 20765-1:2018en,fr,de01-december-2018SIST EN ISO 20765-1:2018SLOVENSKI

STANDARD
SIST EN ISO 20765-1:2018
EUROPEAN STANDARD NORME EUROPÉENNE EUROPÄISCHE NORM
EN ISO 20765-1
September
t r s z ICS
y wä r x r English Version
Natural gas æ Calculation of thermodynamic properties æ Part

sã Gas phase properties for transmission and Gaz naturel æ Calcul des propriétés thermodynamiques æ Partie

sã Propriétés de la phase gazeuse pour des
Erdgas æ Berechnung thermodynamischer Eigenschaften æ Teil
sã Eigenschaften der Gasphase für This European Standard was approved by CEN on
u s August
t r s zä

egulations which stipulate the conditions for giving this European Standard the status of a national standard without any alterationä Upætoædate lists and bibliographical references concerning such national standards may be obtained on application to the CENæCENELEC Management Centre or to any CEN memberä

translation under the responsibility of a CEN member into its own language and notified to the CENæCENELEC Management Centre has the same status as the official versionsä

CEN members are the national standards bodies of Austriaá Belgiumá Bulgariaá Croatiaá Cyprusá Czech Republicá Denmarká Estoniaá Finlandá Former Yugoslav Republic of Macedoniaá Franceá Germanyá Greeceá Hungaryá Icelandá Irelandá Italyá Latviaá Lithuaniaá Luxembourgá Maltaá Netherlandsá Norwayá Polandá Portugalá Romaniaá Serbiaá Slovakiaá Sloveniaá Spainá Swedená Switzerlandá Turkey and United Kingdomä

EUROPEAN COMMITTEE FOR STANDARDIZATION COMITÉ EUROPÉEN DE NORMALISATION EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre:
Rue de la Science 23,
B-1040 Brussels

t r s z CEN All rights of exploitation in any form and by any means reserved worldwide for CEN national Membersä Refä Noä EN ISO

t r y x wæ sã t r s z ESIST EN ISO 20765-1:2018

EN ISO 20765-1:2018 (E) 2 Contents Page European foreword ....................................................................................................................................................... 3

SIST EN ISO 20765-1:2018

EN ISO 20765-1:2018 (E) 3 European foreword The text of ISO 20765-1:2005 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 20765-1:2018 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 20765-1:2005 has been approved by CEN as EN ISO 20765-1:2018 without any modification. SIST EN ISO 20765-1:2018

SIST EN ISO 20765-1:2018
Reference numberISO 20765-1:2005(E)© ISO 2005

INTERNATIONAL STANDARD ISO20765-1First edition2005-09-15Natural gas — Calculation of thermodynamic properties — Part 1: Gas phase properties for transmission and distribution applications Gaz naturel — Calcul des propriétés thermodynamiques — Partie 1: Propriétés de la phase gazeuse utilisée pour des applications de transport et de distribution

SIST EN ISO 20765-1:2018

ISO 20765-1:2005(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 2005 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.

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© ISO 2005 – All rights reserved
SIST EN ISO 20765-1:2018
ISO 20765-1:2005(E) © ISO 2005 – All rights reserved

iiiContents Page Foreword............................................................................................................................................................iv Introduction........................................................................................................................................................v 1 Scope......................................................................................................................................................1 2 Normative references............................................................................................................................1 3 Terms and definitions...........................................................................................................................1 4 Thermodynamic basis of the method..................................................................................................2 4.1 Principle..................................................................................................................................................2 4.2 The fundamental equation of Helmholtz free energy.........................................................................3 4.3 Thermodynamic properties derived from the Helmholtz free energy..............................................5 5 Method of calculation............................................................................................................................8 5.1 Input variables........................................................................................................................................8 5.2 Conversion from pressure to reduced density...................................................................................9 5.3 Implementation......................................................................................................................................9 6 Ranges of application.........................................................................................................................10 6.1 Pressure and temperature..................................................................................................................10 6.2 Pipeline quality gas.............................................................................................................................10 7 Uncertainty...........................................................................................................................................11 7.1 Uncertainty for pipeline quality gas...................................................................................................11 7.2 Impact of uncertainties of input variables........................................................................................14 8 Reporting of results.............................................................................................................................14 Annex A (normative)

Symbols and units........................................................................................................16 Annex B (normative)

The Helmholtz free energy of the ideal gas...............................................................19 Annex C (normative)

The equation for the Helmholtz free energy..............................................................22 Annex D (normative)

Detailed documentation for the equation of state.....................................................24 Annex E (informative)

Assignment of trace components.............................................................................30 Annex F (informative)

Implementation of the method...................................................................................32 Annex G (informative)

Examples.....................................................................................................................35 Bibliography......................................................................................................................................................42

SIST EN ISO 20765-1:2018
ISO 20765-1:2005(E) iv

© ISO 2005 – All rights reserved 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 20765-1 was prepared by Technical Committee 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 The following parts are under preparation: ⎯ Part 2: Single phase properties (gas, liquid and dense-fluid) for extended ranges of application ⎯ Part 3: Two-phase properties (vapour-liquid equilibria) SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) © ISO 2005 – All rights reserved

vIntroduction This part of ISO 20765 specifies methods for the calculation of thermodynamic properties of natural gases, natural gases containing synthetic admixture, and similar mixtures. This part of ISO 20765 has four normative annexes and three informative annexes.

SIST EN ISO 20765-1:2018
SIST EN ISO 20765-1:2018
INTERNATIONAL STANDARD ISO 20765-1:2005(E) © ISO 2005 – All rights reserved

1Natural gas — Calculation of thermodynamic properties — Part 1: Gas phase properties for transmission and distribution applications 1 Scope This part of ISO 20765 specifies a method of calculation for the volumetric and caloric properties of natural gases, natural gases containing synthetic admixture and similar mixtures, at conditions where the mixture can exist only as a gas. The method is applicable to pipeline-quality gases within the ranges of pressure, p, and temperature, T, at which transmission and distribution operations normally take place. For volumetric properties (compression factor and density), the uncertainty of calculation is about ± 0,1 % (95 % confidence interval). For caloric properties (for example enthalpy, heat capacity, Joule-Thomson coefficient, speed of sound), the uncertainty of calculation is usually greater. 2 Normative references The following referenced documents are indispensable 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 31-4, Quantities and units — Part 4: Heat ISO 7504, Gas analysis — Vocabulary ISO 12213-2, Natural gas — Calculation of compression factor — Part 2: Calculation using molar-composition analysis ISO 14532, Natural gas — Vocabulary 3 Terms and definitions For the purposes of this document, the terms and definitions given in ISO 31-4, ISO 7504 and ISO 14532 and the following apply. NOTE See Annex A for the list of symbols and units used in this part of ISO 20765. 3.1 caloric property characteristic of a gas or homogeneous gas mixture which can be calculated from a fundamental equation of state NOTE The caloric properties to which this part of ISO 20765 can be applied are internal energy, enthalpy, entropy, isochoric heat capacity, isobaric heat capacity, Joule-Thomson coefficient, isentropic exponent and speed of sound. SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) 2

© ISO 2005 – All rights reserved 3.2 equation of state mathematical relationship between state variables of a gas or homogeneous gas mixture NOTE In this part of ISO 20765, it is useful to distinguish between two types of equation of state, namely (1) volumetric equation of state, in which the relationship is between the state variables pressure, temperature and the volume occupied by a given amount of substance, and (2) fundamental equation of state, in which the relationship is between the density, temperature and the Helmholtz free energy. 3.3 residual property that part of a thermodynamic property which results from the non-ideal (real-gas) behaviour of a gas or homogeneous gas mixture, i.e. the difference between a thermodynamic property of a real gas or gas mixture and the same thermodynamic property for the same gas or gas mixture, in the ideal state, at the same state conditions of temperature and density 3.4 thermodynamic property volumetric or caloric property 3.5 volumetric property characteristic of a gas or homogeneous gas mixture that can be calculated from a volumetric equation of state NOTE The volumetric properties to which this part of ISO 20765 can be applied are compression factor and density. 4 Thermodynamic basis of the method 4.1 Principle The method recommended is based on the concept that pipeline-quality natural gas is completely characterized for 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 method. In practice, the state variables available as input data are more usually temperature and pressure and, in this case, it is necessary first to convert these to temperature and density. Equations are presented which express the Helmholtz free energy of the gas as a function of density, temperature and composition, from which all of the thermodynamic properties can be obtained in terms of the Helmholtz free energy and its derivatives with respect to temperature and density. The method uses a detailed molar composition analysis in which all components present in amounts exceeding 0,000 05 mole fraction [50 molar ppm 1)] should be represented. 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 natural gases, it may be necessary to take into consideration additional components such as C9 and C10 hydrocarbons, water vapour and hydrogen sulfide. For manufactured gases, hydrogen and carbon monoxide should be considered. More precisely, the method uses a 21-component analysis in which all of the major and minor components of natural gas are included (see 6.2). Any trace component present but not identified as one of the 21 specified components may be reassigned appropriately to a specified component.

1) ppm is a depredated unit. SIST EN ISO 20765-1:2018
ISO 20765-1:2005(E) © ISO 2005 – All rights reserved

34.2 The fundamental equation of Helmholtz free energy 4.2.1 Background The AGA8 equation [1] was published in 1992 by the Transmission Measurements Committee of the American Gas Association, having been designed specifically as a means for the high accuracy calculation of compression factor. In this respect, it is already the subject of ISO 12213-2. Since then it has become increasingly apparent that the equation has excellent potential for use in the calculation of all thermodynamic properties of natural gas, even though the accuracy of calculation is less well documented. In order for the AGA8 equation to become useful for the calculation of all thermodynamic properties, there are two major requirements. a) The equation itself, published initially in a form explicit only for volumetric properties, has to be mathematically recast in a form explicit for the residual Helmholtz free energy. In fact, although not published as such, the original development of the equation was as a fundamental equation in the form of Helmholtz free energy. This formulation [2] is essential in that all residual thermodynamic properties can be calculated from the Helmholtz free energy and its derivatives with respect to the state conditions of temperature and density. b) For the calculation of caloric properties, a formulation is required for the Helmholtz free energy of the ideal gas as a function of temperature. Most previous formulations for the ideal gas have been explicit in the isobaric heat capacity and so, again, the chosen formulation [3], [4] has to be recast so as to be explicit in the Helmholtz free energy. Again, derivatives of the Helmholtz free energy with respect to the state conditions are needed. An important aspect of the formulations chosen for both the ideal and residual parts of the Helmholtz free energy is that the derivatives required for calculating the thermodynamic properties can be given in analytical form. Hence, there is no need for numerical differentiation or integration within any computer program that implements the procedures. As a result, numerical problems are avoided and calculation times are shorter. The method of calculation described is very suitable for use within process simulation programs and, in particular, within programs developed for use in natural gas transmission and distribution applications. 4.2.2 The Helmholtz free energy The Helmholtz free energy, f, of a homogeneous gas mixture at uniform pressure and temperature can be expressed as the sum of a part f o describing the ideal gas behaviour and a part fr describing the residual or real-gas behaviour, as given in Equation (1): ()()()or,,,,,,fXfXfXρΤρΤρΤ=+ (1) which, rewritten in the form of dimensionless reduced free energy ϕ = f/(R⋅T), becomes Equation (2): ()()()or,,,,,,XXXϕδτϕδτϕδτ=+ (2) where X is a vector that defines the composition of the mixture; τ is the inverse (dimensionless) reduced temperature, related to the temperature, T, as given in Equation (3): /LTτ= (3) where L = 1 K. SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) 4

© ISO 2005 – All rights reserved Note that Equations (1) and (2) are written in terms of the molar density, ρ, and reduced density, δ, respectively, not in terms of the more commonly available input variable of pressure, p. This is because, from statistical thermodynamics, the Helmholtz free energy appears as a natural consequence of the number and types of molecular interactions in a mixture and, therefore, becomes a natural function of the molar density and mole fractions of the molecules. The reduced density, δ, is related to the molar density, ρ, as shown in Equation (4): 3Kδρ=⋅ (4) where K is a mixture size parameter. The ideal part, ϕo, of the reduced Helmholtz free energy is obtained from equations for the isobaric heat capacity in the ideal gas state (see 4.2.3), and the residual part, ϕris, from the AGA8 equation of state (see 4.2.4). 4.2.3 The Helmholtz free energy of the ideal gas The Helmholtz free energy of an ideal gas can be expressed in terms of the enthalpy, ho, and entropy, so, as given in Equation (5): ()()()ooo,,

,,fTXhTXRTTsTXρρ=−⋅−⋅ (5) The enthalpy, ho, and entropy, so, can in turn be expressed in terms of the isobaric heat capacity, co,p, of the ideal gas as given in Equations (6) and (7), where the implied limits of integration are T and T: ()oo,po,, dhTXcTh=+∫ (6) o,poo,1(,,)dlnlnlnNiiicTsTXTRRsRxxTTρρρ=⎛⎞⎛⎞=−⋅−⋅+−⋅⋅⎜⎟⎜⎟⎝⎠⎝⎠∑∫ (7) The reference state of zero enthalpy and zero entropy is here adopted as T = 298,15 K and p = 0,101 325 MPa for the ideal unmixed gas. The integration constants, o,h and o,s, are then determined so as to conform to this definition. The reference (ideal) density, ρ, is given by ρ = p/(R⋅T). The reduced Helmholtz free energy ϕo = fo/(R⋅T) can then be written, using Equations (6) and (7), as a function of δ, τ and X, as given in Equation (8): ()o,po,po,o,o21,,d1dlnlnlnNiiicchsXxxRLRRRττδϕδτττττδττ=⋅⎛⎞⎛⎞=−+−+++−+⋅⎜⎟⎜⎟⋅⋅⋅⎝⎠⎝⎠∑∫∫ (8) See Annex B for details of this formulation. 4.2.4 The residual part of the Helmholtz free energy The residual part of the reduced Helmholtz free energy is obtained, for the purposes of this part of ISO 20765, by use of the AGA8 equation. Written for the compression factor as a function of reduced density, inverse reduced temperature and composition, the AGA8 equation has the form of Equation (9): ()()1858313131expnnnnnuubkknnnnnnnnBZCCbckcKδδττδδδ==⋅=+−⋅+⋅⋅−⋅⋅−⋅∑∑ (9) SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) © ISO 2005 – All rights reserved

5where B is the second virial coefficient; bn, cn, kn, un are coefficients of the equation and functions of composition; Cn is a function of composition. The compression factor, Z, is related to the residual part of reduced free energy, ϕr, as given in Equation (10): r,1Zδϕδ=+⋅ (10) where r,ϕδis the partial derivative of ϕr with respect to reduced density at constant 2 and X. Elimination of Z between Equations (9) and (10), and integration with respect to reduced density leads to the Equation (11) for the residual part of the reduced Helmholtz free energy: 1858r31313(,,)exp()nnnnuubknnnnnBXCCcKδϕδτδττδδ==⋅=−⋅+⋅⋅−⋅∑∑ (11) See Annexes C and D for details of this formulation. 4.2.5 The reduced Helmholtz free energy The fundamental equation [Equation (2)] for the reduced Helmholtz free energy, ϕ, makes it possible, through use of Equation (8) for the ideal part, ϕo, and Equation (11) for the residual part, ϕr, to calculate all thermodynamic properties analytically. The reduced Helmholtz free energy, ϕ=, =therefore can be written as given in Equation (12): ()o,po,po,o,21185831313,,d1dlnlnln......exp()nnnnNiiiuubknnnnncchsXxxRRRRBCCcKττδϕδτττττδττδδττδδ===⋅⎛⎞⎛⎞=−+−+++−+⋅+⎜⎟⎜⎟⋅⋅⎝⎠⎝⎠⋅+−⋅+⋅⋅−⋅∑∫∫∑∑ (12) 4.3 Thermodynamic properties derived from the Helmholtz free energy 4.3.1 Background All of the thermodynamic properties can be written explicitly in terms of the reduced Helmholtz free energy, ϕ, and various derivatives thereof. The required derivatives, ϕτ, ϕττ, ϕδ=, ϕδδ and ϕτδ, are defined as given in Equations (13): ,Xϕϕττδ∂⎛⎞=⎜⎟⎝⎠∂

22,Xϕϕτττδ⎛⎞∂=⎜⎟∂⎝⎠
,Xϕϕδδτ∂⎛⎞=⎜⎟⎝⎠∂
22,Xϕϕδδδτ⎛⎞∂=⎜⎟∂⎝⎠

,X,Xϕϕτδτδτδ⎡⎤∂∂⎛⎞=⎢⎥⎜⎟⎝⎠∂∂⎢⎥⎣⎦ (13) Each derivative is the sum of an ideal part (see Annex B) and a residual part (see Annex C). The substitutions given in Equations (14) and (15) help to simplify the appearance of the relevant relationships: ()221,X2δϕϕδϕδϕδδδδδτ⎡⎤∂⋅⎢⎥==⋅⋅+⋅⎢⎥∂⎢⎥⎣⎦ (14) SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) 6

© ISO 2005 – All rights reserved 22,Xδϕϕτδϕτδϕττδδτδδ⎡⎤⋅⎛⎞∂=−=⋅−⋅⋅⎢⎥⎜⎟⎝⎠∂⎣⎦ (15) Detailed expressions for ϕτ, ϕττ, ϕδ, ϕ1 and ϕ2 can be found in Annex C. The relevant general relationships for the various thermodynamic properties are given in 4.3.2.1 to 4.3.2.9 [Equations (17) to (26)]. In Equations (19) to (24), lowercase symbols represent molar quantities (i.e. quantity per mole) and the corresponding upper case symbols represent specific quantities (i.e. quantity per kilogram). Conversion of molar variables to mass-basis variables is achieved by division by the molar mass M. NOTE In these equations, R is the molar gas constant; consequently R/M is the specific gas constant. The molar mass, M, of the mixture is derived from the composition, X, and molar masses, Mi, of the pure substances as given in Equation (16): 1NiiiMxM==⋅∑ (16) Values for molar masses, Mi, of pure substances are given in References [1] and [2]; these values are identical with those given in ISO 6976:1995 [5]. NOTE The values given in ISO 6976 for the molar masses are in most cases not identical with the most recent values adopted by the international community of metrologists. They are, however, the values that were in general use during the development of the AGA8 equation, and are therefore retained here; the differences are in all cases less than 0,001 kg/kmol. In Equations (20), (21) and (23) to (26), the basic expressions for the properties h, s, cp, µ, κ and w have been transformed in several ways, 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. For clarity, the basic thermodynamic relationships are given first in each subclause, and the subsidiary transformations are given below. 4.3.2 Equations for thermodynamic properties 4.3.2.1 Compression factor and density The expression for the compression factor, Z, is given by Equation (17): Zδϕδ=⋅ (17) where ϕδ is the derivative with respect to the reduced molar density of the Helmholtz free energy [see also Equation (10)]. The molar density, ρ, and specific (mass) density, D, are related to pressure as given in Equation (18): ()/DMpZRTρ==⋅⋅ (18) Values of compression factor, Z, calculated in accordance with this part of ISO 20765 should normally be identical with values calculated in accordance with ISO 12213-2. In any case where a requirement for priority is identified, ISO 12213-2 shall take precedence. 4.3.2.2 Internal energy The expression for the internal energy, u, is given by Equation (19): uUMRTRTτϕτ⋅==⋅⋅⋅ (19) SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) © ISO 2005 – All rights reserved

74.3.2.3 Enthalpy The expression for the enthalpy, h, is given by Equation (20): hHMRTRTuUMZZRTRTτϕδϕτδ⋅=⋅⋅=⋅+⋅⋅=+=+⋅⋅ (20) 4.3.2.4 Entropy The expression for the entropy, s, is given by Equation (21): sSMRRuUMRTRTτϕϕϕϕτ⋅==⋅−⋅=−=−⋅⋅ (21) 4.3.2.5 Isochoric heat capacity The expression for the isochoric heat capacity, cv, is given by Equation (22): 2vvcCMRRτϕττ⋅==−⋅ (22) 4.3.2.6 Isobaric heat capacity The expression for the isobaric heat capacity, cp, is given by Equation (23): pp22212v212v21cCMRRcRCMRϕτϕϕϕϕϕϕττ⋅==−⋅+=+⋅=+ (23) 4.3.2.7 Joule-Thomson coefficient The expression for the Joule-Thomson coefficient, µ, is given by Equation (24): ()2122212p12p111RDRMRcRCMµµρϕϕϕτϕϕϕϕϕϕττ⋅⋅⋅⋅=−=−⋅⋅⎛⎞=−⎜⎟⎝⎠⎛⎞=−⎜⎟⋅⎝⎠ (24) SIST EN ISO 20765-1:2018

ISO 20765-1:2005(E) 8

© ISO 2005 – All rights reserved 4.3.2.8 Isentropic exponent The expression for the isentropic exponent, κ, is given by Equation (25): 2212212v212vpp11vvRcZRCMZcCZcZCϕϕτϕκδϕϕϕϕϕϕϕττδ−⋅=⋅+=+⋅==⋅=⋅ (25) 4.3.2.9 Speed of sound The expression for the speed of sound, w, is given by Equation (26): 22212p1vp1vwMRTcZcCCϕϕτϕκϕϕττ⋅=−⋅⋅=⋅== (26) 5 Method of calculation 5.1 Input variables Although the natural formulation of the method presented in this part of ISO 20765 uses reduced density, inverse reduced temperature and molar composition as the input variables, the input variables most usually available for use are the absolute pressure, absolute temperature and the molar composition. In consequence, it is usually necessary first to evaluate the inverse reduced temperature and the reduced density from the available input. The conversion from temperature to inverse reduced temperature is given by Equation (3). The conversion from pressure to reduced density can be carried out as described in 5.2. If, instead of the pressure, p, the (mass) density, D, is available as input, then δ is obtained directly, without the need for the procedure

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