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STANDARDSIST EN ISO 12213-2:20051DGRPHãþD
SIST EN ISO 12213-2:2009
EUROPEAN STANDARD NORME EUROPÉENNE EUROPÄISCHE NORM
EN ISO 12213-2
September 2009 ICS 75.060 Supersedes EN ISO 12213-2:2005English Version
Natural gas - Calculation of compression factor - Part 2: Calculation using molar-composition analysis (ISO 12213-2:2006)
Gaz naturel - Calcul du facteur de compression - Partie 2: Calcul à partir de l'analyse de la composition molaire (ISO 12213-2:2006)
Erdgas - Berechnung von Realgasfaktoren - Teil 2: Berechnungen basierend auf einer molaren Gasanalyse als Eingangsgröße (ISO 12213-2:2006) This European Standard was approved by CEN on 13 August 2009.
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 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 Management Centre has the same status as the official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
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EN ISO 12213-2:2009 (E) 2 Contents Page Foreword .3 SIST EN ISO 12213-2:2009
EN ISO 12213-2:2009 (E) 3 Foreword The text of ISO 12213-2:2006 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 12213-2:2009. 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of an identical text or by endorsement, at the latest by March 2010, and conflicting national standards shall be withdrawn at the latest by March 2010. Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such patent rights. This document supersedes EN ISO 12213-2:2005. 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, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and the United Kingdom. Endorsement notice The text of ISO 12213-2:2006 has been approved by CEN as a EN ISO 12213-2:2009 without any modification.
SIST EN ISO 12213-2:2009
SIST EN ISO 12213-2:2009
Reference numberISO 12213-2:2006(E)© ISO 2006
INTERNATIONAL STANDARD ISO12213-2Second edition2006-11-15Natural gas — Calculation of compression factor — Part 2: Calculation using molar-composition analysis Gaz naturel — Calcul du facteur de compression — Partie 2: Calcul à partir de l'analyse de la composition molaire
SIST EN ISO 12213-2:2009
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© ISO 2006 – All rights reserved
SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
iiiContents Page Foreword.iv 1 Scope.1 2 Normative references.1 3 Terms and definitions.1 4 Method of calculation.2 4.1 Principle.2 4.2 The AGA8-92DC equation.2 4.3 Input variables.3 4.4 Ranges of application.3 4.5 Uncertainty.5 5 Computer program.7 Annex A (normative)
Symbols and units.8 Annex B (normative)
Description of the AGA8-92DC method.10 Annex C (normative)
Example calculations.18 Annex D (normative)
Pressure and temperature conversion factors.19 Annex E (informative)
Performance over wider ranges of application.20 Annex F (informative)
Subroutines in Fortran for the AGA8-92DC method.25 Bibliography.32
SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) iv
© ISO 2006 – 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 12213-2 was prepared by Technical Committee ISO/TC 193, Natural gas, Subcommittee SC 1, Analysis of natural gas. This second edition cancels and replaces the first edition (ISO 12213-2:1997), Table 1 of which has been technically revised. ISO 12213 consists of the following parts, under the general title Natural gas — Calculation of compression factor: ⎯ Part 1: Introduction and guidelines ⎯ Part 2: Calculation using molar-composition analysis ⎯ Part 3: Calculation using physical properties
SIST EN ISO 12213-2:2009
INTERNATIONAL STANDARD ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
1Natural gas — Calculation of compression factor — Part 2: Calculation using molar-composition analysis 1 Scope ISO 12213 specifies methods for the calculation of compression factors of natural gases, natural gases containing a synthetic admixture and similar mixtures at conditions under which the mixture can exist only as a gas. This part of ISO 12213 specifies a method for the calculation of compression factors when the detailed composition of the gas by mole fractions is known, together with the relevant pressures and temperatures. 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, with an uncertainty of about ± 0,1 %. It can be applied, with greater uncertainty, to wider ranges of gas composition, pressure and temperature (see Annex E). More detail concerning the scope and field of application of the method is given in ISO 12213-1. 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 6976, Natural gas — Calculation of calorific values, density, relative density and Wobbe index from composition ISO 12213-1, Natural gas — Calculation of compression factor — Part 1: Introduction and guidelines ISO 80000-4, Quantities and units — Part 4: Mechanics ISO 80000-5, Quantities and units — Part 5: Thermodynamics 3 Terms and definitions For the purposes of this document, the terms and definitions given in ISO 12213-1 apply. SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) 2
© ISO 2006 – All rights reserved 4 Method of calculation 4.1 Principle The method recommended uses an equation based on the concept that pipeline quality natural gas may be uniquely characterized for calculation of its volumetric properties by component analysis. This analysis, together with the pressure and temperature, are used as input data for the method. The method uses a detailed molar-composition analysis in which all constituents present in amounts exceeding a mole fraction of 0,000 05 should be represented. Typically, this includes all alkane hydrocarbons up to about C7 or C8 together with nitrogen, carbon dioxide and helium. For other gases, additional components such as water vapour, hydrogen sulfide and ethylene need to be taken into consideration (see Reference [1] in the Bibliography). For manufactured gases, hydrogen and carbon monoxide are also likely to be significant components. 4.2 The AGA8-92DC equation The compression factor is determined using the AGA8 detailed characterization equation (denoted hereafter as the AGA8-92DC equation). This is an extended virial-type equation. The equation is described in AGA Report No. 8[1]. It may be written as ()()185813131expnnnkbknnnnnnZBCbckcρρρρρ===+−+−−∑∑mrrrr**nnC (1) where Z is the compression factor; B is the second virial coefficient; ρm is the molar density (moles per unit volume); ρr is the reduced density; bn, cn, kn are constants (see Table B.1); nC∗ are coefficients which are functions of temperature and composition. The reduced density ρr is related to the molar density ρm by the equation 3Kρρ=rm (2) where K is a mixture size parameter. The molar density can be written as ()pZRTρ=m (3) where p is the absolute pressure; R is the universal gas constant; T is the absolute temperature. SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
3Z is calculated as follows: first the values of B and nC∗(n = 13 to 58) are calculated, using relationships given in Annex B. Equations (1) and (3) are then solved simultaneously for ρm and Z by a suitable numerical method (see Figure B.1). 4.3 Input variables The input variables required for use with the AGA8-92DC equation are the absolute pressure, absolute temperature and molar composition. The composition is required, by mole fraction, of the following components: nitrogen, carbon dioxide, argon, methane, ethane, propane, n-butane, methyl-2-propane (iso-butane), n-pentane, methyl-2-butane (iso-pentane), hexanes, heptanes, octanes, nonanes, decanes, hydrogen, carbon monoxide, hydrogen sulfide, helium, oxygen and water. NOTE If the mole fractions of the heptanes, octanes, nonanes and decanes are unknown, then use of a composite C6+ fraction may be acceptable. The user should carry out a sensitivity analysis in order to test whether a particular approximation of this type degrades the result. All components with mole fractions greater than 0,000 05 shall be accounted for. Trace components (such as ethylene) shall be treated as given in Table 1. If the composition is known by volume fractions, these shall be converted to mole fractions using the method given in ISO 6976. The sum of all mole fractions shall be unity to within 0,000 1. 4.4 Ranges of application 4.4.1 Pipeline quality gas The ranges of application for pipeline quality gas are as defined below: absolute pressure 0 MPa u p u 12 MPa temperature 263 K u T u 338 K superior calorific value 30 MJ⋅m−3 u HS u 45 MJ⋅m−3 relative density 0,55 u d u 0,80 The mole fractions of the natural-gas components shall be within the following ranges: methane 0,7 u xCH4 u 1,00 nitrogen 0 u xN2 u 0,20 carbon dioxide 0 u xCO2 u 0,20 ethane 0 u xC2H6 u 0,10 propane 0 u xC3H8 u 0,035 butanes 0 u xC4H10 u 0,015 pentanes 0 u xC5H12 u 0,005 hexanes 0 u xC6 u 0,001 heptanes 0 u xC7 u 0,000 5 SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) 4
© ISO 2006 – All rights reserved octanes plus higher hydrocarbons 0 u xC8+ u 0,000 5 hydrogen 0 u xH2 u 0,10 carbon monoxide 0 u xCO u 0,03 helium 0 u xHe u 0,005 water 0 u xH2O u 0,000 15 Any component for which xi is less than 0,000 05 can be neglected. Minor and trace components are listed in Table 1. Table 1 — Minor and trace components Minor and trace component Assigned component Oxygen Oxygen Argon, neon, krypton, xenon Argon Hydrogen sulfide Hydrogen sufide Nitrous oxide Carbon dioxide Ammonia Methane Ethylene, acetylene, methanol (methyl alcohol), hydrogen cyanide Ethane Propylene, propadiene, methanethiol (methyl mercaptan) Propane Butenes, butadienes, carbonyl sulfide (carbon oxysulfide), sulfur dioxide n-Butane Neo-pentane, pentenes, benzene, cyclopentane, carbon disulfide n-Pentane All C6−isomers, cyclohexane, toluene, methylcyclopentanen-Hexane All C7−isomers, ethylcyclopentane, methylcyclohexane, cycloheptane, ethylbenzene, xylenes n-Heptane All C8−isomers, ethylcyclohexane n-Octane All C9−isomers n-Nonane All C10−isomers and all higher hydrocarbons n-Decane The method applies only to mixtures in the single-phase gaseous state (above the dew point) at the conditions of temperature and pressure of interest. 4.4.2 Wider ranges of application The ranges of application tested beyond the limits given in 4.4.1 are: absolute pressure 0 MPa u p u 65 MPa temperature 225 K u T u 350 K relative density 0,55 u d u 0,90 superior calorific value 20 MJ⋅m−3 u HS u 48 MJ⋅m−3 SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
5The allowable mole fractions of the major natural-gas components are: methane 0,50 u xCH4 u 1,00 nitrogen 0 u xN2 u 0,50 carbon dioxide 0 u xCO2 u 0,30 ethane 0 u xC2H6 u 0,20 propane 0 u xC3H8 u 0,05 hydrogen 0 u xH2 u 0,10 The limits for minor and trace gas components are as given in 4.4.1 for pipeline quality gas. For use of the method outside these ranges, see Annex E. 4.5 Uncertainty 4.5.1 Uncertainty for pipeline quality gas The uncertainty of results for use on all pipeline quality gas within the limits described in 4.4.1 is ± 0,1 % (for the temperature range 263 K to 350 K and pressures up to 12 MPa) (see Figure 1). For temperatures above 290 K and at pressures up to 30 MPa the uncertainty of the result is also ± 0,1 %. For lower temperatures, the uncertainty of ± 0,1 % is at least maintained for pressures up to about 10 MPa. This uncertainty level has been determined by comparison with the GERG databank of measurements of the compression factor for natural gases [2], [3]. A detailed comparison was also made with the GRI pVT data on gravimetrically prepared simulated natural-gas mixtures [4], [5]. The uncertainty of the measurements in both databanks used to test the method is of the order of ± 0,1 %. 4.5.2 Uncertainty for wider ranges of application The estimated uncertainties for calculations of compression factors beyond the limits of quality given in 4.4.1 are discussed in Annex E. 4.5.3 Impact of uncertainties of input variables Listed in Table 2 are typical values for the uncertainties of the relevant input variables. These values may be achieved under optimum operating conditions. As a general guideline only, an error propagation analysis using the uncertainties in the input variables produces an additional uncertainty of about ± 0,1 % in the result at 6 MPa and within the temperature range 263 K to 338 K. Above 6 MPa, the additional uncertainties are greater and increase roughly in direct proportion to the pressure. SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) 6
© ISO 2006 – All rights reserved
AGA8-DC92 equation Key p pressure T temperature 1 ∆Z u ± 0,1 % 2 ∆Z ± 0,1 % to ± 0,2 % 3 ∆Z ± 0,2 % to ± 0,5 % NOTE The uncertainty limits given are expected to be valid for natural gases and similar gases with xN2 u 0,20, xCO2 u 0,20, xC2H6 u 0,10 and xH2 u 0,10, and for 30 MJ⋅m−3 u HS u 45 MJ⋅m−3 and 0,55 u d u 0,80. Figure 1 — Uncertainty limits for the calculation of compression factors SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
7Table 2 — Uncertainties of input variables Input variable Absolute uncertainty Absolute pressure ± 0,02 MPa Temperature ± 0,15 K Mole fraction of
inerts ± 0,001 nitrogen ± 0,001 carbon dioxide ± 0,001 methane ± 0,001 ethane ± 0,001 propane ± 0,000 5 butanes ± 0,000 3 pentanes plus higher hydrocarbons ± 0,000 1 hydrogen and carbon monoxide ± 0,001 4.5.4 Reporting of results Results for compression factor and molar density shall be reported to four and to five places of decimals, respectively, together with the pressure and temperature values and the calculation method used (ISO 12213-2, AGA8-92DC equation). For verification of calculation procedures, it is useful to carry extra digits. 5 Computer program Software which implements this International Standard has been prepared. Users of this part of ISO 12213 are invited to contact ISO/TC 193/SC 1, either directly or through their ISO member body, to enquire about the availability of this software. SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) 8
© ISO 2006 – All rights reserved Annex A (normative)
Symbols and units Symbol Meaning Units an Constant in Table B.1 — B Second virial coefficient m3⋅kmol−1 nijB∗ Mixture interaction coefficient [Equations (B.1) and (B.2)] — bn Constant in Table B.1 — cn Constant in Table B.1 — nC∗ Coefficients which are functions of temperature and composition — Ei Characteristic energy parameter for ith component (Table B.2) K Ej Characteristic energy parameter for jth component K Eij Binary energy parameter for second virial coefficient K ijE∗ Binary energy interaction parameter for second virial coefficient (Table B.3) — F Mixture high-temperature parameter — Fi High-temperature parameter for ith component (Table B.2) — Fj High-temperature parameter for jth component — fn Constant in Table B.1 — G Mixture orientation parameter — Gi Orientation parameter for ith component (Table B.2) — Gj Orientation parameter for jth component — Gij Binary orientation parameter — ijG∗ Binary interaction parameter for orientation (Table B.3) — gn Constant in Table B.1 — HS Superior calorific value MJ⋅m−3 K Size parameter (m3/kmol)1/3 Ki Size parameter for ith component (Table B.2) (m3/kmol)1/3 Kj Size parameter for jth component (m3/kmol)1/3 Kij Binary interaction parameter for size (Table B.3) — kn Constant in Table B.1 —
SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
9Symbol Meaning Units M Molar mass kg⋅kmol−1 Mi Molar mass of ith component kg⋅kmol−1 N Number of components in gas mixture
n An integer (from 1 to 58) — p Absolute pressure MPa Q Quadrupole parameter — Qi Quadrupole parameter for ith component — Qj Quadrupole parameter for jth component — qn Constant (Table B.1) — R Gas constant (= 0,008 314 510) MJ⋅(kmol⋅K)−1 Si Dipole parameter for ith component (Table B.2) — Sj Dipole parameter for jth component — sn Constant (Table B.1) — T Absolute temperature K U Mixture energy parameter K Uij Binary interaction parameter for mixture energy (Table B.3) — un Constant in Table B.1 — Wi Association parameter for ith component (Table B.2) — Wj Association parameter for jth component — wn Constant (Table B.1) — xi Mole fraction of ith component in gas mixture — xj Mole fraction of jth component in gas mixture — Z Compression factor — ρ Mass density kg⋅m−3 ρr Reduced density of gas — ρm Molar density kmol⋅m−3 SIST EN ISO 12213-2:2009
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© ISO 2006 – All rights reserved Annex B (normative)
Description of the AGA8-92DC method B.1 General For gas mixtures, the compression factor Z is calculated using the equations given in 4.2. This annex gives a detailed description of the computations and the necessary numerical values. The description is based upon that given in AGA Report No. 8 [1]. A program implementing this description is given in Annex F, and as such provides the correct solution. Other computational procedures are acceptable provided that they can be demonstrated to yield identical numerical results (see Annex C for examples). B.2 Computer implementation of the AGA8-92DC method B.2.1 Overview of the calculation procedure I Input the absolute temperature T, absolute pressure p and mole fraction of each component xi of the mixture. NOTE For pressure and temperature, values known in any other units will first have to be converted precisely to values in megapascals and kelvins, respectively (see ISO 80000-4 and ISO 80000-5 and Annex D for relevant conversion factors). II Compute the equation of state coefficients B and nC∗ (n = 13 to 58) that depend on T and xi. III Solve iteratively for the molar density ρm, using the equation of state rearranged to give the pressure p. IV Output the compression factor after the computed pressure from step III and the input pressure from step I agree within a specified range of convergence (e.g. 1E-06). Figure B.1 shows a flow diagram of these steps. B.2.2 Details of the calculation procedure Step I Input the absolute temperature T, absolute pressure p and mole fraction xi of each constituent in the natural-gas mixture. Step II At the absolute temperature T and the mole fractions xi of the natural gas (as input from step I), compute the composition- and temperature-dependent coefficients B and nC∗ (n = 13 to 58). The second virial coefficient B is given by the following equations: ()1832*111nnNNuunijijijnijnijBaTxxBEKK−====∑∑∑ (B.1) ()()()()()1212*11111nnnnnfgqswnijijnijnnijnijnijBGgQQqFFfSSsWWw=+−+−+−+−+− (B.2) SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
11 Figure B.1 — AGA8-92DC equation — Calculation flow diagram The binary parameters Eij and Gij are calculated using the following equations: ()12*ijijijEEEE= (B.3) ()*2ijijijGGGG=+ (B.4) Note that all values of the binary interaction parameters ijE∗ and ijG∗ are 1,0 except for the values given in Table B.3. SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) 12
© ISO 2006 – All rights reserved The coefficients nC∗ (n = 13 to 58) are given by the equation: ()()()*2111nnnnnqgfuunnnnnCaGgQqFfUT−=+−+−+− (B.5) The mixture parameters U, G, Q and F are calculated using the following conformal solution mixing equations, where in the double sums i ranges from 1 to N - 1 and, for each value of i, j ranges from i + 1 to N: ()()215252551111NNNiijijijiiijiUxExxUEE−===+⎛⎞⎜⎟=+−⎜⎟⎝⎠∑∑∑ (B.6) ()()1*1111NNNiiijijijiijiGxGxxGGG−===+=+−+∑∑∑ (B.7) 1NiiiQxQ==∑ (B.8) 21NiiiFxF==∑ (B.9) It should be noted that all values of the binary interaction parameters Kij, ijE∗, ijG∗ and Uij are 1,0 except for the values given in Table B.3. Also note that Fi is zero for all components except hydrogen, for which F(H2) = 1,0, and that Wi is zero for all components except water, for which W(H2O) = 1,0. Step III In the computation of the compression factor Z, the composition of the gas is known, the absolute temperature T of the gas is known and the absolute pressure is known. The problem then is to compute the molar density ρm, using the equation of state expression for the pressure p. For this purpose, the definition of the compression factor Z as given in Equation (1) (see 4.2) is substituted into Equation (3) to obtain an equation for the pressure as given in Equation (B.10): ()()1858**13131expnnnkbknnnnnnnnpRTBCCbckcρρρρρρ==⎡⎤⎢⎥=+−+−−⎢⎥⎣⎦∑∑mmrrrr (B.10) Equation (B.10) is solved using standard equation of state density search algorithms. Having obtained an equation for the pressure p [Equation (B.10)], the problem is then to search for the value of the molar density ρm that will yield the pressure that is within a preset limit (e.g. 1 × 10−6) equal to the input pressure. The reduced density ρr is related to the molar density ρm by the mixture size parameter [see Equation (2) in 4.2]. The mixture size parameter K is calculated using the following equation: ()()2152552511121NNNiiijijijiijiKxKxxKKK−===+⎛⎞⎜⎟=+−⎜⎟⎝⎠∑∑∑ (B.11) Note that in the summations the subscript i refers to the ith component in the gas mixture and the subscript j refers to the jth component in the gas mixture. The quantity N is the number of components in the mixture. Thus, in the single summation, i ranges over the integer values from 1 to N. For example, for a mixture of 12 components, N = 12 and there would be 12 terms in the single sum. In the double summation, i ranges SIST EN ISO 12213-2:2009
ISO 12213-2:2006(E) © ISO 2006 – All rights reserved
13from 1 to N − 1 and, for each value of i, j ranges from i + 1 to N. For example, for a mixture of 12 components, there would be 66 terms in the double summation if all values of Kij differed from 1,0. However, because many of the values of Kij are 1,0, the number of non-zero terms in the double summation is small for many natural-gas mixtures. Note that all values of Kij are 1,0 except for the values given in Table B.3. Step IV Once the molar density ρm has been obtained in step III, the compression factor is calculated in step IV using the pressure, temperature, molar density and gas constant: ()ZpRTρ=m (B.12) NOTE The density ρ (mass per unit volume) can be calculated as follows: ρ = Mρm (B.13) where M is calculated from the equation: 1NiiiMxM==∑ (B.14) Report the density to three places of decimals. Table B.1 — Equation of state parameters n an bn cn kn un gn qn fn sn wn 1 0,153 832 600 1 0 0 0,0 0 0 0 0 0 2 1,341 953 000 1 0 0 0,5 0 0 0 0 0 3 − 2,998 583 000 1 0 0 1,0 0 0 0 0 0 4 − 0,048 312 280 1 0 0 3,5 0 0 0 0 0 5 0,375 796 500 1 0 0 − 0,5 1 0 0 0 0 6 − 1,589 575 000 1 0 0 4,5 1 0 0 0 0 7 − 0,053 588 470 1 0 0 0,5 0 1 0 0 0 8 0,886 594 630 1 0 0 7,5 0
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