Microsoft Word C040263e doc Reference number ISO 20765 1 2005(E) © ISO 2005 INTERNATIONAL STANDARD ISO 20765 1 First edition 2005 09 15 Natural gas — Calculation of thermodynamic properties — Part 1 G[.]
Principle
The recommended method relies on the principle that the thermodynamic properties of pipeline-quality natural gas can be fully characterized through component analysis This analysis, combined with the state variables of temperature and density, supplies the essential input data required for the method.
In practical applications, the primary state variables used as input data are typically temperature and pressure Therefore, it is essential to first convert these variables into temperature and density.
The Helmholtz free energy of a gas can be expressed through equations that incorporate density, temperature, and composition, enabling the derivation of all thermodynamic properties based on these variables.
Helmholtz free energy and its derivatives with respect to temperature and density
The method involves a comprehensive molar composition analysis, requiring representation of all components present in amounts greater than 0.00005 mole fraction (50 molar ppm) In the case of typical natural gas, this analysis typically includes alkane hydrocarbons up to C7 or C8, along with nitrogen, carbon dioxide, and helium.
Typically, isomers for alkanes above C 5 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 C 9 and
C 10 hydrocarbons, water vapour and hydrogen sulfide For manufactured gases, hydrogen and carbon monoxide should be considered
The method employs a comprehensive 21-component analysis that encompasses all significant and minor constituents of natural gas Any trace components not classified among the 21 specified can be accurately reassigned to an appropriate designated component.
The fundamental equation of Helmholtz free energy
The AGA8 equation [1] was published in 1992 by the Transmission Measurements Committee of the American
The Gas Association has developed a method specifically for accurately calculating the compression factor, which is recognized under ISO 12213-2 This equation has shown significant potential for determining various thermodynamic properties of natural gas, although the accuracy of these calculations is not as thoroughly documented.
To effectively utilize the AGA8 equation for calculating thermodynamic properties, two key requirements must be met First, the equation, originally published solely for volumetric properties, needs to be reformulated to explicitly express the residual Helmholtz free energy Notably, the original development of the equation was intended as a fundamental equation in this form.
The Helmholtz free energy is crucial for calculating all residual thermodynamic properties through its derivatives concerning temperature and density To determine caloric properties, it is necessary to express the Helmholtz free energy of an ideal gas as a function of temperature Previous formulations have primarily focused on isobaric heat capacity, necessitating a reformulation to explicitly represent the Helmholtz free energy Additionally, derivatives of the Helmholtz free energy with respect to state conditions are essential for accurate calculations.
The formulations selected for the ideal and residual components of the Helmholtz free energy allow for the necessary derivatives to be expressed analytically This eliminates the need for numerical differentiation or integration in computer programs, thereby avoiding numerical issues and reducing calculation times.
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
The Helmholtz free energy, denoted as \( f \), for a homogeneous gas mixture under constant pressure and temperature, can be represented as the sum of two components: \( f^o \), which accounts for the ideal gas behavior, and \( f^r \), which reflects the residual or real-gas behavior, as illustrated in Equation (1).
( , , ) o ( , , ) r ( , , ) f ρ Τ X = f ρ Τ X + f ρ Τ X (1) which, rewritten in the form of dimensionless reduced free energy ϕ = f/(R⋅T), becomes Equation (2):
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
Equations (1) and (2) are expressed in terms of molar density, \$\rho\$, and reduced density, \$\delta\$, rather than the more commonly used pressure, \$p\$ This approach stems from statistical thermodynamics, where the Helmholtz free energy naturally arises from the types and number of molecular interactions in a mixture, making it a function of molar density and the mole fractions of the constituent molecules.
The reduced density, δ, is related to the molar density, ρ, as shown in Equation (4):
K 3 δ = ⋅ρ (4) where K is a mixture size parameter
The ideal component, \$\phi_o\$, of the reduced Helmholtz free energy is derived from the equations related to isobaric heat capacity in the ideal gas state In contrast, the residual component, \$\phi_{ris}\$, is obtained using the AGA8 equation of state.
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, h o , and entropy, s o , as given in Equation (5):
The enthalpy (\$h_o\$) and entropy (\$s_o\$) of an ideal gas can be expressed using the isobaric heat capacity (\$c_{o,p}\$) This relationship is defined by Equations (6) and (7), with the integration limits set between temperatures \$T_\theta\$ and \$T\$.
The reference state for zero enthalpy and zero entropy is established at a temperature of 298.15 K and a pressure of 0.101325 MPa for an ideal unmixed gas Integration constants, \( h_{o,\theta} \) and \( s_{o,\theta} \), are calculated to align with this definition The ideal reference density, \( \rho_{\theta} \), is expressed as \( \rho_{\theta} = \frac{p_{\theta}}{R \cdot T_{\theta}} \).
The reduced Helmholtz free energy ϕ o = f o /(R⋅T) can then be written, using Equations (6) and (7), as a function of δ, τ and X, as given in Equation (8):
See Annex B for details of this formulation
4.2.4 The residual part of the Helmholtz free energy
The residual component of the reduced Helmholtz free energy is derived using the AGA8 equation, as specified in ISO 20765 This equation expresses the compression factor in relation to reduced density, inverse reduced temperature, and composition, and is represented in the form of Equation (9).
B is the second virial coefficient; b n , c n , k n , u n are coefficients of the equation and functions of composition;
The compression factor, Z, is related to the residual part of reduced free energy, ϕ r , as given in Equation (10):
Z = + ⋅δ ϕ δ (10) where ϕ r, δ is the partial derivative of ϕ r with respect to reduced density at constant τ 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:
See Annexes C and D for details of this formulation
4.2.5 The reduced Helmholtz free energy
The fundamental equation for the reduced Helmholtz free energy, ϕ, enables the analytical calculation of all thermodynamic properties by utilizing the ideal part, ϕ o, and the residual part, ϕ r Consequently, the reduced Helmholtz free energy can be expressed as outlined in Equation (12).
Thermodynamic properties derived from the Helmholtz free energy
All thermodynamic properties can be expressed explicitly using the reduced Helmholtz free energy, denoted as ϕ, along with its various derivatives The necessary derivatives include ϕ τ, ϕ ττ, ϕ δ, ϕ δδ, and ϕ τδ, which are defined accordingly.
Each derivative consists of an ideal component and a residual component The substitutions outlined in Equations (14) and (15) facilitate the simplification of the associated relationships.
Detailed expressions for ϕ τ , ϕ ττ , ϕ δ , ϕ 1 and ϕ 2 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, M i , of the pure substances as given in Equation (16):
Values for molar masses, M i , of pure substances are given in References [1] and [2]; these values are identical with those given in ISO 6976:1995 [5]
The molar masses specified in ISO 6976 may not align with the latest values recognized by the international metrology community However, these values were widely used during the formulation of the AGA8 equation and are preserved in this context, with discrepancies being minimal.
Equations (20), (21), and (23) to (26) present fundamental expressions for the thermodynamic properties h, s, c_p, à, κ, and w, which have been modified to utilize previously derived values for simplifying further calculations This method is particularly beneficial in scenarios where multiple or all thermodynamic properties need to be evaluated To enhance understanding, the primary thermodynamic relationships are outlined at the beginning of each subclause, followed by the relevant transformations.
The expression for the compression factor, Z, is given by Equation (17):
Z = ⋅δ ϕ δ (17) where ϕ δ 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
The compression factor values, denoted as Z, derived from this section of ISO 20765 should typically match those obtained from ISO 12213-2 However, if a priority requirement arises, ISO 12213-2 will take precedence.
The expression for the internal energy, u, is given by Equation (19): u U M
The expression for the enthalpy, h, is given by Equation (20): h H M
The expression for the entropy, s, is given by Equation (21): s S M
The expression for the isochoric heat capacity, c v , is given by Equation (22): v v 2 c C M
The expression for the isobaric heat capacity, c p , is given by Equation (23): p p
The expression for the Joule-Thomson coefficient, à, is given by Equation (24):
The expression for the isentropic exponent, κ, is given by Equation (25):
The expression for the speed of sound, w, is given by Equation (26):
Input variables
The method outlined in ISO 20765 typically utilizes reduced density, inverse reduced temperature, and molar composition as input variables However, the most commonly available input variables are absolute pressure, absolute temperature, and molar composition Therefore, it is often necessary to first calculate the inverse reduced temperature and reduced density from the available data The conversion from absolute temperature to inverse reduced temperature is specified in Equation (3).
The conversion from pressure to reduced density can be carried out as described in 5.2
When the mass density, D, is available as input instead of pressure, p, the value of δ can be directly calculated using the formula δ = D⋅K^3/M, where M represents the molar mass as defined in Equation (16).
The required composition includes mole fractions of 21 components: nitrogen, carbon dioxide, methane, ethane, propane, n-butane, iso-butane, n-pentane, iso-pentane, n-hexane, n-heptane, n-octane, n-nonane, n-decane, hydrogen, oxygen, carbon monoxide, water, hydrogen sulfide, helium, and argon For the applicable mole fraction ranges, refer to section 6.2 Any trace components not listed among these 21 can be appropriately assigned to one of them, as detailed in Annex E.
The total of all mole fractions must equal one; if it does not, the composition is either incorrect or incomplete Users should refrain from proceeding until they have identified and resolved the underlying issue.
If the mole fractions of heptanes, octanes, nonanes and decanes are unknown, then the use of a composite
C 6+ 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
NOTE If the composition is known by volume fractions, these will need to be converted to mole fractions using the method given in ISO 6976 [5].
Conversion from pressure to reduced density
Combination of Equations (4), (9) and (18) results in Equation (27):
To determine the reduced molar density, δ, the input variables such as pressure, inverse reduced temperature, and composition can be utilized to solve Equation (27) The variable quantities B(τ,X), C n (X), K(X), along with the coefficients b n, c n, k n, and u n in Equation (27), can be derived from the equations and tables provided in Annex D.
(Equations (D.1), (D.6) and (D.11), and Table D.1, respectively) for these quantities Numerical values for all pure-component and binary interaction parameters that are also required for the evaluation of Equations (D.1),
(D.6) and (D.11) are given in Tables D.2 and D.3, respectively
A standard equation-of-state density-search algorithm is often the most effective method for obtaining solutions These algorithms typically start with an initial density estimate, commonly using the ideal-gas approximation, and then iteratively calculate pressure (p) and density (δ) The goal is to determine the density value that matches the known pressure to a specified accuracy In this context, a suitable criterion is that the pressure derived from the calculated reduced molar density, δ, should align with the input pressure to within 1 part in 10^6.
Implementation
The revised set of input variables, including reduced density (\$δ\$), inverse reduced temperature (\$τ\$), and composition (\$X\$), is now available for calculations This allows for the determination of the reduced Helmholtz free energy and other thermodynamic properties using the fundamental equation Specifically, Equation (12) expresses the reduced Helmholtz free energy as \$ϕ = ϕ_o + ϕ_r\$, while Equation (11) outlines the formulation of the residual part.
Helmholtz free energy ϕ r as a function of reduced density, δ, inverse reduced temperature, τ, and the molar composition, X The ideal part, ϕ o , formulated in Equation (8), may be developed as given by Equation (B.3) of
Annex B so as to express ϕ as given in Equation (28):
( ) ( ) ln ln sinh( ) ln cosh( )
ln sinh( ) ln cosh( ) ln ln ln
Values for all of the coefficients (A o,1 ) i , (A o,2 ) i and B o,i to J o,i for the ideal gas are given in Annex B for all of the 21 possible component gases
Derivatives of ϕ with respect to reduced density and inverse reduced temperature are essential for evaluating thermodynamic properties, as outlined in Equations (C.2) to (C.6) in Annex C The evaluation of these thermodynamic properties can be performed using Equations (17) to (26) Additionally, Annex D provides the values for the coefficients \(b_n\), \(c_n\), \(k_n\), and \(u_n\), along with the quantities \(C_n\), which depend on composition.
A more detailed description of the implementation procedure is given in Annex F
Pressure and temperature
The method outlined in ISO 20765 is specifically applicable to pipeline quality gases, as detailed in section 6.2, and is relevant for the typical pressure and temperature ranges encountered during transmission and distribution operations.
Table 1 outlines the applicable ranges of pressure and temperature for the method, which is specifically designed for gaseous mixtures It is important to note that this method is not valid for conditions where the calculated compression factor falls below 0.5.
Table 1 — Ranges of application for pipeline quality gas
Pipeline quality gas
Pipeline quality gas shall be taken as a natural (or similar) gas with mole fractions of the various major and minor components that fall within the ranges given in Table 2
Possible trace components of natural gases, and details of how to deal with these, are discussed in Annex E
The total combined content of all trace components shall not exceed 0,000 5 mole fraction
Table 2 — Ranges of mole fractions for major and minor components of natural gas
Number i Component Range mole fraction
Uncertainty for pipeline quality gas
Figures 1 to 3 illustrate uncertainty diagrams for pure methane, focusing on the compression factor, speed of sound, and enthalpy for mixtures with a mole fraction of methane near unity The uncertainty is expressed as a 95% confidence limit, representing the maximum value within each region, which includes the uncertainty from well-documented reference data and the discrepancy between this data and the property values calculated using the method outlined in ISO 20765 The reference data utilized are derived from equations presented in Reference [6].
Y pressure, expressed in megapascals a Region of uncertainty of ± 0,08 % b Region of uncertainty of ± 0,04 %
Figure 1 — Uncertainty diagram for Z , the compression factor of methane
Y pressure, expressed in megapascals a Region of uncertainty of ± 0,20 % b Region of uncertainty of ± 0,05 %
Figure 2 — Uncertainty diagram for w , the speed of sound of methane
Y pressure, expressed in megapascals a Region of uncertainty of ± 3 kJ/kg b Region of uncertainty of ± 2 kJ/kg c Region of uncertainty of ± 1 kJ/kg
Figure 3 — Uncertainty diagram for H , the enthalpy of methane
7.1.2 Uncertainty diagrams for natural gas
Figures 4, 5, and 6 illustrate the uncertainty diagrams for the compression factor, speed of sound, and enthalpy of natural gases, providing a comprehensive guide to the expected uncertainties in these properties.
The 95% confidence limit for each region indicates the maximum discrepancy between the directly measured properties of various natural gases and the values calculated using the method outlined in ISO 20765 Directly measured values for the compression factor are sourced from Reference [7], while speed of sound values are obtained from Reference [8], and enthalpy values are referenced accordingly.
NOTE 1 For all gases, the uncertainty diagram for density is identical in form to that for compression factor
NOTE 2 For all gases, the uncertainty in the isentropic exponent is approximately twice the uncertainty in speed of sound
Y pressure, expressed in megapascals a Region of uncertainty of ± 0,4 % b Region of uncertainty of ± 0,2 % c Region of uncertainty of ± 0,1 %
Figure 4 — Uncertainty diagram for Z , the compression factor of natural gas
Y pressure, expressed in megapascals a Region of uncertainty of ± 2,0 % b Region of uncertainty of ± 0,8 % c Region of uncertainty of ± 0,2 %
Figure 5 — Uncertainty diagram for w , the speed of sound of natural gas
Y pressure, expressed in megapascals a Region of uncertainty of ± 3 kJ/kg b Region of uncertainty of ± 2 kJ/kg
Figure 6 — Uncertainty diagram for H , the enthalpy of natural gas
Due to the lack of high-quality experimental data for properties such as density, speed of sound, and enthalpy—similar to the situation with the compression factor—definitive numerical estimates of uncertainty cannot be provided However, some guidelines can still be suggested.
For gases at low pressures (below 1 MPa) exhibiting nearly ideal behavior (compression factor greater than 0.95), the caloric properties can be predicted with low uncertainty This is due to the fact that most of these properties are derived from the ideal part of the Helmholtz free energy, which is based on high-accuracy data for the ideal isobaric heat capacity Consequently, properties such as density, compression factor, speed of sound, isochoric and isobaric heat capacities, and isentropic exponent can be determined with high accuracy.
Joule-Thomson coefficient are probably all predicted within 0,1 %.
Impact of uncertainties of input variables
Users must understand that uncertainties in input variables, such as pressure, temperature, and mole fraction composition, can significantly impact the uncertainty of calculated results In applications where this additional uncertainty is critical, it is essential to conduct sensitivity tests to assess its magnitude.
Thermodynamic properties must be reported using the units specified in Annex A, with the number of decimal places as outlined in Table 3 The report should clearly state the temperature, pressure (or density), and detailed composition relevant to the results Additionally, the calculation method must be referenced, such as ISO 20765-1.
For validation of computational procedures, it can be useful to carry extra digits (see example calculations in
Table 3 — Reporting of results Symbol Property Units Decimal places
Z compression factor — 4 ρ molar density kmol/m 3 3
D density kg/m 3 2 u molar internal energy kJ/kmol 0
U specific internal energy kJ/kg 1 h molar enthalpy kJ/kmol 0
H specific enthalpy kJ/kg 1 s molar entropy kJ/(kmolãK) 2
S specific entropy kJ/(kgãK) 3 c v molar isochoric heat capacity kJ/(kmolãK) 2
C v specific isochoric heat capacity kJ/(kgãK) 3 c p molar isobaric heat capacity kJ/(kmolãK) 2
C p specific isobaric heat capacity kJ/(kgãK) 3 à Joule-Thomson coefficient K/MPa 2 κ isentropic exponent — 2 w speed of sound m/s 1
Symbol Meaning Source of values Units a n constants in Equations (D.2) and (D.6) Table D.1 -
(A o,1 ) i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
(A o,2 ) i coefficient in the ideal gas equation [(Equation (B.3)] Table B.1 - b n constants in the real gas equation [(Equation (9)] Table D.1 -
B second virial coefficient Equation (D.1) m 3 /kmol
B o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 - n *
B quantities in Equation (D.1) Equation (D.2) - nij *
B binary interaction parameter in Equation (D.2) Equation (D.3) - c n constants in the real gas equation [Equation (9)] Table D.1 - c p molar isobaric heat capacity Equation (23) kJ/(kmolãK) c v molar isochoric heat capacity Equation (22) kJ/(kmolãK)
C o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
C n coefficients in the real gas equation [Equation (9)] Equation (D.6) -
C p specific isobaric heat capacity Equation (23) kJ/(kgãK)
C v specific isochoric heat capacity Equation (22) kJ/(kgãK)
D specific (mass) density Equation (18) kg/m 3
D o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
E i energy parameter in Equations (D.4) and (D.7) Table D.2 -
E o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
E ij binary interaction energy parameter in Equation (D.2) Equation (D.4) - nij *
E binary interaction energy parameter in Equation (D.4) Table D.3 - f molar Helmholtz free energy Equation (1) kJ/kmol f n constants in Equations (D.3) and (D.6) Table D.1 -
F i high temperature parameter in Equations (D.3) and (D.10) Table D.2 -
F o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 - g n constants in Equations (D.3) and (D.6) Table D.1 -
G i orientation parameter in Equations (D.5) and (D.8) Table D.2 -
G o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
G ij binary interaction orientation parameter in Equation (D.3) Equation (D.5) - nij *
G binary interaction orientation parameter in
Table D.3 - h molar enthalpy Equation (20) kJ/kmol
H specific enthalpy Equation (20) kJ/kg
H o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
I o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 -
J o,i coefficient in the ideal gas equation [Equation (B.3)] Table B.1 - k n constants in the AGA8 equation [Equation (9)] Table D.1 -
K mixture size parameter in Equation (9) Equation (D.11) (m 3 /kmol) 1/3
K i component size parameter in Equations (D.2) and (D.11) Table D.2 (m 3 /kmol) 1/3
K ij binary interaction size parameter in Equation (D.11) Table D.3 -
M molar mass of a mixture Equation (16) kg/kmol
M i molar mass of component i Table D.2 kg/kmol
N number of components in gas mixture input - p pressure input MPa q n constants in Equations (D.3) and (D.6) Table D.1 -
Q i quadrupole parameter in Equations (D.3) and (D.9) Table D.2 -
R molar gas constant R = 8,314 510 kJ/(kmolãK)
The molar gas constant value provided is not the latest accepted by the international metrology community, but it reflects the value commonly used during the development of the AGA8 equation The discrepancy between this value and the current accepted value is less than five parts per million Additionally, the molar entropy is expressed in kJ/(kmol·K), and the constants in Equation (D.3) can be found in Table D.1.
S specific entropy Equation (21) kJ/(kgãK)
S i dipole parameter in Equation (D.3) Table D.2 -
T temperature input K u molar internal energy Equation (19) kJ/kmol u n constants in the AGA8 equation [Equation (9)] Table D.1 -
U specific internal energy Equation (19) kJ/kg
V ij binary interaction parameter in Equation (D.7) Table D.3 - w speed of sound Equation (26) m/s w n constants in Equation (D.3) Table D.1 -
W i association parameter in Equation (D.3) Table D.2 - x mole fraction input -
X mixture mole fraction vector (x 1 , x 2 , x 3 , x i , x 21 ) input -
Z compression factor Equation (17) - δ reduced density Equation (4) - ϕ reduced Helmholtz free energy Equation (2) - κ isentropic exponent Equation (25) - à Joule-Thomson coefficient Equation (24) K/MPa ρ molar density Equation (18) kmol/m 3 τ inverse reduced temperature Equation (3) -
In the context of a binary interaction, subscripts are used to denote components, with \(i\) representing the first component (where \(i = 1\) to \(N\)) and \(j\) for the second component (where \(j = 2\) to \(N\)) The equation of state coefficient is indexed by \(n\) (ranging from \(1\) to \(58\)), while \(o\) refers to the ideal-gas state and \(r\) indicates the residual part The symbol \(\delta\) denotes the partial derivative with respect to reduced molar density, and \(\theta\) represents the reference state, defined at \(T_\theta = 298.15 \, \text{K}\) and \(p_\theta = 0.101325 \, \text{MPa}\) Additionally, \(\tau\) signifies the partial derivative with respect to inverse reduced temperature.
The Helmholtz free energy of the ideal gas
B.1 Calculation of the Helmholtz free energy of the ideal gas a) The ideal isobaric heat capacity of a single component [3, 4] may be written as in Equation (B.1)
Equation (B.1) may be generalized to the case of an N-component mixture by use of Equation (B.2): o,p o,p
R =∑ = x R (B.2) b) This formulation for the ideal isobaric heat capacity, c p,o , may be inserted into Equation (8) for the reduced Helmholtz free energy, ϕ o The integrations in Equation (8) may then be performed, yielding
( ) ( ) ln ln sinh( ) ln cosh( )
ln sinh( ) ln cosh( ) ln ln ln
In Equation (B.3), the constants (A o,1 ) i and (A o,2 ) i are related to the integration constants (s o,θ ) i and
(h o,θ) i of Equation (8) in accordance with Equations (B.4) and (B.5): o,1 o,θ o,
In Equation (B.3), it is crucial to note that ϕ o depends on the reduced molar density, δ, of the real gas, rather than that of the ideal gas Consequently, a complete evaluation of this equation requires an available value for δ, as detailed in Annex D This equation will be utilized for calculating the reduced properties.
The Helmholtz free energy of an ideal gas is analyzed, with recent studies utilizing the best available values for the ideal isobaric heat capacity, \( c_{o,p} \) These values have been re-correlated to yield numerical constants \( A_o \) to \( J_o \) for 21 pure component gases, as outlined in ISO 20765.
Values of the constants (A o,1 ) i and (A o,2 ) i and constants B o,i to J o, i for use in Equation (B.3) are given in
B.2 Derivatives of the Helmholtz free energy of the ideal gas
To calculate certain thermodynamic properties, it is essential to determine the first and second partial derivatives of the reduced Helmholtz free energy, \( \phi^o \), from the ideal gas equation The necessary mathematical expressions for these calculations are provided in Equations (B.6) and (B.7).
( ) sinh( ) cosh( ) cosh( ) sinh( ) sinh( ) cosh( ) x i
Table B.1 presents the parameter values for the Helmholtz free energy of various ideal gas components, including nitrogen, carbon dioxide, methane, ethane, propane, n-butane, iso-butane, n-pentane, iso-pentane, n-hexane, n-heptane, n-octane, n-nonane, n-decane, hydrogen, oxygen, carbon monoxide, water, hydrogen sulfide, helium, and argon Each component is characterized by specific coefficients (A, B, C, D, E, F, G, H, I, J) that define its thermodynamic properties, essential for understanding gas behavior in various applications.
The equation for the Helmholtz free energy
C.1 Calculation of the Helmholtz free energy
The reduced Helmholtz free energy for the specified gas mixture shall be calculated from Equation (C.1):
= + ⋅ − ∑ ⋅ + ∑ ⋅ ⋅ − ⋅ (C.1) where ϕ o shall be calculated in accordance with the procedure described in Annex B That part of
Equation (C.1) represents the residual component of the Helmholtz free energy, excluding the term ϕ o, as derived from Equation (11) The calculations for the variable quantities B(τ, X), C n (X), and K(X) in Equation (C.1) are detailed in D.1, while the values of the constants used in Equation (C.1) can be found in D.2.
C.2 Derivatives of the Helmholtz free energy
Calculating thermodynamic properties involves the use of first and second partial derivatives related to the inverse reduced temperature and reduced density of the Helmholtz free energy The necessary mathematical expressions are outlined in Equations (C.2) to (C.6).
Equations and constants required for the evaluation of Equations (C.2) to (C.6) are given in Annex D
Detailed documentation for the equation of state
To evaluate Equation (C.1), it is essential to assign values to the second virial coefficient, B(τ, X), the quantities C n (X), and the size parameter K(X) Additionally, since Equation (C.1) explicitly involves the reduced molar density, δ, a method to relate this quantity to pressure is required These aspects are addressed in sections D.1 a) to D.1 d) The second virial coefficient will be calculated using Equations (D.1) to (D.5) along with relevant constants from Tables D.1 to D.3.
G =G ⋅ G +G (D.5) b) The quantities C n (for n = 13 to 58) shall be calculated using Equations (D.6) to (D.10), together with the values of the relevant constants from Tables D.1 to D.3:
In Table D.2, the interaction parameter \( F_i \) is zero for all components except hydrogen, where \( F_{15} = 1.0 \), while \( W_i \) is zero for all components except water, with \( W_{18} = 1.0 \) Additionally, many interaction parameters listed in Table D.3 are equal to unity The mixture size parameter \( K(X) \) can be calculated using Equation (D.11) along with the relevant constants from Tables D.2 and D.3.
Many values of K ij have a value of unity d) The reduced molar density, δ (p, τ, X), shall be determined as the solution of Equation (D.12)
(Equation (27) of the main text rewritten so as to be explicit in pressure)
Equation (D.12) may be solved using a standard equation-of-state density-search algorithm
D.2 Values of constants for the equation of state
This section lists values for all the constants needed to implement the AGA8 equation of state through
Equations (C.1) and (D.1) to (D.12) are supported by Table D.1, which lists constants relevant to the overall structure of the equation, while Table D.2 provides constants associated with the properties of specific components.
Table D.3 gives values for those constants that relate to the properties of pair-wise (binary) unlike molecule interactions between components
Table D.1 — Constants of the equation of state n a n b n c n k n u n g n q n f n s n w n
Table D.2 — Pure component characterization parameters i Component
Table D.3 — Binary interaction parameters i j Component pair *
15 17 hydrogen carbon monoxide 1,100 000 1,0 1,0 1,0 The interaction parameters for any pair of components not listed in Table D.3 shall all have the value 1,0
To calculate the thermodynamic properties of natural gas or similar mixtures containing trace components not listed in Table 2, it is essential to assign each trace component to one of the 21 major and minor components defined in the AGA8 equation Table E.1 provides recommendations for these assignments.
Recommendations are based on an assessment of assignments that optimize the accuracy of thermodynamic properties, considering factors such as molar mass, energy parameters related to critical temperature, and size parameters linked to critical volume Since no single assignment can satisfy all properties equally, users may prefer alternative assignments for specific applications requiring only one property Therefore, these recommendations are not normative, and implementations involving trace components should be thoroughly documented.
NOTE The set of components completed by the addition of those in Table E.1 to those in Table 2 is the same as the set of components included in ISO 6976:1995 [5]
Table E.1 — Assignment of trace components Trace component Formula Recommended assignment i
2,3-dimethylbutane C 6 H 14 n-hexane 10 ethylene (ethene) C 2 H 4 ethane 4 propylene (propene) C 3 H 6 propane 5
1-butene C 4 H 8 n-butane 6 cis-2-butene C 4 H 8 n-butane 6 trans-2-butene C 4 H 8 n-butane 6