Extended corresponding states equation of state for natural gas systems

Fluid Phase Equilibria 183–184 (2001) 21–29

Extended corresponding states equation of state for natural gas systems夽 J.F. Estela-Uribe a,∗ , J.P.M. Trusler b a

b

Facultad de Ingenier´ıa, Universidad Javeriana — Cali, Calle 18 118–250, Cali, Colombia Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK

Abstract We present an extended corresponded states model for the prediction of thermodynamic properties of natural gas systems. This model is based on shape factors expressed as generalised functions of the reduced temperature and density, the acentric factor and the critical compression factor. A single set of coefficients was optimised for 12 major components of natural gases. The extension to multicomponent systems was carried out according to the one-fluid model with temperature- and density-dependent binary interaction parameters. Compression factors and speeds of sound of natural gases were predicted with average deviations within ±0.036%. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Corresponding states; Density; Compressibility factor; Speed of sound; Natural gas

1. Introduction Accurate prediction of the thermodynamic properties of natural gases is required for flow metering purposes under custody transfer and pipeline transmission conditions. Current standards call for the gas density ρ and speed of sound u to be predicted with uncertainty within ±0.1% in the interval 273 K ≤ T ≤ 333 K and p ≤ 12 MPa. The thermodynamic modelling of natural gas systems represents a demanding compromise between accuracy and flexibility. The desired accuracy is currently achieved by pure-component equations of state (EoS) based on the Helmholtz free energy such as the recent reference equation for methane [1], but these models are not suitable for multicomponent applications. Extended corresponding states (ECS) models offer a viable option for they may be both highly accurate for pure fluids and applicable to mixtures. The accuracy of any ECS model depends upon proper enforcement of conformality between 夽

Paper presented at the Fourteenth Symposium on Thermophysical Properties, 25–30 June 2000, Boulder, CO, USA. Corresponding author. Tel.: +57-2-3218-390; fax: +57-2-5528-23. E-mail address: [email protected] (J.F. Estela-Uribe). ∗

0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 4 1 7 - 4

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the mixture constituents and the reference fluid, application of a suitable mixture model and use of an accurate reference-fluid EoS. Several accurate ECS models have been reported in the last two decades including McCarty’s model for liquefied natural gas [2], the model of Huber and Ely for refrigerants [3] and that of Clarke et al. for the ternary system (N2 + Ar + O2 ) [4]. The objective of this work was to develop an ECS model for natural gas under custody transfer conditions. This model should meet the required accuracy in both ρ and u in contrast to currently available models that cannot always achieve high accuracy in both thermal and caloric properties. For instance, the MGERG-88 model [5] is not suitable for caloric-property calculations and the AGA8-DC92 model [6] exhibits deviations in u as large as 0.82% at high pressures and low temperatures when compared with experimental data [7,8]. 2. Theory 2.1. Extended corresponding states relations The corresponding states condition between two fluids is stated as follows: 
T Z(T , ρ) = Z0 , hρ f
 res res T Φ (T , ρ) = Φ0 , hρ f

(1) (2)

Here, Z is the compression factor, Φ res = Ares /nRT the dimensionless residual Helmholtz free energy, and the subscript ‘0’ refers to reference-fluid properties. The equivalent substance reducing ratios f and h map the properties of the fluid of interest onto those of the reference fluid. The simple corresponding state principle (CSP) asserts that the configurational properties of two fluids must be equal at the same reduced conditions; thus, f and h are the simple ratios f = T c /T0c and h = ρ0c /ρ c . This is valid only for small groups of very similar molecules such as the light noble gases. To extend the simple CSP to more complex systems, temperature- and density-dependent shape factors θ and ϕ are introduced such that,
c T fii = θii (Tr , ρr ) (3) T0c
c ρ0 hii = ϕii (Tr , ρr ) (4) ρc Here, double subscripts are introduced to allow for extension to mixture nomenclature. It is possible to choose values of the shape factors such that Eqs. (1) and (2) are obeyed exactly but, in practice, θ and ϕ are usually approximated by empirical correlations in terms of reduced temperature and density. 2.2. Application to mixtures To apply the ECS formalism to mixtures, we adopt the van der Waals one-fluid model [9] whereby the configurational properties of the mixture are equated with those of a single hypothetical fluid. Thus, the

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configurational properties of the mixture can be calculated from those of the reference fluid as follows:
 T (5) Zx (T , ρ) = Z0 , hxρ fx
 T res res Φx (T , ρ) = Φ0 , hx ρ (6) fx Here, the subscript ‘x’ refers to the mixture properties and fx and hx are the scaling parameters of the mixture which we obtain from the van der Waals mixing rules  hx = (7) xi xj hij i

j

 xi xj fij hij fx hx = i

(8)

j

The unlike terms which appear here are calculated from the conventional Lorentz–Berthelot combining rules with binary interaction parameters ξ ij and ηij fij = ξij (fii fjj )1/2 1/3

(9) 1/3

hij = ηij 81 (hii + hjj )3

(10)

As each of the mixture components is separately in corresponding states with the reference fluid, the arguments of Eqs. (3) and (4) for fii and hii are Tr = (T θii )/(fx T0c ) and ρr = (hx ρ)/(ϕii ρ0c ). 3. Extended corresponding states model The first stage in the development of our ECS model was to obtain the behaviour of θ and ϕ for the main components of natural gas from experimental volumetric data on the pure substances. Since the method used to obtain θ and ϕ from experimental data is described in detail by Estela-Uribe [10], we give only a brief account here. The reference fluid was methane for which we used the reference EoS of Setzmann and Wagner [1]. Experimental compression factors along isotherms were used to calculate Φ res . To solve for θ and ϕ, Eqs. (1) and (2) were expressed more conveniently as Z(τ, δ) = Z0 (θτ, ϕδ)

(11)

Φ res (τ, δ) = Φ0res (θτ, ϕδ)

(12)

Here, τ = T c /T is the inverse reduced temperature and δ = ρ/ρ c is the reduced density. Shape factors were obtained for ethane, propane, the two butanes, nitrogen, carbon dioxide and normal hydrogen. The data sources are listed in Table 1. We next correlated θ and ϕ independently as functions of τ and δ and obtained generalised expressions for the parameters in terms of the acentric factor ω and the critical compression factor Zc . The shape factor correlations are θ(τ, δ) = 1 + (ω − ω0 )(A1 (τ ) + A2 (τ ) exp(−δ 2 ) + Ψθ (τ, δ))
c Z0 {1 + (ω − ω0 )(A3 (τ ) + A4 (τ ) exp(−δ 2 ) + Ψϕ (τ, δ))} ϕ(τ, δ) = Zc

(13) (14)

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Table 1 Coefficients of shape-factor correlations Generalised coefficients

a1,1 a1,2 a2,1 a2,2 a3,1 a3,2 a4,1 a4,2 b1 b2 c1 c2

0.058990275 0.930466257 −0.088091565 0.181659444 0.004360921 0.410925734 −0.644414001 0.213433722 0.013017218 0.285879454 −0.121476965 0.123927229

Substance-dependent coefficients Component

ω

Zc

CH4 C 2 H6 C 3 H8 i-C4 H10 n-C4 H10 i-C5 H12 n-C5 H12 n-C6 H14 N2 CO2 CO H2

0.011406 0.099349 0.15124 0.18675 0.20086 0.227 0.16082 0.299 0.039988 0.21651 0.062971 −0.21515

0.28629 0.27933 0.27872 0.27787 0.27413 0.27082 0.26970 0.26618 0.28584 0.27391 0.28651 0.31515

Here, ω0 = 0.011406 is the acentric factor of methane calculated with the ancillary vapour pressure equation reported by Setzmann and Wagner [1]. The temperature-dependent coefficients are given by the general expression Ai (τ ) = ai,1 − ai,2 lnτ

(15)

The functions Ψ θ and Ψ ϕ were included to improve the representation of the properties of pure substances in the critical region. They are given by Ψθ (τ, δ) = b1 δ exp(−b2 ∆2 )

(16)

Ψϕ (τ, δ) = c1 δ exp(−c2 ∆2 )

(17)

where ∆ is a distance function defined by ∆(τ, δ) = (δ − 1)2 + (τ −1 − 1)2

(18)

These correlations improved on those introduced by Leach et al. [11]. The functions of Eqs. (16) and (17) were introduced to enhance the representation of the shape factors in the critical region and the corresponding effect was detailed by Estela-Uribe [10]. Exact thermodynamic consistency is not observed by this method because each shape factor is correlated separately [12]. A method to enforce consistency was detailed by Estela-Uribe and Trusler [13] and applied in the development of an alternative ECS model for natural gas mixtures [10]. However, the extent of the inconsistencies in the present model are quite small and we avoid them entirely by calculating all properties from Φ res (τ ,δ) and its partial derivatives. The values of the coefficients in the model were obtained in a simultaneous fit to about 1500 data points comprising compression factors, vapour pressures and saturated vapour and liquid densities of ethane [14], propane [15], i-butane [16] and n-butane [17], and also speeds of sound of ethane [18]. Initially, literature values of ω and Zc were used but later ω and Zc were re-optimised as empirical constants on a substance-specific basis for the 12 most important components of interest. This improved the prediction

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of compression factors of components other than ethane and propane. Table 1 lists the coefficients. The shape-factor correlations are valid for the intervals δ ≤ 2.5 and 0.3 ≤ τ ≤ 1.9. Temperature- and density-dependent binary interaction parameters were correlated as ξij = kij,1 (1 + kij,2 T )(1 + kij,3 ρ)

(19)

ηij = kij,4 (1 + kij,5 T )(1 + kij,6 ρ)

(20)

The coefficients kij were fitted to about 1300 binary-mixture Z and u data points. The volumetric data were taken from the GERG databank [19] for the systems given in Table 2. The coefficients kij for the systems (CH4 + i-C4 H10 ), (C2 H6 + C3 H8 ), (C2 H6 + n-C4 H10 ), (N2 + i-C4 H10 ) and (CO2 + C3 H8 ), for which there are no binary-mixture data reported in reference [19], were fitted to 980 natural-gas Z and u data points [19,7,8]. The acoustic data of Trusler and co-workers were used for the systems (CH4 + C2 H6 ) [20,8] and (CH4 + C3 H8 ) [21], and data of Younglove et al. [7] were used for the systems (CH4 + C2 H6 ), (CH4 + C3 H8 ), (CH4 + N2 ), (CH4 + CO2 ) and (N2 + CO2 ). We also included new speeds of sound measured in the mixtures (0.45783CH4 + 0.55217N2 ), (0.80037CH4 + 0.19966N2 ) and (0.80253CH4 + 0.19747CO2 ) at 170 K ≤ T ≤ 450 K and p ≤ 20 MPa [10]. The kij values are given in Table 2. The perfect-gas contribution to the heat capacities was taken from the generalised correlation of Jaeschke and Schley [22].

Table 2 Coefficients of binary-interaction-parameter correlations System

kij ,1

kij ,2 (K−1 )

kij ,3 (dm3 mol−1 )

kij ,4

kij ,5 (K−1 )

CH4 + C2 H6 CH4 + C3 H8 CH4 + I-C4 H10 CH4 + n-C4 H10 CH4 + n-C5 H12 CH4 + n-C6 H14 CH4 + N2 CH4 + CO2 CH4 + CO CH4 + H2 N 2 + C 2 H6 N 2 + C 3 H8 N2 + i-C4 H10 N2 + n-C4 H10 N2 + CO2 N2 + CO N 2 + H2 CO2 + C2 H6 CO2 + C3 H8 C 2 H6 + C 3 H8 C2 H6 + n-C4 H10 C 2 H6 + H 2

1.121270 1.485892 0.870274 1.758056 1.243834 1.114716 1.077358 1.287222 1.855661 0.698570 1.436560 0.939234 1.434347 1.940711 2.321621 2.682645 −0.213644 0.709211 1.261747 1.013778 0.864051 1.690073

0.000149 −0.000556 1.354E-06 −0.000398 −0.000522 −0.001163 −0.000267 −0.000429 −0.000783 0.002443 −0.000580 −2.74E-05 1.738E-06 −0.001699 −0.001067 −0.002480 −0.023559 0.000550 −0.001058 0.000101 8.09E-07 −0.000385

−0.003394 −0.008885 8.72E-07 −0.016937 −0.005902 0.010242 −0.003015 −0.004999 −0.032548 −0.014070 −0.014331 0.002408 7.295E-07 0.000178 −0.014726 −0.029520 0.061909 0.002115 −0.047923 0.000889 9.925E-07 −0.004100

0.907079 0.657832 1.468744 0.597285 0.690195 1.875570 0.888969 0.681383 2.007959 −0.678092 0.752174 1.134551 0.428527 0.733865 0.414111 0.409948 −0.041627 1.593573 0.661367 0.650150 1.404569 0.577230

−4.65E-06 0.000335 7.77E-07 −0.000426 0.000419 0.000430 1.47E-05 0.000359 −0.002845 −0.005867 −0.000433 −0.000116 1.21E-06 −0.000368 0.000275 0.000382 −0.00647 −0.000668 −0.000660 0.000542 7.78E-07 0.001482

kij ,6 (dm3 mol−1 ) 0.002440 0.004868 8.224E-07 0.012941 0.019841 −0.018421 0.004559 0.001346 0.067864 0.030102 0.019222 −0.008203 1.329E-06 0.009798 0.011544 0.022396 15.371659 0.007145 0.169665 0.014338 9.658E-07 0.005820

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4. Results We summarise the representation of pure-component compression factors in Table 3. For the light hydrocarbons up to n-butane, the absolute average percentage deviations (AAD) are similar to, or better than, those obtained with MBWR equations of state [23]. As the aim of this work was to address the prediction of natural gas properties, we limit the presentation of results accordingly and defer a thorough presentation of results for binary mixtures to a later stage. Furthermore, the data used to fit the mixture model were largely restricted to the intervals 270 K ≤ T ≤ 330 K and p ≤ 12 MPa, thus, the model should be used with caution outside those intervals. The deviations in calculated Z of natural gases are given in Table 3 for 4473 data points taken from reference [19]. The composition groups are those defined by GERG [5]. The root-mean-square (RMS) error or our deviations is 0.046% whereas that reported by Jaeschke et al. [5] for the MGERG-88 model is 0.055%. Our results also compare favourably with the AAD of 0.035% and the RMS error of 0.050% reported by Starling and Savidge [6] for the AGA8-DC92 model. Fig. 1 shows deviations in calculated Table 3 Statistical analysis of absolute average percentage deviations in compression factors and speeds of sound of pure components and natural gases Component

Overall

Super-critical

AAD in compression factors of pure components C 2 H6 0.12 0.10 C 3 H8 0.06 0.04 i-C4 H10 0.29 0.21 n-C4 H10 0.33 0.29 n-C5 H12 1.16 1.04 N2 0.59 0.47 CO2 0.17 0.096 CO 0.27 0.19 H2 0.21 0.18 Gas

AAD

Bias

Near-critical 0.10 0.08 0.32 0.36 1.86 0.97 0.20 0.92 0.57 Maximum deviation

Deviations in compression factors of natural gases Group 1 0.040 −0.009 Group 2 0.041 −0.022 Group 3 0.034 0.015 Group 4 0.025 0.009 Group 5 0.031 0.010 Group 6 0.027 0.010 Overall 0.033 0.005

−0.16 −0.18 0.28 −0.13 0.17 −0.14 0.28

Deviations in speeds of sound of natural gases Gulf Coast 0.016 0.003 Amarillo 0.014 −0.0003 Statoil Dry 0.049 −0.046 Statvordgass 0.074 −0.008 Synthetic 0.043 0.032 Overall 0.036 −0.002

0.061 0.072 −0.13 0.37 0.15 0.37

Sub-critical

Data source

0.26 0.04 0.31 0.37

[14] [15] [16] [17] [24,25] [26–28] [29–31] [32] [33]

0.62 0.21 1.22

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Fig. 1. Relative deviations Z/Z in predicted compression factors Z of an Ekofisk natural gas (Group 3, N10) at 273.15 K with respect to experimental data [19]. (—) This work; (- - -) MGERG-88; (· · · ) AGA8-DC92.

Fig. 2. Relative deviations u/u in predicted speeds of sound u of a Gulf Coast natural gas at 275 K with respect to experimental data [7]. (—) This work; (· · · ) AGA8-DC92.

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Z for an Ekofisk gas [19] using our model and those obtained with the two models mentioned above; for the sake of simplicity the results are presented only for the lowest isotherm in the corresponding data set. Table 3 also gives the deviations in calculated u for 272 data points taken from four natural gases [7] and a synthetic gas [8] in the interval 275 K ≤ T ≤ 350 K and pressures up to 20 MPa for the gas of reference [8]. Our results compare quite favourably with those obtained with the AGA8-DC92 model for which we calculated an AAD of 0.056%. Fig. 2 contains deviations in calculated u for a Gulf Coast gas [7]. The overall AAD in calculated u for the three mixtures of reference [10] is 0.027% with a maximum deviation of 0.14% for the system (0.80253CH4 + 0.19747CO2 ) at 225 K and 3.35 MPa. For the same mixtures we obtained an AAD of 0.050% with the AGA8-DC92 model with a maximum deviation of 0.70% for the system (0.45783CH4 + 0.55217N2 ) at 170 K and 4.55474 MPa. 5. Conclusions The results we have obtained show that ECS models can achieve quite high accuracy in both density and speed of sound when applied to either pure substances or multicomponent natural-gas mixtures. Although the mixture model was fitted mainly to a rather limited range of temperatures and pressures, we have shown that the model behaviour remains acceptable under extrapolation. The extension of the model to broader ranges of temperature, pressure and to other data sets is the matter of continuing research. List of symbols a coefficient in temperature-dependent term of shape-factor correlations A Helmholtz free energy or temperature-dependent term in shape-factor correlations. b,c coefficients of critical-enhancement functions in shape-factor correlations f,h equivalent substance reducing ratios k coefficient in binary interaction parameter correlation n amount of substance p pressure R universal gas constant (R = 8.31451 J mol−1 K−1 ) T temperature V volume x mole fraction Z compression factor Greek letters ∆ distance function Φ dimensionless residual Helmholtz free energy, Ares /nRT δ reduced density (ρ/ρ c ) η,ξ binary interaction parameters θ,ϕ shape factors ρ amount-of-substance density (n/V) τ inverse reduced temperature, (Tc /T) ω acentric factor Ψ critical enhancement function

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Superscripts c critical property res residual property Subscripts 0 reference-fluid property i,j component indices x mixture property References [1] [2] [3] [4] [5]

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