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Geochimica et Cosmochimica Acta 219 (2017) 74–95

Prediction of the PVTx and VLE properties of natural gaseswith a general Helmholtz equation of state. Part I: Application

to the CH4–C2H6–C3H8–CO2–N2 system

Shide Mao a,⇑, Mengxin Lu a, Zeming Shi b

aState Key Laboratory of Geological Processes and Mineral Resources, and School of Earth Sciences and

Resources, China University of Geosciences, Beijing 100083, ChinabCollege of Earth Sciences, Chengdu University of Technology, Sichuan Province 610059, China

Received 14 November 2016; accepted in revised form 12 September 2017; Available online 19 September 2017

Abstract

A general equation of state (EOS) explicit in Helmholtz free energy has been developed to predict the pressure–volume-temperature-composition (PVTx) and vapor-liquid equilibrium (VLE) properties of the CH4–C2H6–C3H8–CO2–N2 fluid mix-tures (main components of natural gases). This EOS, which is a function of temperature, density and composition, with fourmixing parameters used, is based on the improved EOS of Sun and Ely (2004) for the pure components (CH4, C2H6, C3H8,CO2 and N2) and contains a simple generalized departure function presented by Lemmon and Jacobsen (1999). Comparisonwith the experimental data available indicates that the EOS can calculate the PVTx and VLE properties of the CH4–C2H6–C3H8–CO2–N2 fluid mixtures within or close to experimental uncertainties up to 623 K and 1000 bar within full range ofcomposition. Isochores of the CH4–C2H6–C3H8–CO2–N2 system can be directly calculated from this EOS to interpret the cor-responding microthermometric and Raman analysis data of fluid inclusions. The general EOS can calculate other thermody-namic properties if the ideal Helmholtz free energy of fluids is combined, and can also be extended to the multi-componentnatural gases including the secondary alkanes (carbon number above three) and none-alkane components such as H2S, SO2,O2, CO, Ar and H2O. This part of work will be finished in the near future.� 2017 Elsevier Ltd. All rights reserved.

Keywords: CH4–C2H6–C3H8–CO2–N2; Equation of state; PVTx; Phase equilibria; Fluid inclusion

1. INTRODUCTION

Fluid inclusions are the material trapped and sealed inthe growing minerals, and they have apparent phase bound-ary with the host minerals. In the studies of the oil-gas fluidinclusions, a frequently used method to estimate trappingtemperatures and pressures, is from the isochore intersec-tion of the H2O-salt inclusions and natural oil–gas inclu-sions entrapped at the same time (Mi et al., 2004; George

https://doi.org/10.1016/j.gca.2017.09.025

0016-7037/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (S. Mao).

et al., 2008; Liu et al., 2009). The H2O-salt inclusions areoften approximated by the NaCl–H2O system, whose iso-chores can be calculated from the pressure-temperature-volume-composition (PVTx) models or equations of state(EOS) of the NaCl–H2O system (Driesner, 2007; Maoand Duan, 2008; Sun and Dubessy, 2012; Mao et al.,2015b). However, the natural oil-gas inclusions are of com-plex mixtures, whose main components involve CH4, C2H6,C3H8, or CO2, and N2, and minor components include H2S,SO2, O2, CO, Ar, H2, H2O and some secondary alkanes(carbon number above three). To calculate the isochoresof the oil-gas inclusions, an accurate EOS is required to


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Table 2Critical parameters of pure fluids.

i T ci ðKÞ qci ðmol � dm�3ÞCH4 190.564 10.1393427C2H6 305.322 6.8708545C3H8 369.89 5.0000431CO2 304.1282 10.6249787N2 126.192 11.1839000

Table 3Exponents of Eq. (6).

m im jm km

1 1 1.500 02 1 0.250 03 1 1.250 04 3 0.250 05 7 0.875 06 2 1.375 07 1 0.000 18 1 2.375 19 2 2.000 110 5 2.125 111 1 3.500 212 1 6.500 213 4 4.750 214 2 12.50 3

S. Mao et al. / Geochimica et Cosmochimica Acta 219 (2017) 74–95 75

predict the PVTx and vapor-liquid equilibrium (VLE)properties of the oil-gas mixtures.

During the past several decades, a large number of EOSshave been developed to calculate thermodynamic propertiesof natural fluid mixtures, which can be roughly divided intofour kind of types (Table 1): cubic EOSs, virial EOSs, sta-tistical mechanical EOSs, and multi-parameter complexEOSs. The cubic EOSs have simple forms and can be usedto predict the VLE properties of the non-polar or the weak-polar fluids, but they are poor in yielding the liquid volume.The virial EOSs usually have good representation of vol-umes, but not of phase equilibria. The statistical mechanicalEOSs can calculate the VLE properties of the polar andnon-polar fluids, but they are poor in reproducing the den-sities of fluids at low temperatures. The multi-parametercomplex EOSs can reproduce the experimental phase equi-librium and volumetric data provided that precise experi-mental data are available and a good optimizationmethod is used. One of the popular multi-parameter EOSsis the Helmholtz free energy EOS, which can reproduce allthermodynamic data of common polar and non-polar fluidsup to high temperatures and pressures. For pure fluids,some Helmholtz free energy EOSs are recommended asthe standard EOSs by National Institute of Standardsand Technology (NIST), e.g., EOS of CH4 (Setzmann andWagner, 1991), C2H6 (Bucker and Wagner, 2006), CO2

(Span and Wagner, 1996), N2 (Span et al., 2000), H2O(Wagner and Pruß, 2002). Sun and Ely (2004) establisheda general Helmholtz free energy EOS (SE2004) for the purenon-polar and polar fluids, but large deviations were foundfor the saturated liquid density, vapor density and pressurein the near-triple-point region and/or the near-criticalregion. Alexandrov et al. (2013) extended the generalizedEOS of Sun and Ely (2004) to pure normal alkanes (C5-C50). Span and Wagner (2003a) proposed a simultaneouslyoptimized algorithm, then developed a general form of EOSexplicit in Helmholtz free energy for the pure non-polar flu-ids (Span and Wagner, 2003b) and another form for thepure polar fluids (Span and Wagner, 2003c). For fluid mix-tures, Lemmon and Jacobsen (1999) developed a general-ized EOS explicit in Helmholtz free energy to predict thethermodynamic properties of mixtures containing CH4,C2H6, C3H8, n-C4H10, i-C4H10, C2H4, N2, Ar, O2 andCO2 within the estimated accuracy of experimental data.In the model of Lemmon and Jacobsen, EOSs of pure fluidswere from those that NIST recommended before 1999.Kunz et al. (2007) presented a Helmholtz free energy EOS(GERG2004) to calculate the thermodynamic properties

Table 1Types of equations of state (EOS) and key references.

Types Key references

Cubic EOSs Redlich and Kwong (1949), Soave (1972),(2004), Lin et al. (2006)

Virial EOSs Duan et al. (1992), Pitzer et al. (1992), SoStatistical mechanicalEOSs

Zhang et al. (2000), Churakov and Gottsc

Multi-parameter complexEOSs

Setzmann and Wagner (1991), Span and W(2002), Span and Wagner (2003a,b,c), Sun

of natural gases of eighteen components, and Kunz andWagner (2012) extended the EOS to the fluid mixtures oftwenty-one natural gas components (GERG2008), wherethe forms of EOSs of pure fluids are not the same and con-tain large deviations for the VLE properties of the gas-water systems. Gernert and Span (2016) proposed animproved Helmholtz energy mixture model for the humidgases and Carbon Capture and Storage (CCS) mixturesincluding CO2, H2O, N2, O2, Ar and CO, where the usedEOSs of pure fluids are from those NIST recommended.

Therefore, it is possible to accurately predict the thermo-dynamic properties, especially the PVTx and VLE proper-ties, of both pure fluids and fluid mixtures with a generalHelmholtz free energy EOS, which will make computer cal-culation and application easier and more convenient thanthose EOSs using different forms for pure fluids and mix-tures. In this work, firstly the generalized EOS of Sun andEly (2004) was improved to calculate the PVTx and VLEproperties of pure fluids, such as CH4, C2H6, C3H8, CO2,

Peng and Robinson (1976), Patel and Teja (1982), Duan and Hu

ave (1999), Trusler (2000), Kedge and Trebble (2004)halk (2003a,b), Sun and Dubessy (2010, 2012), Sun et al. (2014)

agner (1996), Lemmon and Tillner-Roth (1999), Wagner and Prußand Ely (2004), Kunz et al. (2007), Kunz and Wagner (2012)

76 S. Mao et al. / Geochimica et Cosmochimica Acta 219 (2017) 74–95

N2, which belong to the main components of natural gases.Then the generalized mixture model of Lemmon andJacobsen (1999) was used to calculate the PVTx and VLEproperties of fluid mixtures of up to five components, wheremixing parameters were fitted to the experimental volumet-ric and phase equilibrium data of binary mixtures. Finally,the PVTx and VLE properties of the multi-component fluidmixtures were tested by the general EOS, from which theisochores of the CH4–C2H6–C3H8–CO2–N2 system can becalculated to interpret the microthermometric and Ramanspectrometric data on fluid inclusions.

Table 5The calculated deviations from the data of NIST for pure fluids.

AAD(q) % AAD(Ps) %

This work SE2004 This work SE2004

CH4 0.11 0.14 0.01 0.07C2H6 0.08 0.11 0.21 0.68C3H8 0.14 0.19 0.14 0.82CO2 0.08 0.19 0.01 0.04N2 0.20 0.39 0.02 0.02

Notes: SE2004 refers to the EOS of Sun and Ely (2004), AAD is the avrange of 91–620 K, 90–675 K, 86–650 K, 216–650 K and 63–650 K for CHbar, Ps is the saturated pressure, q0 is the saturated liquid density, and q

Table 4Coefficients of Eq. (6) for CH4, C2H6, C3H8, CO2 and N2.

am

m CH4 C2H6 C3H8

1 1.8322924E+00 1.4106992E+00 8.9344622E�012 8.7789882E�01 9.5273464E�01 9.5710115E�013 �3.4460887E+00 �3.2800111E+00 �2.9259359E+004 6.2657087E�02 7.4487238E�02 8.1174579E�025 2.5615427E�04 2.7682993E�04 2.9457602E�046 �6.5955074E�02 �7.0645927E�02 �6.3671091E�027 �5.9610303E�02 �7.7537532E�02 �6.1187921E�028 �3.1811362E�01 2.5014685E�02 2.5506442E�019 �8.9264371E�02 2.2167878E�01 4.7835582E�0110 �2.0997734E�02 �1.5589363E�02 �8.4243641E�0311 �5.9425962E�02 �2.2362512E�01 �3.2310881E�0112 4.3560418E�03 �3.1480583E�04 3.1307119E�0313 1.9129741E�03 �4.7949598E�02 �7.7576991E�0214 �1.7371647E�02 �1.7902771E�02 �2.5417583E�02m CO2 N2

1 5.8478135E�01 1.8744499E+002 9.5777220E�01 9.3065366E�013 �2.6415042E+00 �3.5352103E+004 7.4935594E�02 6.4576333E�025 2.1519215E�04 2.4160195E�046 �2.6666316E�02 �5.8692620E�027 1.3294773E�02 �6.0786818E�028 1.2546908E�01 �3.3794167E�019 4.2015287E�01 �1.3110278E�0110 �1.2243601E�02 �2.5010756E�0211 �2.5984348E�01 �2.1459359E�0212 �1.7963087E�02 1.4669049E�0313 �6.1607536E�02 1.8683393E�0214 �1.9553446E�02 �1.7252238E�02

2. EQUATION OF STATE OF PURE FLUIDS

The EOS of pure fluid (CH4, C2H6, C3H8, CO2 or N2) isin terms of dimensionless Helmholtz free energy ai definedas

ai ¼ Ai

RTð1Þ

where Ai is the molar Helmholtz free energy of purecomponent i, R is the molar gas constant

(8:314472 J � mol�1 � K�1), and T is the temperature in K.The dimensionless Helmholtz free energy ai is repre-

sented by

aiðd; sÞ ¼ a0i ðd; sÞ þ ar

i ðd; sÞ ð2Þwhere a0

i and ari are the ideal-gas part and the residual part

of dimensionless Helmholtz free energy of component i,respectively, and d and s are the reduced parameters definedas

d ¼ qqci

ð3Þ

s ¼ T ci

Tð4Þ

where q is the density, and qci and T ci are the critical densityand temperature of component i, respectively. The criticalparameters of the five components considered in this workare listed in Table 2.

The a0i in Eq. (2) can be calculated from the spectro-

scopic data or theoretical models, and it has a general formas

a0i ðd; sÞ ¼ ln dþ funcðsÞ ð5Þ

where funcðsÞ is a function of s. Jaeschke and Schley (1995)and Kunz et al. (2007) established respectively a general

form of funcðsÞ for natural pure fluids. a0i has contribution

to the thermodynamic properties of fluids except for theVLE properties.

The ari in Eq. (2) usually has different forms which are of

semi-empirical, dependent on the different temperature-density conditions. Sun and Ely (2004) developed a generalform of ar

i for the pure non-polar and polar fluids asfollows:

ari ðd; sÞ ¼

X6

m¼1

amsjmdim þ

X14

m¼7

amsjmdime�dkm ð6Þ

AAD(q0) % AAD(q00) %

This work SE2004 This work SE2004

0.04 0.10 0.23 0.440.05 0.18 0.19 0.600.28 0.16 0.23 0.890.08 0.29 0.18 0.200.03 0.16 0.24 0.27

erage absolute deviation, q is the density covering the temperature

4, C2H6, C3H8, CO2 and N2, respectively, and pressure up to 200000 is the saturated vapor density.


where exponents (im, jm, km) are the same for all pure fluids(Table 3), and values of parameter am are obtained by fit-ting to the thermodynamic data of each fluid. The EOSof Sun and Ely (2004) is limited to engineering applicationswith valid pressure generally below 1000 bar, and somelarge deviations were found for the saturated liquid density,vapor density and pressure in the near-triple-point regionand/or the near-critical region. Therefore, in order to betterapply the general Eq. (6) to fluid inclusion studies, am in Eq.

Fig. 1. The deviations of single-phase density, saturated pressure, saturaLeft is from this work and right is from Sun and Ely (2004), T is the tesaturated liquid density, q00 is the saturated vapor density, subscript calreferenced data are from National Institute of Standards and Technolog

(6) were refitted simultaneously to the PVTx and VLE dataof each fluid obtained from the Chemistry WebBook ofNIST (2017) up to 650 K and 2000 bar.

Regressed coefficients of Eq. (6) for CH4, C2H6, C3H8,CO2 and N2 are listed in Table 4. The density or molar vol-ume of pure fluids can be calculated from the followingequation with the Newton iterative method:

P ¼ qRT ½1 þ daridðd; sÞ� ð7Þ

ted liquid density and saturated vapor density for the CH4 system:mperature, P is the pressure, Ps is the saturated pressure, q’ is thedenotes the calculated values, and subscript NIST denotes that they.


where P is the pressure, and arid is the derivative of ar

i withrespect to d. If fluid is in vapor or supercritical state, the ini-tial density of fluid can be set equal to that of ideal gas. Iffluid is in liquid state, the initial density can be set as thesaturated liquid density of pure fluids. Molar Gibbs freeenergy of component i (Gi) can be obtained from this Helm-holtz free energy EOS:

Gi ¼ RT ð1 þ a0i þ ar

i þ daridÞ ð8Þ

The saturated properties of pure fluids as a function oftemperature can be calculated from this EOS by a reliable

Fig. 2. The deviations of single-phase density, saturated pressure, saturatLeft is from this work and right is from Sun and Ely (2004), and the me

and highly efficient approach similar to that used by Maoet al. (2011) for pure H2O. At vapor-liquid phase equilib-rium, the molar Gibbs free energy of component i in liquidphase G0

i equals to that of component i in vapor phase G00i .

Combining Eqs. (5) and (8), one single VLE equation canbe obtained:

lnd0 þ ari ðs;d0Þ þ d0ar

idðs;d0Þ � ½lnd00 þ ari ðs;d00Þþ d00ar

idðs;d00Þ� ¼ 0

ð9Þwhere d0 ¼ q0=qci and d00 ¼ q00=qci, and q0 and q00 denote thesaturated liquid and vapor densities, respectively. Because

ed liquid density and saturated vapor density for the C2H6 system:anings of variables and subscripts are the same as those in Fig. 1.

Fig. 3. The deviations of single-phase density, saturated pressure, saturated liquid density and saturated vapor density for the C3H8 system:Left is from this work and right is from Sun and Ely (2004), and the meanings of variables and subscripts are the same as those in Fig. 1.


pressures at VLE are identical for the coexisting phases,another equilibrium equation can be obtained fromEq. (7):

d0½1 þ d0aridðs; d0Þ� � d00½1 þ d00ar

idðs; d00Þ� ¼ 0 ð10ÞCombining Eqs. (9) and (10), the Newton iteration

method can be used to calculate the saturated propertiesof each pure fluid at a given temperature: selecting the val-ues of q0 and q00 from the auxiliary equations (which wereprovided in the standard EOSs recommended by NIST)as initial density values and the density function as an iter-

ative function, and calculating d0 and d00 with the followingsimultaneous equations:

d0ðkþ1Þ ¼ d0ðkÞ þ 1

K½Kðs; d00Þ � Kðs; d0Þ�J dðs; d00Þf

�½Jðs; d00Þ � Jðs; d0Þ�Kdðs; d00Þg ð11Þ

d00ðkþ1Þ ¼ d00ðkÞ þ 1

K½Kðs; d00Þ � Kðs; d0Þ�J dðs; d0Þf

�½Jðs; d00Þ � Jðs; d0Þ�Kdðs; d0Þg ð12Þwhere J ;K; J d;Kd, and K are defined as

Fig. 4. The deviations of single-phase density, saturated pressure, saturated liquid density and saturated vapor density for the CO2 system:Left is from this work and right is from Sun and Ely (2004), and the meanings of variables and subscripts are the same as those in Fig. 1.


Jðs; dÞ ¼ d½1 þ daridðs; dÞ� ð13Þ

Kðs; dÞ ¼ daridðs; dÞ þ ar

i ðs; dÞ þ ln d ð14Þ

J dðs; dÞ ¼ @J@d

� �s

¼ 1 þ 2darid þ d2ar

idd ð15Þ

Kdðs; dÞ ¼ @K@d

� �s

¼ 2arid þ dar

idd þ1

dð16Þ

K ¼ J dðs; d00ÞKdðs; d0Þ � J dðs; d0ÞKdðs; d00Þ ð17Þ

The method can calculate the saturated properties ofpure fluids from the temperature in triple point to that incritical point. In the calculation, the following convergencecondition can be set:

jKðs; d00Þ � Kðs; d0Þj þ jJðs; d00Þ � Jðs; d0Þj < 10�8 ð18ÞWith the improved EOS and the Newton iteration meth-

ods mentioned above, the single-phase density, saturatedpressure, saturated liquid and vapor densities can be calcu-lated for each pure fluid. The average absolute deviations of

Fig. 5. The deviations of single-phase density, saturated pressure, saturated liquid density and saturated vapor density for the N2 system: Leftis from this work and right is from Sun and Ely (2004), and the meanings of variables and subscripts are the same as those in Fig. 1.


these properties for unitary CH4, C2H6, C3H8, CO2 and N2

fluid are listed in Table 5, where the SE2004 EOS is alsocompared with the data of NIST. It can be seen fromTable 5 that the average absolute deviations of thesingle-phase density and saturated properties from thisimproved EOS are smaller than those from the SE2004EOS. Figs. 1–5 show the deviations of single-phase density,saturated pressure, saturated liquid and vapor densities forCH4, C2H6, C3H8, CO2 and N2, respectively. As shown inFig. 1, the saturated properties of CH4 have been improved

in the near-critical region. For C2H6, the saturated pressureand vapor density have been improved in the near-triple-point region (Fig. 2b and d), so do the saturated den-sities in the near-critical region (Fig. 2c and d). For C3H8,the saturated pressure and vapor density near the triple-point region (Fig. 3b and d), and the saturated vapor den-sity in the near-critical region (Fig. 3d) from this EOS are inagreement with the NIST data, better than the SE2004EOS. The saturated densities of CO2 calculated from thisimproved EOS and the SE2004 EOS are in good agreement

Table 6Coefficients and exponents of Eq. (23).

k Nk dk tk

1 �2.4547627E�02 1 22 �2.4120612E�01 1 43 �5.1380195E�03 1 �24 �2.3982483E�02 2 15 2.5977234E�01 3 46 �1.7201412E�01 4 47 4.2949003E�02 5 48 �2.0210859E�04 6 09 �3.8298423E�03 6 410 2.6992331E�06 8 �2


with the NIST data (Fig. 4c and d), but the calculated sat-urated pressure from this EOS has been improved in theentire VL region (Fig. 4b). Compared with the SE2004EOS, the saturated liquid density of N2 from this EOSagrees well with the NIST data in the near-critical region(Fig. 5c).

3. EQUATION OF STATE OF THE CH4–C2H6–C3H8–

CO2–N2 FLUID MIXTURES

The EOS of the CH4–C2H6–C3H8–CO2–N2 fluid mix-tures is in terms of dimensionless Helmholtz free energya, which has the same form as Eq. (1) provided that Ai isreplaced by the molar Helmholtz free energy of mixturesA. The a is represented by

a ¼ aid þ aE ð19Þwhere aid is the dimensionless Helmholtz free energy of an

ideal mixture and aE is the excess dimensionless Helmholtz

free energy of mixture. aid results directly from the funda-mental equations of pure fluids and can be written as

aid ¼Xn

i¼1

xi a0i ðq; T Þ þ lnðxiÞ

� �þXn

i¼1

xiari ðd; sÞ ð20Þ

where xi is the mole fraction of component i. The super-scripts ‘‘id”, ‘‘0” and ‘‘r” denote the ideal mixing, theideal-gas part and the residual part of dimensionless Helm-

Table 7Binary parameter values of the mixtures (F ij, nij, 1ij, bij).

Binary mixtures F ij nij

CH4–C2H6 4.6288463E�01 �1.7283355CH4–C3H8 1.4900000E+00 �7.9000002CH4–CO2 8.7949338E�01 2.5558121ECH4–N2 2.4313265E�01 3.1304356EC2H6–C3H8 �1.1875279E�01 �4.2436073C2H6–CO2 �8.9551612E�01 2.1842743EC2H6–N2 4.0365476E�01 1.0900950EC3H8–CO2 �2.8403907E�01 1.1814517EC3H8–N2 1.3097399E+00 3.8532717ECO2–N2 1.2781440E+00 2.5761769E

Note: Subscripts i and j refer to the first component and the second com

holtz free energy, respectively. The subscripts i denotes thecomponent. Here n = 5, and the subscripts 1–5 refer toCH4, C2H6, C3H8, CO2 and N2, respectively. d and s arethe reduced parameters, which have the same forms asEqs. (3) and (4), and the only difference is that qci and T ci

are replaced by the pseudo-critical density qc and tempera-ture T c of the mixtures. qc and T c are defined as

qc ¼Xn

i¼1

xi

qci

þXn�1

i¼1

Xn

j¼iþ1

xixjfij

" #�1

ð21Þ

T c ¼Xn

i¼1

xiT ci þXn�1

i¼1

Xn

j¼iþ1

xbij

i x/ij

j 1ij ð22Þ

where fij, 1ij, bij and /ij are the mixture-dependent binary

parameters associated with components i and j. The valueof /ij is one except for the C3H8–CO2 system, where

/ij ¼ 1:4036159. If the value of /ij for the C3H8–CO2 sys-

tem is set as one, big deviations of the PVTx and VLEproperties will yield in the fitting.

The aE in Eq. (19) is given by a general form as

aE ¼Xn�1

i¼1

Xn

j¼iþ1

xixjF ij

X10

k¼1

Nkddkstk ð23Þ

where Nk, dk and tk are the general parameters independentof fluids, which can be derived from the model of Lemmonand Jacobsen (1999) (Table 6), and F ij is a binary parameterof components i and j.

The residual part of dimensionless Helmholtz freeenergy of the CH4–C2H6–C3H8–CO2–N2 fluid mixtures ar

is defined by

ar ¼Xn

i¼1

xiari ðd; sÞ þ aEðd; s; xÞ ð24Þ

Here the ari of pure fluids is calculated from the general Eq.

(6). Values of the binary parameters (fij, 1ij, bij and F ij) in

above equations for the fluid mixtures are determined bya non-linear regression to the selected experimental PVTxand VLE data of binary systems. In general, the PVTx

and VLE data change regularly with temperature and pres-sure. If some data deviate largely from the adjacent data atthe same or approximated conditions, they are deemed to

1ij bij

E�03 6.5429302E+00 1.2959001E+00E�04 1.7100000E+01 1.8600000E+00�03 �3.8924385E+01 1.0713116E+00�03 �1.5111974E+01 9.7944322E�01E�03 7.4867815E+00 9.1876947E�01�03 �4.8279795E+01 9.1890867E�01�03 �7.0746547E+00 1.1676007E+00�03 �7.7249028E+01 1.0017273E+00�02 �1.2729955E+01 3.5925694E+00�03 �1.2488803E+01 7.5722203E�01

ponent, respectively.

Table 8Calculated PVTx and VLE deviations for the binary mixtures.

References Number of data points T-P-x range AAD (%)

Styles Total Used T (K) P (bar) x This work GERG2004 SE2004

xCH4 þ ð1 � xÞC2H6

Sage and Lacey (1939) PVTx 741 428 294–394 1.0–310 0.32–0.9 0.89 0.91 0.91Michels and Nederbragt (1939) PVTx 97 89 273–323 10–82 0.20–0.8 0.47 0.86 0.97Rodosevich and Miller (1973) PVTx 19 19 91–116 0.23–1.49 0.05–0.3 0.04 0.05 0.11Pan et al. (1975) PVTx 8 8 91–115 0.24–1.20 0.51–0.7 0.05 0.05 0.13Hiza et al. (1977) PVTx 20 20 105–130 0.26–2.62 0.35–0.7 0.06 0.07 0.13Haynes et al. (1985) PVTx 414 408 100–320 17–359 0.35–0.7 0.54 0.50 0.54Blanke and Weiss (1995) PVTx 129 129 274–333 20–71 0.75–0.9 0.17 0.07 0.07Hou et al. (1996) PVTx 226 135 300–320 1.1–111 0.0–1.0 0.11 0.07 0.13Wichterle and Kobayashi (1972a) VLE 121 59 158–200 0.19–46 0–1.0 2.61 1.54 2.69Davalos et al. (1976) VLE 9 7 250 13–66.6 0–0.55 0.62 0.82 1.42Miller et al. (1977) VLE 25 20 160–180 0.21–32.9 0–1.0 2.01 2.22 3.27Wei et al. (1995) VLE 54 47 210–270 3.34–65 0–0.86 0.88 0.85 1.21Vrabec and Fischer (1996) VLE 29 25 160–250 0.22–63.1 0–0.9 6.35 6.46 4.27Raabe et al. (2001) VLE 10 8 240–270 15.2–66.3 0.06–0.6 0.61 0.64 1.70Zhang and Duan (2002) VLE 30 28 160–280 6–61.3 0.02–0.9 2.65 2.79 2.80Janisch et al. (2007) VLE 28 24 140–270 4.9–66.3 0.05–0.9 0.80 1.20 1.74Han et al. (2012b) VLE 68 44 126–140 1.48–6.3 0.26–1.0 2.76 2.14 3.60

xCH4 þ ð1 � xÞC3H8

Huang et al. (1967) PVTx 140 140 120–310 34–344 0.2–0.75 0.79 0.98 3.18Shana’a and Canfield (1968) PVTx 5 5 108.15 0.32–0.69 0.2–0.85 0.39 1.30 0.15Rodosevich and Miller (1973) PVTx 12 12 91–114 28–140 0.9–0.97 0.98 1.26 1.17Pan et al. (1975) PVTx 8 8 91–115 0.31–1.35 0.8979 1.92 1.94 1.93Hiza et al. (1977) PVTx 20 20 105–130 0.26–2.62 0.3–0.86 0.50 0.04 0.11May et al. (2001) PVTx 58 58 290–313 20–100 0.7–0.93 0.22 0.18 0.31May et al. (2002) PVTx 14 14 297–313 30–95 0.84 0.29 0.18 0.11Richter and McLinden (2014) PVTx 238 238 248–373 1–60 0.2–0.75 0.05 0.24 4.35Sage et al. (1934) VLE 55 43 293–363 10–96 0.08–0.7 2.47 3.24 3.84Reamer et al. (1950) VLE 122 101 278–361 7–102 0.25–1.0 0.80 0.97 1.11Akers et al. (1954) VLE 80 66 158–273 3–97 0–1.0 2.29 3.15 2.88Cheung and Wang (1964) VLE 25 0 114–122 0.4–2.2 0.1–1.0 13.76 5.55 12.51Wichterle and Kobayashi (1972b) VLE 106 52 130–214 0–64 0–1.0 2.53 1.02 3.03Poon and Lu (1974) VLE 22 0 114–122 0.4–2.2 0.1–0.9 2.94 5.03 3.89Joffe (1976) VLE 13 12 227–344 34–90 0.1–0.96 0.89 0.89 0.70Webster and Kidnay (2001) VLE 74 66 230–270 1–84 0–0.92 2.68 1.86 1.80Kandil et al. (2005) VLE 10 0 315–340 28–68 0.08–0.3 1.06 6.27 2.20

xCH4 þ ð1 � xÞCO2

Arai et al. (1971) PVTx 145 145 253–288 24–145 0.04–0.6 1.34 1.35 1.70Tong and Liu (1984) PVTx 24 21 382–452 80–128 0.49–0.5 0.70 0.66 0.66Magee and Ely (1988) PVTx 91 91 255–400 21–358 0.02 0.24 0.14 0.28Brugge et al. (1989) PVTx 155 135 300–320 2–92 0.1–0.9 0.07 0.05 0.15Seitz and Blencoe (1996) PVTx 94 94 673.15 199–999 0.1–0.9 0.68 0.67 0.63Seitz et al. (1996a) PVTx 196 44 323–573 199–999 0.1–0.9 0.43 0.18 0.74Hwang et al. (1997) PVTx 228 228 255–350 18–449 0.1–0.9 0.37 0.25 0.57Mondejar et al. (2012) PVTx 314 314 250–400 9–200 0.4–0.8 0.21 0.12 0.17Yang et al. (2016) PVTx 181 0 301–306 19.9–119.66 0.05 0.84 1.50 0.98Donnelly and Katz (1954) VLE 78 50 200–271 11–81 0.0–0.92 3.03 3.50 5.71Neumann and Walch (1968) VLE 18 18 233–283 37–82 0.07–0.5 3.40 2.15 4.94Hwang et al. (1976) VLE 126 0 153–219 12–65 0.77–1.0 2.57 2.46 2.75Davalos et al. (1976) VLE 33 33 230–270 15–85 0.01–0.6 3.21 2.82 5.13Joffe (1976) VLE 12 11 209–271 34–73 0.1–0.64 1.57 3.41 4.79Mraw et al. (1978) VLE 59 25 153–219 9–65 0.1–1.0 1.96 1.26 3.85Somait and Kidnay (1978) VLE 12 9 270 38–84 0.02–0.3 2.23 2.80 4.47Al-Sahhaf et al. (1983) VLE 29 27 219–270 9–85 0.0–0.55 3.00 2.53 5.11Xu et al. (1992) VLE 21 0 288–293 56–82 0.01–0.2 1.78 1.29 2.93Bian et al. (1993) VLE 6 0 301 70–77 0.0–0.03 1.22 1.11 1.07Wei et al. (1995) VLE 57 49 230–270 10–84 0.07–0.6 2.32 2.06 4.59Vrabec and Fischer (1996) VLE 19 19 230–270 13–78 0.02–0.3 4.25 4.34 6.34Webster and Kidnay (2001) VLE 40 36 230–270 12–84 0.02–0.6 2.41 1.83 4.03Nasir et al. (2015) VLE 262 0 240.35–297.15 18.3–83.7 0.02–0.64 2.73 0.96 3.58


Table 8 (continued)



xCH4 þ ð1 � xÞN2

Liu and Miller (1972) PVTx 33 7 91–115 1–12 0.15–0.5 0.53 0.37 0.58Rodosevich and Miller (1973) PVTx 8 0 91–115 43–454 0.8–0.95 3.86 3.75 3.92Pan et al. (1975) PVTx 7 7 91–115 1–11 0.5–0.86 0.08 1.84 0.13Hiza et al. (1977) PVTx 21 21 95–140 9–21 0.5–0.95 0.09 1.15 0.12Da Ponte et al. (1978) PVTx 182 182 110–120 15–1380 0.3–0.68 0.16 0.24 0.24Straty and Diller (1980) PVTx 581 570 82–320 6–356 0.32–0.7 0.55 0.50 0.32Haynes (1983) PVTx 85 85 140–320 10–164 0.3–0.71 0.13 0.03 0.11Seitz et al. (1996a) PVTx 190 190 323–573 99–999 0.1–0.9 0.35 0.38 0.44Seitz and Blencoe (1996) PVTx 43 0 673.15 199–999 0.1–0.9 5.53 5.03 6.36Ababio et al. (2001) PVTx 83 83 308–333 9–120 0.5–0.78 0.13 0.13 0.16Chamorro et al. (2006) PVTx 242 242 230–400 9–192 0.8–0.9 0.16 0.21 0.15Janisch et al. (2007) PVTx 17 0 129–180 15–50 0.4–0.9 2.57 1.67 2.08Li et al. (2013) PVTx 27 27 17–270 1.3–16 0.90 0.05 0.07 0.04Brandt and Stroud (1958) VLE 21 0 137–175 34 0.3–0.95 2.76 2.27 3.19Cheung and Wang (1964) VLE 20 0 91.6–164 0.2–6 0.85–1.0 3.39 3.28 4.28Fuks and Bellemans (1967) VLE 20 0 84.2–90.8 1–2 0.3–0.9 2.61 2.68 5.43Parrish and Hiza (1973) VLE 48 48 95–120 2–20 0.1–0.9 1.30 0.66 2.42Miller et al. (1973) VLE 22 0 112 2–13 0.2–0.97 1.79 6.78 2.98Stryjek et al. (1974b) VLE 9 0 183.15 36–48 0.88–1.0 0.60 0.83 7.36McClure et al. (1976) VLE 8 8 90.7 1–3 0.1–0.8 1.98 0.29 3.45Jin et al. (1993) VLE 10 8 123 4–26 0.1–0.95 2.25 1.29 3.66Janisch et al. (2007) VLE 16 0 130–180 0.5–5 0.4–0.96 2.21 1.55 2.74Han et al. (2012a) VLE 77 60 110–123 4–13 0.7–1.0 2.24 3.99 2.33

xC2H6 þ ð1 � xÞC3H8

Kahre (1973) PVTx 10 10 288.75 10–29 0.1–0.86 0.18 0.92 0.50Parrish (1984) PVTx 315 315 283–322 28–97 0.3–0.95 0.17 0.78 0.49Matschke and Thodos (1962) VLE 65 47 311–366 14–52 0.0–0.93 0.63 0.77 0.76Djordjevich and Budenholzer (1970) VLE 50 34 128–255 0.01–15 0.1–0.95 5.39 4.61 8.48Kahre (1973) VLE 10 10 288.75 10–29 0.1–0.86 0.78 0.54 0.47Miksovsky and Wichterle (1975) VLE 102 32 303–269 12–51 0.0–1.0 1.67 1.59 2.04Clark and Stead (1988) VLE 22 22 260–280 4–26 0.1–0.95 4.22 3.28 3.99Blanc and Setier (1988) VLE 151 137 195–270 0.3–20 0.0–0.92 1.68 1.49 2.32xC2H6 þ ð1 � xÞCO2

Reamer et al. (1945) PVTx 1835 798 311–478 28–689 0.1–0.9 0.71 0.60 1.15Gugnoni et al. (1974) PVTx 57 56 240–283 10–50 0.2–1.0 0.78 2.11 2.10Sherman et al. (1989) PVTx 94 94 270–400 32–348 0.01 0.31 0.16 0.29Brugge et al. (1989) PVTx 194 194 300–320 1–68 0.1–0.9 0.38 0.11 0.18Weber (1992) PVTx 265 0 290–320 1–122 0.2–0.8 1.70 2.80 6.52Lau et al. (1997) PVTx 280 187 240–350 15–346 0.1–0.75 0.40 0.32 1.23Goodwin and Moldover (1997) PVTx 32 0 284–297 47–630 0.2–0.8 4.97 2.64 7.27Duarte-Garza and Magee (2001) PVTx 585 585 220–400 28–355 0.2–0.75 0.50 0.41 1.17Hamam and Lu (1974) VLE 39 25 222–289 7–51 0.1–0.9 1.84 2.07 2.26Fredenslund and Mollerup (1974) VLE 65 39 223–293 6–63 0.0–1.0 1.61 1.14 2.87Nagahama et al. (1974) VLE 13 12 253 16–23 0.1–0.9 1.20 1.51 1.00Davalos et al. (1976) VLE 13 11 259 14–21 0.0–0.96 0.37 1.32 0.91Ohgaki and Katayama (1977) VLE 64 42 283–298 35–66 0.0–0.9 1.95 0.80 5.12Vrabec and Fischer (1996) VLE 24 17 223–283 7–50 0.1–0.9 1.32 2.24 3.38Brown et al. (1988) VLE 136 102 207–270 3–36 0.0–1.0 1.81 2.71 1.93Clark and Stead (1988) VLE 11 11 260 19–28 0.0–0.94 1.80 2.04 0.76Wei et al. (1995) VLE 76 53 207–270 4–36 0.0–1.0 2.76 2.85 2.75Vrabec and Fischer (1996) VLE 24 17 223–283 7–50 0.1–0.9 1.32 2.24 3.38

xC2H6 þ ð1 � xÞN2

Hiza et al. (1977) PVTx 4 4 105–120 4–6 0.94 0.04 0.09 0.41Archtermann et al. (1991) PVTx 480 480 270–350 2–287 0.25–0.75 0.20 0.04 0.25Raabe and Kohler (2004) PVTx 27 27 115–139 6–33 0.0–0.96 1.44 0.69 1.52Janisch et al. (2007) PVTx 14 14 150–270 10–100 0.6–0.9 0.55 1.81 0.69Stryjek et al. (1974a) VLE 48 47 139–194 3–135 0.29–0.99 1.73 1.73 3.89Grauso and Fredenslund (1977) VLE 35 30 200–290 7–132 0.43–0.99 1.37 2.79 5.10

(continued on next page)


Table 8 (continued)



Brown et al. (1989) VLE 24 23 220–270 10–120 0.59–0.98 2.83 1.86 4.05Raabe et al. (2001) VLE 11 11 210–270 11–90 0.74–0.98 3.54 1.01 3.45Raabe and Kohler (2004) VLE 21 21 120–138 9–33 0.73–0.95 3.69 3.81 9.32Janisch et al. (2007) VLE 22 22 150–270 5–80 0.8–0.9 2.81 1.64 5.35

xC3H8 þ ð1 � xÞCO2

Galicia-Luna et al. (1994) PVTx 170 170 323–400 70–400 0.7–0.97 0.20 0.80 0.23de Dios et al. (2003) PVTx 559 532 294–344 1–704 0.1–0.9 0.38 0.40 0.55Blanco et al. (2009) PVTx 364 364 308.15 1–200 0.1–0.9 1.23 2.38 1.97Feng et al. (2010) PVTx 225 225 300–400 1–77 0.2–0.5 0.33 0.26 0.36Miyamoto (2014) PVTx 281 281 280–440 100–2000 0.2–0.8 0.29 0.34 0.25Reamer et al. (1951) VLE 74 33 278–344 7–69 0.02–0.99 2.52 2.01 4.15Akers et al. (1954) VLE 9 4 233–277 3–28 0.18–0.95 15.54 13.10 22.61Nagahama et al. (1974) VLE 20 10 253–273 3–34 0.05–0.99 8.55 4.56 8.99Hamam and Lu (1976) VLE 21 10 244–266 5–26 0.18–0.9 5.63 2.49 8.46Acosta et al. (1984) VLE 289 0 211–350 0.6–59 0.56–0.9 4.47 5.82 4.87Niesen and Rainwater (1990) VLE 90 44 311–361 15–67 0.22–0.99 2.50 1.21 2.01Yucelen and Kidnay (1999) VLE 31 8 240–330 2–59 0.2–0.98 5.93 3.13 2.87Webster and Kidnay (2001) VLE 63 25 230–270 1–31 0.04–0.97 9.10 6.70 11.12

xC3H8 þ ð1 � xÞN2

Watson et al. (1954) PVTx 69 59 399–422 6–421 0.0–0.83 1.86 1.75 2.16Hiza et al. (1977) PVTx 6 6 100–100 4–9 0.93–0.98 0.59 0.07 0.38Cheung and Wang (1964) VLE 6 0 92–128 1–6 0.9–1.0 – 25.35 –Schindler et al. (1966) VLE 60 59 103–353 7–138 0.7–1.0 3.35 4.57 3.94Poon and Lu (1974) VLE 32 23 114–122 2–28 0.9–1.0 4.99 6.38 7.74Grauso and Fredenslund (1977) VLE 33 32 230–290 4–219 0.4–1.0 4.93 3.79 3.77Hudziak et al. (1984) VLE 127 0 188–343 0.8–58 0.1–0.74 6.63 6.37 7.30Yucelen and Kidnay (1999) VLE 36 28 240–330 12–51 0.7–1.0 2.83 3.54 3.86

xCO2 þ ð1 � xÞN2

Arai et al. (1971) PVTx 158 135 253–288 24–145 0.4–0.9 1.33 1.71 1.78Hacura et al. (1988) PVTx 256 110 323–348 490–2740 0.25–0.74 0.45 0.41 0.85Ely et al. (1989) PVTx 79 71 300–320 230–331 0.98 0.44 0.62 0.94Brugge et al. (1989) PVTx 196 190 300–320 2–78 0.1–0.9 0.11 0.10 0.15Seitz and Blencoe (1996) PVTx 51 51 673.15 199–999 0.1–0.9 0.52 0.48 1.55Seitz et al. (1996a) PVTx 192 192 323–573 99–999 0.1–0.9 0.61 0.63 0.82Brugge et al. (1997) PVTx 749 749 225–450 10–692 0.1–0.9 0.27 0.50 0.39Mondejar et al. (2011) PVTx 138 0 250–400 9–200 0.1–0.15 0.09 0.05 0.25Mantovani et al. (2012) PVTx 197 0 303–383 10–200 0.9–0.96 1.32 1.10 1.32Mondejar et al. (2012) PVTx 209 209 250–400 9–200 0.2–0.5 0.14 0.26 0.20Yang et al. (2015) PVTx 132 0 298–423 110–310 0.01–0.05 0.09 0.02 0.16Al-Sahhaf et al. (1983) VLE 27 27 220–240 15–167 0.5–1.0 5.23 2.03 9.48Yorizane et al. (1985) VLE 34 7 273–298 45–115 0.6–1.0 27.87 12.33 24.37Brown et al. (1989) VLE 55 14 220–270 10–130 0.8–1.0 3.08 1.47 5.09Xu et al. (1992) VLE 18 6 288–293 60–97 0.85–1.0 0.53 1.22 1.81Yucelen and Kidnay (1999) VLE 19 6 240–270 17–130 0.75–1.0 4.32 3.40 3.68Fandino et al. (2015) VLE 66 0 218.15–288.15 5.6–150.28 0.16–1 4.73 1.80 6.65

Notes: T is the temperature, P is the pressure, and x is the composition in mole fraction. Deviations of the PVTx data are the densitydeviations, and deviations of the VLE data are the deviations of the bubble (or dew) point pressures. GERG2004 is the EOS of Kunz et al.(2007), and SE2004 is the EOS of pure fluids from Sun and Ely (2004) combined with the mixing parameters of Lemmon and Jacobsen (1999).


be unreliable and are rejected from the fitting. These datacan be distinguished by the iso-T or iso-P curves. Theexperimental PVTx and VLE data of binary systems after2014 are not used in the parameterization but to verifythe validity of this general EOS in this work. In the fitting,the objective function is set as the sum of relative deviationof density and fugacity difference of each component

between vapor and liquid phases. Regressed parametersare listed in Table 7.

Similar to Eq. (7), the density or molar volume of theCH4–C2H6–C3H8–CO2–N2 mixtures can be calculated fromthe following equation by the Newton iterative method:

P ¼ qRT ½1 þ dard� ð25Þ

00.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

T=180 KP (b

ar)

xCH4

This work Miller et al. (1977) Wei et al. (1995) Zhang and Duan (2002)

T=210 KT=230 KT=270 K

a

CH4-C2H6

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

120

140

b

CH4-C3H8 Webster and Kidnay (2001) This work

P (b

ar)

xCH4

T=230 K

T=270 K

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

60

70 C2H6-C3H8

T=235 K

P (b

ar)

xC2H6

This work Matschke and Thodos (1962) Blanc and Setier (1988) T=311 KT=344 K

T=280 K

cFig. 6. Vapor-liquid phase equilibria of binary CH4–C2H6, CH4–C3H8 and C2H6–C3H8 systems: T is the temperature, P is thepressure, xCH4

is the mole fraction of CH4, and xC2H6is the mole

fraction of C2H6.

00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.710

20

30

40

50

60

70

80

90

100

T=283.85 K T=265.35 K

P (b

ar)

xCH4

This workNasir et al. (2015)

T=240.35 K

a

CH4-CO2

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60CH4-N2

T=91 K

P (b

ar)

xN2

This work Han et al.(2012) Janisch et al. (2007) McClure et al. (1976)

T=121 K

T=170 K

bFig. 7. Vapor-liquid phase equilibria of binary CH4–CO2 andCH4–N2 systems: T is the temperature, P is the pressure, xCH4

is themole fraction of CH4, and xN2

is the mole fraction of N2. Thecalculated critical pressures and compositions (xCH4

) for the CH4–CO2 system at 283.85 K, 265.35 K and 240.35 K are (87 bar, 0.22),(90 bar, 0.37), (81.6 bar, 0.57), respectively.


where ard is the derivative of ar with respect to d. If the mix-

ture is in vapor or supercritical state, the initial density ofmixture can be set for that of ideal gas. If the mixture isin liquid state, the saturated liquid density of pure compo-nent with the highest critical temperature can be set as theinitial density.

Fugacity and fugacity coefficient of component i can becalculated from the following equations:

f i ¼ xiqRT exp@nar

@ni

� �T ;V ;nj

ð26Þ

lnui ¼@nar

@ni

� �T ;V ;nj

� lnð1 þ dardÞ ð27Þ

@nar

@ni

� �T ;V ;nj

¼ ar þ n@ar

@ni

� �T ;V ;nj

ð28Þ

n@ar

@ni

� �T ;V ;nj

¼ dard 1 � 1

qc

@qc

@xi

� �xj

�Xn

k¼1

xk@qc

@xk

� �xj

" #" #

þ sars

1

T c

@T c

@xi

� �xj

�Xn

k¼1

xk@T c

@xk

� �xj

" #

þ arxi�Xn

k¼1

xkarxk

ð29Þwhere f i is the fugacity of component i, n is the total molenumber, V is the total volume, ni is the mole number ofcomponent i, nj is the mole number of component j and sig-nifies that all mole numbers are held constant except ni, ui isthe fugacity coefficient of component i, and ar

s, arxi

and arxk

are the derivatives of ar with respect to s, xi and xk,respectively.

The VLE properties of the CH4–C2H6–C3H8–CO2–N2

mixtures at given T and P can be calculated by an iterativealgorithm (Michelsen, 1993; Gernert et al., 2014). Here theiterative algorithm of Michelsen (1993) has been used(Appendix A).

00.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35 Webster and Kidnay (2001) This work

P (b

ar)

xCO2

T=230 K

T=270 K

a

C3H8-CO2


With the mixing parameters and above iterative algo-rithms, the PVTx and VLE properties of the CH4–C2H6–C3H8–CO2–N2 mixtures can be calculated. The calculatedPVTx and VLE deviations from experimental data of theten binary mixtures (CH4–C2H6, CH4–C3H8, CH4–CO2,CH4–N2, C2H6–C3H8, C2H6–CO2, C2H6–N2, C3H8–CO2,C3H8–N2, and CO2–N2) are listed in Table 8, where thedeviations of PVTx data are the density deviations, andthe deviations of VLE data are the deviations of the bubble(or dew) point pressures. The GERG2004 and SE2004EOSs are also compared in Table 8, where SE2004 denotesthat the EOS of pure fluids is from Sun and Ely (2004) com-bined with the mixing parameters of Lemmon and Jacobsen(1999). It can be seen that the average absolute density devi-ations of this EOS from each binary datum set are smallerthan those of the SE2004 EOS in most cases, with similaraccuracy of the GERG2004 EOS. So are the same to theaverage absolute deviations of saturated pressures of eachbinary datum set.

The calculated VLE curves of the CH4–C2H6, CH4–C3H8 and C2H6–C3H8 mixtures at different temperaturesare shown in Fig. 6, from which it can be seen that the cal-culated pressure-composition curves are in good agreementwith experimental data. It should be noted that composi-tion in this work refer to mole fraction. Fig. 7 shows the cal-culated VLE curves of the CH4–CO2 and CH4–N2

mixtures. For the CH4–N2 system, the calculatedpressure-composition curves are in good agreement withexperimental data, whereas the calculated pressure-

00.0 0.2 0.4 0.6 0.8 1.05

10

15

20

25

30

P (b

ar)

xC2H6

This work Wei et al. (1995) Brown et al.(1988) Davalos et al. (1976)

T=230 K

T=250 K

a

C2H6-CO2

0.0 0.2 0.4 0.6 0.8 1.00

20406080

100120140160180200

P (b

ar)

xN2

This work Stryjek et al.(1974) Grauso et al.(1977) Brown et al.(1989)

T=172 K

T=200 K

T=220 K

T=270 KT=290 K

C2H6-N2

bFig. 8. Vapor-liquid phase equilibria of binary C2H6–CO2 andC2H6–N2 systems: T is the temperature, P is the pressure, xC2H6

isthe mole fraction of C2H6, and xN2

is the mole fraction of N2.

composition curves of the CH4–CO2 system are in approx-imate agreement with the newly-published experimentaldata of Nasir et al. (2015). The partial disagreement of thismodel with the experimental data for the CH4–CO2 systemat low temperatures should be responsible for the simpleforms of the general EOSs of pure CH4 and CO2 fluids(Mao et al., 2016). The calculated VLE curves of theC2H6–CO2 and C2H6–N2 mixtures at different temperaturesare shown in Fig. 8, where the calculated pressure-composition curves of the C2H6–CO2 system are in goodagreement with experimental data, but the calculatedpressure-composition curves of the C2H6–N2 system deviatefrom experimental data in the near-critical region. Fig. 9

0.0 0.2 0.4 0.6 0.8 1.00

100

200

300

400

P (b

ar)

xN2

This work Yucelen and Kidnay (1999) Grauso et al.(1977) T=260 K

T=270 K

T=290 K

T=330 K

b

C3H8-N2

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

250

300

350

P (b

ar)

xN2

This work Al-Sahhaf and Kidnay (1983) Yucelen and Kidnay (1999) Brown et al. (1989)

T=240 K

T=270 K

CO2-N2

cFig. 9. Vapor-liquid phase equilibria of binary C3H8–CO2,C3H8–N2 and CO2–N2 systems: T is the temperature, P is thepressure, xCO2

is the mole fraction of C3H8, and xN2is the mole

fraction of N2.

Table 9Calculated density deviations for some ternary fluid mixtures.

Composition (mole fraction) Density (mol�dm�3)

T (K) P (bar) xCH4xC2H6

xC3H8qexp qcal Dev (%)

Rodosevich and Miller (1973)100.01 0.49 0.8466 0.1025 0.0509 25.98280 26.09390 0.43108.01 0.81 0.8466 0.1025 0.0509 25.43235 25.65545 0.88108.02 0.84 0.8466 0.1025 0.0509 25.42847 25.69165 1.03115 1.28 0.8466 0.1025 0.0509 24.94077 25.40197 1.85Pan et al. (1975)91 0.303975 0.848 0.1008 0.0512 26.58867 26.53890 �0.19100 0.48636 0.848 0.1008 0.0512 25.98753 25.93290 �0.21108 0.881528 0.848 0.1008 0.0512 25.42588 25.38330 �0.17115 1.367888 0.848 0.1008 0.0512 24.93144 24.89070 �0.16


xN2qexp qcal Dev (%)

Rodosevich and Miller (1973)91 0.68 0.8409 0.1086 0.0505 27.35005 27.35000 0.00100 1.1 0.8409 0.1086 0.0505 26.66738 26.66300 �0.02108 1.81 0.8409 0.1086 0.0505 26.04302 26.04000 �0.01115.01 2.75 0.8409 0.1086 0.0505 25.47316 25.47130 �0.01Pan et al. (1975)91 0.48636 0.8675 0.0909 0.0416 27.48763 27.47370 �0.05100 0.881528 0.8675 0.0909 0.0416 26.79528 26.77600 �0.07108 1.45908 0.8675 0.0909 0.0416 26.16431 26.13830 �0.10115 2.208885 0.8675 0.0909 0.0416 25.66735 25.56300 �0.41Jaeschke and Humphreys (1991)273.02 3.962 0.768 0.072 0.16 0.1762 0.1761 �0.01273.02 6.983 0.768 0.072 0.16 0.3127 0.3126 �0.03273.02 12.241 0.768 0.072 0.16 0.5551 0.5549 �0.04273.02 21.261 0.768 0.072 0.16 0.9853 0.9848 �0.04273.02 36.372 0.768 0.072 0.16 1.7490 1.7488 �0.01273.02 60.821 0.768 0.072 0.16 3.1047 3.1089 0.14273.02 99.382 0.768 0.072 0.16 5.5104 5.5305 0.36


xN2qexp qcal Dev (%)

Rodosevich and Miller (1973)91 70 0.9055 0.0497 0.0447 27.27397 27.19420 �0.29108 185 0.9055 0.0497 0.0447 25.94236 25.86140 �0.31

Notes: T is the temperature, P is the pressure, xi is the mole fraction of component i, qcal is the calculated density, qexp is the experimentaldensity, and Dev ¼ 100ðqcal=qexp � 1Þ.

00 10 20 30 40 50 60 70 80-5.0

-2.5

0.0

2.5

5.0

100(

ρ cal-ρ

exp)ρ

exp

P (bar)

Hou et al. (1996) McElroy et al. (2001)

CH4-C2H6-CO2

Fig. 10. The deviations of the single-phase density for the CH4–C2H6–CO2 system: T is the temperature, P is the pressure, qcal is thecalculated density, and qexp is the experimental density.

00 200 400 600 800 1000-5.0

-2.5

0.0

2.5

5.0

CH4-CO2-N2

100(

ρ cal-ρ

exp)ρ

exp

P (bar)

Jaeschke and Humphreys (1991) Seitz et al. (1996)

Fig. 11. The deviations of the single-phase density for the CH4-CO2-N2 system: T is the temperature, P is the pressure, qcal is thecalculated density, and qexp is the experimental density.


00 10 20 30 40 50 60 70-5.0

-2.5

0.0

2.5

5.0

Webster and Kidnay (2001)100(

P cal-P

exp)/

P exp

P (bar)

T=230 K

Fig. 12. The deviations of bubble point pressures for the CH4–C3H8–CO2 system: T is the temperature, P is the pressure, pcal is the calculatedpressure, and pexp is the experimental pressure.

Fig. 13. The deviations of bubble point pressures for the CH4–C2H6–CO2 system: T is the temperature, P is the pressure, pcal is the calculatedpressure, and pexp is the experimental pressure.


shows the calculated VLE curves of the C3H8–CO2, C3H8–N2 and CO2–N2 mixtures. For the C3H8–CO2 system, thecalculated pressure-composition curves are in good agree-ment with experimental data of Webster and Kidnay(2001), but the calculated pressure-composition curves ofthe C3H8–N2 and CO2–N2 systems deviate from experimen-tal data (Al-Sahhaf et al., 1983; Yucelen and Kidnay, 1999)in the near-critical region.

Ten deviation figures of each binary system for thesingle-phase density, and the saturated vapor or liquid com-positions are placed in the Supplementary data.docx file(see Appendix B).

Table 9 lists the density deviations of this EOS fromexperimental data for some ternary fluid mixtures: CH4–C2H6–C3H8 (Rodosevich and Miller, 1973; Pan et al.,1975), CH4–C2H6–N2 (Rodosevich and Miller, 1973; Pan

et al., 1975; Jaeschke and Humphreys, 1991) and CH4–C3H8–N2 (Rodosevich and Miller, 1973). It can be seen thatthe calculated densities are in good agreement with thosedata. The density deviations of the CH4–C2H6–CO2 systemfrom experimental data (Hou et al., 1996; McElroy et al.,2001) are shown in Fig. 10, from which it can be seen thatthe densities calculated from this EOS are in good agree-ment with the data of Hou et al. (1996), with slightly largedeviations from the data of McElroy et al. (2001). The den-sity deviations of this EOS from experimental data for theCH4–CO2–N2 system (Jaeschke and Humphreys, 1991;Seitz et al., 1996b) are shown in Fig. 11, from which itcan be seen that the calculated densities agree well withthe experimental data. The deviations of the bubble pointpressures from experimental data for the CH4–C3H8–CO2

system (Webster and Kidnay, 2001) and the CH4–C2H6–


CO2 system (Davalos et al., 1976; Wei et al., 1995) areshown in Figs. 12 and 13, respectively, which are usuallywithin 5%.

4. ISOCHORES OF THE CH4–C2H6–C3H8–CO2–N2

FLUID MIXTURES

In the studies of fluid inclusions, the isochores (pressure-temperature relation at constant density and composition)are frequently used to estimate the trapping temperaturesand pressures, and they can be calculated from this EOSof the CH4–C2H6–C3H8–CO2–N2 fluid mixtures by com-bining with microthermometric and/or Raman spectromet-ric data. For fluid inclusions of unary systems, only thehomogenization temperature and homogenization modeare the needed parameters. If homogenization temperature(including homogenization mode) is obtained by themicrothermometric analysis, homogenization pressure anddensity (or molar volume) can be calculated from theimproved EOS of pure fluids by the Newton iterativemethod aforementioned in Section 2, and Eq. (7) can beused to calculate the isochores. For fluid inclusions of theCH4–C2H6–C3H8–CO2–N2 mixtures, experimental iso-chores have not been reported till now, so the predictiveEOS is the best choice to calculate the isochores of fluidinclusions. First, microthermometric and Raman analysisare done to obtain the homogenization temperature andfluid composition. Once the homogenization temperatureand fluid composition are known, the iterative algorithm

3300 400 500 600 700 8000

200

400

600

800

1000

P (b

ar)

T (K)

V m = 100 cm

3 ⋅mol-1

125

150

200250

300

xCH4= 0.2

xC2H6= 0.2

xC3H8= 0.2

xCO2= 0.2

xN2= 0.2

a

300 400 500 600 700 8000

200

400

600

800

1000

300250200150125

Vm = 100 cm

3 ⋅mol-1xCH4

= 0.6xC2H6

= 0.1xC3H8

= 0.1xCO2

= 0.1xN2

= 0.1

P (b

ar)

T (K)bFig. 14. Calculated isochores of the CH4–C2H6–C3H8–CO2–N2

system: T is the temperature, P is the pressure, x is the molefraction, and Vm is the molar volume in cm3�mol�1.

of Michelsen (1993) can be used to calculate the homoge-nization pressure and the homogenization density (or molarvolume) from this EOS. Then Eq. (25) can be used to calcu-late the isochores of the fluid mixtures. Fig. 14 showsthe calculated isochores of two compositions for theCH4–C2H6–C3H8–CO2–N2 mixtures. It can be seen thatthe calculated isochores are slightly curved and can beapproximated as straight lines.

5. CONCLUSIONS

A general Helmholtz free energy EOS of the CH4–C2H6–C3H8–CO2–N2 fluid mixtures has been developedby using four mixing parameters, from which the PVTx

and VLE properties can be obtained by thermodynamicrelations. The EOS can satisfactorily reproduce the experi-mental volumetric and vapor-liquid phase equilibrium dataavailable up to 623 K and 1000 bar, with or close to exper-imental accuracy. The average absolute deviation of densi-ties is below 0.4%, and the average absolute deviation of thebubble (dew) point pressures is about 2%.

The isochores of the CH4–C2H6–C3H8–CO2–N2 fluidinclusions can be calculated by combining with microther-mometric and/or Raman spectrometric data, and they canbe used to estimate trapping temperatures and pressures.The general EOS can calculate other thermodynamic prop-erties provided that the ideal Helmholtz free energy of fluids(Jaeschke and Schley, 1995) is combined, and can also beextended to predict the thermodynamic properties of themulti-component natural gases, including the secondaryalkanes and none-alkane components such as H2S, SO2,O2, CO, Ar and H2O. It should be noted that experimentalvolumetric data at high temperatures and pressures (e.g.,above 623 K and 1000 bar) are still lacking for the CH4–C2H6–C3H8–CO2–N2 fluid mixtures, and future experimen-tal research is needed to further improve this general EOS.

ACKNOWLEDGEMENTS

We thank Professor Hurai and the other two anonymousreviewers for their detailed and helpful comments, which improvedgreatly the quality of manuscript. This work is jointly supported bythe National Natural Science Foundation of China (Grant num-bers: 41673065 and 41373120) and the National Key BasicResearch Development Program (Grant number: 2015CB452606).

APPENDIX A. THE ITERATIVE ALGORITHM OF

MICHELSEN (1993) FOR THE VLE CALCULATION

AT GIVEN T AND P

The VLE compositions and densities at given T and Pcan be calculated by the iterative algorithm of Michelsen(1993) as employed in our previous studies (Mao et al.,2015a, 2016). Assuming that the total mole number of mix-ture equals unity, the bulk composition of component i is

xBulki , the mole number of vapor phase is nV, and the vapor

and liquid compositions of component i are xi and yi, thenxi and yi at a given T and P can be calculated in the follow-ing steps:


Step 1: Give a group of initial reasonable guess values

(between 0 and 1) for xBulki , xi and yi.

Step 2: First calculate the vapor and liquid densitiesfrom Eq. (25), then calculate the fugacity coefficient of com-

ponent i in vapor phase (uVi ) and liquid phase (uL

i ) from Eq.(27).

Step 3: Define an equilibrium factor ki ¼ yi=xi ¼ uLi =u

Vi ,

then calculate ki from uVi and uL

i .

Step 4: Calculate nV from the normalized equationPni¼1x

Bulki ðki � 1Þ=ð1 � nV þ nVkiÞ ¼ 0.

Step 5: Calculate xi and yi from equations

xi ¼ xBulki =ð1 � nV þ nVkiÞ and yi ¼ kixBulk

i =ð1 � nV þ nVkiÞ,respectively.

Step 6: Go to Step 2, and recalculate uVi , uL

i , ki, nV, xi

and yi in turn until the calculated nV remains constant.Then xi and y i are the VLE compositions, and the calcu-lated densities are the saturated densities. It should benoted that when T and P approach the critical point, initial

values for xBulki , xi and y i lie in a narrow range, which can be

frequently set by experience.

APPENDIX B. SUPPLEMENTARY MATERIAL

Supplementary data related to this article can be foundin the Supplementary data.docx file, where ten figures showthe deviations of the single-phase density, and the saturatedvapor and liquid compositions for each binary system(CH4–C2H6, CH4–C3H8, CH4–CO2, CH4–N2, C2H6–C3H8, C2H6–CO2, C2H6–N2, C3H8–CO2, C3H8–N2 andCO2–N2). Supplementary data associated with this articlecan be found, in the online version, at https://doi.org/10.1016/j.gca.2017.09.025.

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