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Ind. Eng. Chem. Res. 2004, 43, 638-644
Prediction of the Natural Hydrocarbon Vapor Phase PVT Properties Angeles Yackow* and Henri Planche Unite´ Chimie et Proce´ de´ s, Ecole Nationale Supe´ rieure de Techniques Avance´ es, 32 bd Victor 75739 Paris Cedex 15, France
This paper presents a new model for the prediction of the compressibility factor Z ) PV/RT of natural gases, either dry or condensate gases. The only requirement is the knowledge of a lumped composition. The model is relevant for temperatures and pressures that both fit petroleum engineering surface design and reservoir simulation. Predictions are expected to be within 1.7% or less for compositions including the most usual methane rich gases, but also the few cases where the light hydrocarbons are replaced by major amounts of mineral gases such as N2, CO2, and H2S (special reservoirs or reinjection processes). This model was fitted on 207 natural gas data and is compared to the Lee-Kesler (LK) equation on a set of 220 natural gas data. For natural gas, the new model better predicts the compensations between the mixture second virial matrix coefficient contributions. As a consequence, below 500 bar the two models give similar precision accuracy except for the gas compositions where LK poorly returns the mixture mean second virial coefficient. In this latter case, the new model is much better. 1. Introduction Petroleum thermodynamics trigger important physical controls in production, transport, and refining of petroleum fluids. Phase behavior, fluid composition and density are required to determine how much oil or gas is present, how much can be recovered, how fast it can be recovered, and the reservoir management and strategies. These properties should be also modeled for downstream engineering: surface installations are designed and sized on the basis of the fluid physical characteristics. The equipment installed (pumping systems, storage) must guarantee finding the product specifications while minimizing energy consumption and required investment. Petroleum fluid properties can be obtained by the following: (1) laboratory analysis, measured data on fluid samples at reservoir conditions; (2) black oil correlations, calculated data using validated correlations on specific oil databases; and (3) compositional modes/ equation of state (EOS), calculated data using consistent thermodynamic models based on fluid compositions. One of the fluid properties required both for reservoir engineering calculations and for dimensioning surface pumping systems is the gas compressibility factor Z. Many models1-18 have been proposed for petroleum thermodynamics. But most of them only provide a mathematical formalism and empirical adjustable parameters to be fitted, case by case, on field-specific experimental data. The trend is thus to enlarge their extrapolation ability in order to provide more predictive tools. This paper comes within this objective, although it is restricted to the prediction of the compressibility factor Z ) PV/RT of vapor phases. 2. Theoretical Section 2.1. Lumping Compositions and Experimental Data Sets. A petroleum fluid is a complex mixture of * To whom correspondence should be adressed. Tel.: +33 145525427. Fax: +33 145528322. E-mail: angeles.yackow@ ensta.org.
hydrocarbons, sulfur, and oxygen bearing hydrocarbons and variable amounts of minerals: acid gas (CO2 and H2S), nitrogen, and water. Condensates and oil prone fluids contain thousands of molecular species. It is thus impossible to handle that complexity in engineering models. Lumping compositions is necessary. The condensate gas main characteristic is that bottomhole temperature isotherm cuts the fluid dew point curve. This means that depletion during the production process results in the deposit of condensate. That liquid build-up focuses lumping compositions on the principle of a volatility hierarchy, defining four major classes: group I or supercritical gases, C1 and N2; group II or subcritical gases, CO2, H2S, C2, C3, iC4, C4; group III or volatile liquids, C5-C10 and H2O; group IV or heavy liquids, C11+. Note that natural alkenes, alkynes, or cyclic hydrocarbons with less than five carbons are nearly absent from natural gases. Thus, the C2 to C4 fraction only contains four saturate molecular species. Group I-IV components can be found either in vapor or liquid phases. Proportions depend on the pressure and temperature conditions; i.e., in general, group IV is nearly absent in the vapor phases below 150 bar for typical temperatures of petroleum production processes. For higher pressures, both vapor and liquid phases are present, and even group IV could be significant in a nearly critical vapor. The composition lumping we propose for our compressibility factor modeling includes in molar percent each molecular species of groups I and II; carbon to carbon lumped fractions for group III, C5, C6, C7, C8, C9 and C10; and the lumped group IV The most usual procedure for building a thermodynamic property mixture model is to first represent each pure component behavior in the full extent of the available data. Then a mixing rule is defined to be validated on model binary or ternary mixtures, and finally, the ability of that mixture rule to predict real complex mixtures is discussed. This procedure implicitly assumes that the complex mixture phenomenology is straightforward and predictable once the pure component or model mixture phenomenology is identified.
10.1021/ie030450a CCC: $27.50 © 2004 American Chemical Society Published on Web 12/13/2003
Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 639 Table 1. Data Description data group
description
no. of points
condensate gasa (1984-2001) Biswas et al. (1990)21 Matthews et al. (1942)22 Reamer et al. (1944)23 Reamer et al. (1951)24 Angus et al. (1978)25 Angus et al. (1977)26 Din (1961)27 Angus et al. (1976)28
Experimental Data PVT reports from TOTAL (32 data setsb) PVT of 2 simulated natural gas (4 data sets) Z of 9 natural gases (11 data sets) PVT of 4 mixtures of CH4 and CO2 (33 data sets) PVT of 5 mixtures of CH4 and H2S (37 data sets) Z of CH4 (9 data sets) Z of N2 (8 data sets) PVT of N2 (7 data sets) Z of CO2 (10 data sets)
207 68 152 336 277 58 169 91 56
ethane29 propane29 butanes (i- and n-)29
Pure Component Data NIST modeling of exptl data (10 data sets) NIST modeling of exptl data (10 data sets) NIST modeling of exptl data (14 data sets)
89 76 95
pressure range (bar)
temp range (K)
1-519 1.2-160 34-409 1-700 1-700 1-400 1-500 1-500 1-500
298-430 298 & 323 278-367 311-511 278-444 225-473 250-600 220-550 373-580
1-500 1-500 1-500
200-573 248-600 220-600
a TOTAL oil company confidential data. b A data set is defined as a group of data where either the temperature or pressure has constant value.
Generally, this assumption is quite badly supported, meaning that some information of the complex mixtures’ composition class is not implicitly included in the simple mixtures or pure components’ composition classes. For this reason, we proceed here through an inversion of the most usual model building procedure. We first identify an efficient mixture rule inside the restricted natural gas subclass; then, we extend these rules to pure components and model mixtures. Thus, natural gases could be grouped into a specific composition class because petroleum gas mixtures exhibit a limited range of theoretical possible compositions. It is well-known that the most usual tendency of abundances is C1 . C2 > C3 > C4, and so forth, where single methane molar percent is at least 60-70%, and C2, C3, and C4 mass fractions are first order, similar to condensate gases. In the C6 to C10 cut of condensate gases, the aromatic fraction composition is generally between 10% and 20%. Some C11+ components (resins and asphaltens) can hardly dissolve in a gas phase. Therefore, considering the compressibility factor of pure methane vapors, we have first defined an efficient mixture rule to return accurate values for an experimental database (reports of TOTAL petroleum company), scanning the most commonly observed natural gases. This database contains vapor phase properties measured during standard PVT (presssure-volumetemperature) experiments on bottom hole fluid samples. These standards are called CMD (constant mass depletion) and CVD (constant volume depletion). For the socalled bottom hole dry gases, the depleted vapor’s composition is conservative. For bottom hole condensate gases, retrograde condensation happens during CMD and CVD. The depleted vapor’s composition thus evolves with pressure. Therefore, industrial data sets are expected to be more informative than literature databases because they scan a wider variety of compositions. Second, the mixing rule was reviewed to accurately represent scarcer, but possibly observable, cases, i.e., where the mineral gases N2, CO2, and/or H2S turn out to be major, or the case where methane is highly depleted and replaced by ethane or even propane as the major hydrocarbon. For the pure components ethane, propane, and butanes, we did not turn back to the original experimental data. We used SBWR calculated values of Z fitted by NIST29 (National Institute of Standards and Technology). Table 1 presents the experimental and the pure component calculated data used for fitting the new model.
Figure 1. Equation’s domain of validity.
Figure 2. Compressibility factor of a condensate gas (77 mol % C1, 10.5 mol % C2, 8.4 mol % C3-C5, 0.3 mol % C9+, 0.5 mol % N2, 1.1 mol % CO2). Data obtained from CVD test (TOTAL).
In Figure 1, the temperature-pressure plane exhibits the different data sets we have used and the area of petroleum production interest. The black solid symbol (b), corresponding to 1 bar and 298.15 K, outlines the standard storage conditions. Generally, petroleum operating conditions could be represented by a diagonal drawn between any bottomhole conditions (O) and the standard storage conditions. Figure 1 points out that our natural gas database is limited to about 500 bar, whereas there would exist gas-phase reserves at higher pressures (up to 2000 bar). It should be also noted that
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no experimental data containing water was included. Actually, we have found some data20 where added water is present, but PVT measurements are then realized on a mixture of vapor phase and aqueous liquid phase. Furthermore, the true water content in the restricted vapor phase is not available. 2.2. Predictive Modeling of the Gas Compressibility Factor. State-of-the-Art. Many EOS have been proposed1-15,17 to represent petroleum mixture properties. All of them were originally developed for chemical engineering, then extended for petroleum engineering. They can be separated into two main classes. The socalled “cubics” are derived from van der Waals (1873). On this track, the Peng-Robinson5 or Redlich-KwongSoave1-3 equations are probably the best known for petroleum engineers. These equations allow us to represent experimental data on phase equilibrium compositions (P, T, x, y), but they are quite poor for volumetric properties. Several modifications have been proposed for overcoming this problem, among them we find Peneloux’s translation10 for the liquid phase volume. But this correction is not relevant to our work, because we only deal with vapor phases. The second class of EOS is derived from the Benedict-Webb-Rubin equation.11 The original equation is purely heuristic, and it claims to have only 8 empirical parameters for a pure component, but 20-35 empirical parameters should be fitted to represent every thermodynamic function using partial derivatives of the same numerical model. On the track of BWR, the Lee-Kesler4 (LK) equation is the best known among petroleum engineers. This equation accurately represents pure component phase behaviors. But in the case of mixtures, it fails to represent VLE (vapor-liquid equilibria) compositions, although it is accepted as the most accurate not only for representing, but for predicting, vapor volumetric properties. A natural gas focused LK equation improvement was thus chosen as the best state-of-the-art reference to this work. Lee and Kesler proposed to express the compressibility factor of any hydrocarbon using methane as the first reference Z(0) (nil acentric factor) and normal octane for the second one Z(r) (acentric factor ) 0.3978):
Z ) Z(0) +
ω
(Z(r) - Z(0))
(r)
(1)
ω
Lee-Kesler’s model gives satisfactory results for the calculation of volumetric properties and for the departure from ideality of enthalpy of pure substances. On the other hand, it is only rarely used for calculating their VLE. It can be applied to nonpolar and slightly polar substances. The success of this method depends on the accuracy of the values of the critical temperatures, critical pressures, and acentric factors. This equation can be also applied to mixtures, by use of empirical mixing rules on the acentric factor, and by defining pseudocritical properties for any mixture. The simplest of these rules is that of Kay: a simple molar weighting of critical properties of the constituents. One of the best rules known for hydrocarbons is that proposed by Lee and Kesler. We’ll prefer in this work the version revised by Plo¨cker.9 This mixing rule introduces an adjustable binary parameter, kij, characteristic of the i-j binary. Given the thousands of molecular components in petroleum fluids, an equation that needs a matrix of binary parameters associated
with a classical kij mixing rule requires some predictive method in order to be numerically tractable. Many authors proposed different ways of predicting and/or calculating this parameter. Depending on the binary mixture, the parameter kij could be a function of temperature, molecular mass, solubility, or acentric factor, or it could be just a constant. Montel19 suggests three correlations that are the most suitable for calculating the binary coefficients for reservoir fluids, and these are the ones that we have used.
hydrocarbon-hydrocarbon: kij ) 0.93722 + 0.04372Q - 0.0007202Q2 (2) nitrogen-hydrocarbon: kij ) 0.94733 + 0.03675Q - 0.0003845Q2 CO2 or H2S-hydrocarbon: kij ) 0.90751 + 0.01685Q + 0.0002778Q2 where
Q)
TcjVcj TciVci
(3)
Equation 2 implies that nine empirical parameters have been fitted using petroleum gas experimental data for completion of the “predictive” version of the LeeKesler model we used as the state-of-the-art reference. 2.3. The PVT Modeling of Natural Hydrocarbons Vapor Phases. 2.3.1. First Draft of the Model: Restricted Natural Gases Modeling. In most natural gases, the molar percent of methane exceeds 60% or even 70%. Thus, we propose to consider the functional PVT for a natural gas as a deformation of the functional PVT of pure methane vapors. Lee and Kesler suggested a similar idea, with a deformation principle based on the corresponding states. Here, the model’s principal characteristic is that the PVT functional relation for a methane-prone natural gas could be modeled as a phase containing a pure free methane vapor embedding a nanoaerosol of higher molar mass condensed components. Thus, at constant pressure, the volume V of the pure methane functional should be replaced by V*. We assume that the difference (V* - V) is the vapor phase fraction occupied by condensed components. Postulating the density of these condensed C2+ components to be first-order independent of their molar mass, (V* - V) is temperature dependent and turns to zero for pure methane, and the weighting factor of any component i is expected to be first-order proportional to its mass, by preference to its molar percent
(V* - V) )
1 AxC2+ T*2
(4)
where
T* ) T exp[axC2+]
(5)
10
xC2+ )
∑1 (i - 1)xi + 16xC
11+
(6)
with xi the molar percent of the cut i (for i ) 4 we consider the sum of the molar percent of n-butane and i-butane, the same for i ) 5). The numerical value 16
Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 641
in eq 6 is a mean carbon number for the condensate C11+ fractions. Fitting this value did not significantly improve the model. Because most of natural gases contain at least some percent of the mineral gases nitrogen and CO2, we generalize, respectively, eqs 4 and 5 as
1 (V* - V) ) 2[AxC2+ + BxN2 + CxCO2] T*
(7)
Z(P,T) )
PV ) RT
Ψ1(T*) + Ψ2(T*)exp[Ψ3*(T*)] +
V* - V ) AxC/ 2+ + B C
(8)
By replacing V and T by V* and T* in the BWR equation for pure methane, and fitting the six empirical constants a, b, c, A, B, and C for our 207 H2S free natural gas data (a mean 34 experimental points per empirical parameter), the database is fitted within 1.8% in the whole temperature and pressure range. This is markedly efficient for such a simple mixing rule. But, this model is much poorer for vapor phases of pure components except methane and ethane, or for most model binary mixtures (5-10% accuracy). This inconsistency materializes the fact that extrapolating a formal mixing rule on a component i from a system where molar percents of components i are typically below 5-10% (infinite dilution type) to a system, where those i components are the mixture’s majors, is by no means obvious. 2.3.2. Final Extension to Pure Components and Model Mixtures. Keeping our modeling of the PVT functional for the natural gas class, the formalism is also extended to represent pure component vapors or simple model mixtures. This gives the final expressions we propose for the compressibility factor of natural gases. Pure methane data are first used to calculate Z ) PV/RT as a function of variables P and T for the restricted vapor phase, instead of the more usual variables V and T. It comes for pure methane
PV ) Ψ1(T) + Ψ2(T) exp[Ψ3(T)] (9) RT
ZCH4(P,T) )
T*2T
(13)
where
and
T* ) T exp[axC2+ + bxN2 + cxCO2]
P(V* - V)
( ) { T* 100
0.563
( ) T* 100
0.522
( ) T* 100
ΣCO2 + D
[
ΣN2 + 1.853
ΣH2S +
]
PT -81.070 3/2 358.226 exp P ΣCO2 + T* T*2
[
( ) ( ) ( ) ( )
xC/ 2+ ) x2
T*
-4.103
+ 2x3
100
3xi4
T*
(14)
-0.854
T*
+
100
-0.379
100
] }
-75.548 3/2 P ΣH2S T*2
372.540 exp
+ 3xn4
T*
-0.584
+
100
10
∑5 (i - 1)xi + 16x
11+
(15)
The Σ functions have been empirically designed in order to allow for a good integration of natural traces, model mixtures, and pure component data on the mineral gases, nitrogen, carbon dioxide, and hydrogen sulfide.
ΣN2 ) xN2[1 + 7.922 exp(-2.24010-2xN2)] ΣCO2 ) xCO2[1 + 5.224 exp(-1.46910-1xCO2)]
(16)
ΣH2S ) xH2S[1 + 1.252 exp(-6.24410-2xH2S)] Finally the definitions for T* and Ψ3* are
T* ) T exp[axC//2+ + bΣN2 + cΣCO2 + dΣH2S] (17)
where
Ψ1(T) ) 1 + P
[(
[
Ψ2(T) ) P γ0 +
[
Ψ3(T) ) P 0 +
)
)]
(
β1 β2 R1 R2 P + 2 + β0 + + 2 T T 1000 T T
(
)]
γ1 γ2 δ 1 δ2 P + + δ + + T T 1000 0 T T
(
)]
1 2 κ1 κ2 P + 2+ κ0 + + 2 T T 1000 T T
xC//2+ ) 1.101x2 + 2.118x3 + 2.778xi4 + 2.892xn4 + 10
∑5 (i - 1)xi + 16x
(10)
11+
(11)
(12)
Functions Ψ1, Ψ2, and Ψ3 are nondimensional; therefore, the units of the empirical parameters R, β, γ, δ, , and κ should be consistent with this condition. The numerical values consistent with Kelvin and bar units for temperature and pressure are given in Table 2. This expression for pure methane is then transformed into an expression capable of calculating the compressibility factor of condensate gas, model mixtures, and pure components. The procedure consists of changing T into T*, Ψ3 into Ψ3*, and V into V* according to
(18)
1 [0.899x2 + 0.037x3 T* 0.331xi4 - 0.359xn4 - 1.861ΣN2 + 0.112ΣCO2 +
Ψ*3(T*) ) Ψ3(T*) +
1.609ΣH2S] (19) Explicit numerical coefficients in eqs 14, 15, 18, and 19 have been fitted only using pure component data. Explicit numerical coefficients in eq 16 are mainly sensitive on pure and binary data. All of these explicit numerical coefficients are consistent with Kelvin and bar units. Only parameters a, b, c, A, B, and C are fitted on the class of natural gases. Parameters d and D for H2S are fitted on model mixtures containing H2S because none of our 207 natural gas data contained information on hydrogen sulfide behavior. Thus, finally,
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Table 2. Constants for the Functions Ψi (i ) 1-3) R γ
β δ κ
0 [bar-1]
1 [bar-1K]
2 [bar-1K2]
-1.838 × 10-3 1.632 × 10-3
7.907 × 10-1 1.436 -2.776
-224.040 -468.560 575.870
0 [bar-2]
1 [bar-2K]
2 [bar-2K2]
1.271 × 10-3 -7.120 × 10-3 -3.781 × 10-2
-1.399 5.346 32.525
378.370 -812.530 -6959.200
Table 3. Parameters Fitted for Natural Gases parameter
value
A B C D a b c d
65.58 21.08 -118 -56 -3.124 × 10-3 1.746 × 10-4 -3.211 × 10-3 -3.737 × 10-3
Table 4. Comparison between Lee-Kesler and Results from This Work RMS% data group TOTAL PVT data Biswas et al. (1990) Matthews et al. (1942) Reamer et al. (1944) C1-CO2 Reamer et al. (1951) C1-H2S methane ethane propane butanes (i- and n-) nitrogen carbon dioxide all the data a
Fitted data.
a
no. data Lee-Kesler points EOS this work 207 68 152 342 222 58 90 76 95 260 56 1626
4.65b 1.02b 2.25b 1.35b 0.75b 0.42a 2.37b 4.24b 0.93b 0.75b 1.15b 2.30b
1.84a 1.05b 2.16b 1.49a 2.59a 0.58a 1.33a 1.64a 1.77a 0.76a 0.84a 1.65
Predicted data.
only 6 parameters have been fitted using experimental information on natural gases. They are presented in Table 3. The formal definition of these 6 parameters is very close to that experienced in the first attempt to build a natural gas mixing rule. 3. Results Predictions of the LK equation have been calculated for comparison on the same data sets. Defining the rootmean-square deviation (RMS) as
x
RMS% ) 100 qi is defined as
N
qi2 ∑ i)1 N
(20)
Figure 3. Lee-Kesler’s systematic deviation trend correlated to cut C3 + C4 + C5 (a), to cut C9+ (b).
qi ) (experimental value - calculated value)i (21) and N is the number of data points. The qi functions were calculated for all the data used during this work, meaning natural gas, model mixtures, and pure components. Our model was fitted over 1406 data (condensate gas, 207; model mixtures, 564; and pure components, 635), and it was tested in prediction for 220 natural gas data. In Table 4, we present a summary of the comparison between our model and Lee-Kesler’s. The mean RMS on the total data set is lower for our model compared to LK. More detailed results show that the differences between the two models are more important on mixtures than on pure components. Figures 2-4 focus on several cases extracted from the natural gas data. These figures show that even in the mid-pressure region where the mixture second virial coefficient still controls the model’s predictions, LK may be poorer than the new model. This means that the second virial coefficient parameter matrix for mixtures is poorly represented by LK. For some natural gas compositions, these wrong coefficients cancel their influence so that the LK prediction is good. But occasionally they badly equilibrate, and the LK prediction is then poor. A systematic deviation tendency can be identified for the LK equation in Figure 3. The natural mixtures’ mean second virial coefficient is increasingly too high
Figure 4. New model’s trend with cut C3 + C4 + C5 (a), with cut C9+ (b), for comparison with LK.
for rising contents in the fraction C3 + C4 + C5. Simultaneously, it is increasingly too small for rising contents of the C9+ fraction. Of course, between the two there is a mean path for LK accurate predictions. No such systematic deviation tendency has been identified with the new correlation (Figure 4). We have also checked the pure components’ second virial coefficient representation versus temperature for the two models. With our model, the main deviation occurs with ethane. For a typical natural gas (C2 molar percent 10%)
Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 643
at 100 °C, this induces a 0.001 absolute shift on Z at 100 bar. With Lee-Kesler’s model, the main deviation is on methane, inducing a 0.005 shift. For the gas compositions where LK is poorer, discrepancies with experimental data may reach a decade higher than what was inherited from the pure components’ fits. No doubt therefore that LK fails on some of the nondiagonal coefficients of the mixture virial coefficient matrix. Considering the function (Z - 1)/P at constant composition and temperature, it should linearly turn to a constant value when decreasing pressure to zero. Plotting values calculated from the experimental data sets allows the estimation of the effective experimental data precision. This value is within 0.2-0.6%. With a mean 1.2% accuracy for fitted data, and then turning up to 1.8-2% for predictions, the new model still does not reach the experimental accuracy. Considering the mean accuracy of pure predictions in Table 4, for both of the two models, it comes out that for compositions where LK is good (at 1% deviation), the new model gives similar accuracy. 4. Conclusion The new model’s good accuracy for representing both middle and high pressure regions for natural mixtures and the strikingly small number of parameters which have been fitted for that purpose (4 parameters for N2 + CO2, and only 2 for hydrocarbons) give promise of reliable extrapolations out of the original database. Which of the two models, the LK equation or the new one, is best for predictions? The LK correlation of Montel19 was fitted on a TOTAL database, but not the same one we used in this work for fitting our model. At first sight, this easily supports why in Table 4 the mean LK precision on TOTAL data is 4.65 (LK predictions) whereas our precision is 1.84 (new model fit). And this seems furthermore confirmed when using the two models for pure predictions: their accuracy is similar. Results reported here would thus not allow us to answer the question of the best model. But taking into account more details of the physical and mathematical context, it comes that the question could be decided. As pointed out in Figures 3 and 4, the poorer return of the LK model on our TOTAL database is directly correlated for any pressure to the poor prediction of the second virial coefficient. Natural gases alternatively contain either too much gasoline (C3-C5) or too much C9+ condensates out of a mean path composition where LK predictions are right. Although our database scans the most common natural gas compositions, we do not denote any systematic deviation trend with the new equation. These facts would mark what happens. The published reference of LK we used here has been fitted by Montel on an original natural gas data set. But either the gas composition domain experienced in this original data was smaller, so some of the LK parameters have been poorly determined, or the composition field in the LK fitting had a similar extent as the one we used for the new model. In the latter case, LK would poorly represent Z for some more fundamental reason on its mixture rule design. Whatever the very reason for the poorer predictions of LK reference in this work, we can conclude that the natural gas composition domain used for fitting the original LK was either mathematically equivalent or included in the composition field of the TOTAL database
used here. As a consequence, which of the two models is best for predictions in petroleum industry? Seemingly the one which does not give systematic deviations for some local parts of the widest composition field experienced. Thus, the new model is better than the stateof-the-art reference LK. Acknowledgment The authors wish to thank the CNRS, and TOTAL for the condensate gas experimental database supply. Nomenclature a, b, c, d ) adjustable parameters of our equation A, B, C, D ) adjustable parameters of our equation kij ) binary interaction parameters P ) absolute pressure, bar T ) temperature, K V ) volume, m3/mol x ) molar percent Z ) compressibility factor Greek Letters R, β, γ, δ, , κ ) constants in Ψ functions ω ) Pitzer acentric factor Ψ, Σ ) auxiliary functions of our equation Subscripts c ) critical property i, j ) component i, j Superscripts 0 ) simple fluid (Lee-Kesler equation) r ) reference fluid (Lee-Kesler equation)
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Received for review May 27, 2003 Revised manuscript received September 30, 2003 Accepted October 9, 2003 IE030450A
