Modeling of the Surface Tension of Multicomponent Mixtures with the

Ind. Eng. Chem. Res. 2005, 44, 3321-3329

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Modeling of the Surface Tension of Multicomponent Mixtures with the Gradient Theory of Fluid Interfaces Christelle Miqueu,* Bruno Mendiboure, Alain Graciaa, and Jean Lachaise Laboratoire des Fluides Complexes- UMR 5150 (CNRS-Total-UPPA), Universite´ de Pau et des Pays de l’Adour, B.P. 1155, 64013 Pau Cedex, France

The gradient theory of fluid interfaces is for the first time applied, without any lumping, to complex mixtures of more than three components, here made up of hydrocarbons and of a high proportion of carbon dioxide, nitrogen, or methane. It is combined with the volume-corrected Peng-Robinson equation of state. No adjustable parameters are used in the influence parameters mixing rule, which allows use of the gradient theory in a predictive manner. It gives very good estimates of the surface tension of the complex mixtures studied. In any case, it is found to be much superior to the traditional parachor method. The gradient theory is also used to compute the density profiles of the mixture components in the interface; it confirms that the low interfacial tensions of the systems studied are principally induced by a local accumulation of carbon dioxide, nitrogen, or methane in the interface. 1. Introduction Interfacial tension is a physical parameter which plays an important role in many processes in a number of industrial applications, such as heat transfer under boiling conditions, mass transfer during extraction, etc.. In the petroleum industry, processes such as gas condensate recovery, near-critical fluids recovery, and secondary and tertiary crude oil recovery, in particular, by gas injection, bring very low surface tensions into play. These surface tensions (i.e. liquid/vapor interfacial tensions) must be accurately known because of their dominating influence on capillary pressures, relative permeabilities, and residual liquid saturations. To avoid expensive measurements, one needs reliable theoretical estimates of them. There are many approaches for computing the liquidvapor surface tension of simple fluids or mixtures. To mention only a few, let us quote simple correlations, such as the parachor method1,2 and its derivatives, the corresponding-states principle,3,4 and other thermodynamic correlations.5-7 Let us mention also theories that take into account the density gradients in the interface, such as the perturbation theory,8 integral and density functional theories,9-12 or the gradient theory of fluid interfaces.13,14 The gradient theory originates in the work of van der Waals13 and was reformulated in 1958 by Cahn and Hilliard.14 It converts statistical mechanics of inhomogeneous fluid into nonlinear boundary value problems which must be solved to calculate density and stress distributions through the interface. It has been applied to a wide variety of fluids: hydrocarbons and their mixtures,15-25 polar compounds and their mixtures,20,23,26,27 polymer and polymer melts,28-31 near critical interfaces,32-36 and other liquid-liquid interfaces.37,38 Until now, its use has been restricted to binary and ternary mixtures because of the numerical difficulties encountered with mixtures containing a higher number of components.23 Among the other approaches * Corresponding author. Tel.: +33 5 59 40 76 79. Fax: +33 5 59 40 76 95. E-mail: [email protected].

that we have just cited, the only one which has been used for calculating the surface tension of mixtures of more than three components containing supercritical species is the traditional parachor method or one of its derivatives. But unfortunately, they proved to be very often inefficient,39 so because petroleum fluids are composed of many more than three synthetic components, calculation of their low surface tensions remains unresolved. The present paper is the third part of a systematic application of the gradient theory of fluid interfaces to pure fluid and mixtures. In the first paper,40 we derived a simple correlation to compute the influence parameters of nonpolar pure fluids, necessary to apply the gradient theory to these fluids. In the second paper,41 the gradient theory allowed us to predict correctly the surface tension of synthetic binary and ternary mixtures of these nonpolar pure fluids. Specific algorithms were proposed and validated. In the present paper, we intend to extend the application of the gradient theory to mixtures of more than three components. Mixtures made up of hydrocarbons and a high proportion of carbon dioxide, nitrogen, or methane are considered. These mixtures can be used to model the behavior of real petroleum mixtures, particularly during gas injection processes. The paper is organized as follows: In Section 2, we recall the main features of the gradient theory necessary to understand the method that we propose for calculating the surface tensions of these complex mixtures. In Section 3, we present our method, and we apply it to the chosen mixtures. Finally, we show how the computation of the density profiles through the interface can help us to better understand the surface tension behavior at high pressure. 2. The Gradient Theory for Multicomponent Mixtures 2.1. Outlines of the Theory. The gradient theory has been described extensively in the literature.12,14,18,42-44 Therefore, only the equations that are principally used in this work will be recalled. One is referred to our

10.1021/ie049086l CCC: $30.25 © 2005 American Chemical Society Published on Web 03/25/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005

previous article40 for a more detailed description and bibliographic analysis of the theory. In the absence of any external potential, the gradient theory states that the surface tension of the planar interface of a mixture is given by

σ)

∫-∞ ∑∑ i j +∞

cij

dni dnj dz dz

dz

(1)

where dni/dz represents the local gradient in density of component i. By analogy with the bulk energy parameters, aij ) (1 - kij)xaiaj, and as proposed for the first time by Carey,16 the crossed influence parameters cij are related to the geometric mean of the pure component influence parameters ci and cj by

cij ) (1 - βij)xcicj

(2)

βij being binary interaction coefficients. Stability of the interface requires βij to be included between 0 and 1.16 When βij ) 0, the mixing rule is reduced to the geometric mean. The pure component influence parameters could be obtained from a theoretical expression derived independently by Bongiorno et al.42 and Yang et al.;33 however, this expression is not suitable for the majority of the systems presenting a practical interest because it uses a direct correlation function of the homogeneous fluid, which is not easily accessible. Therefore, either a model must be used to estimate the influence parameters from other measurable or computable quantities, or these parameters must be obtained by fitting the surface tensions computed for pure components to the experimental ones. Computation of surface tension with eq 1 also requires knowledge of the gradients in density through the interface. These latter can be obtained by minimizing the Helmholtz free energy of the planar interface.

F)

[

1

i

j

]

dni dnj

∫-∞+∞ f0(n) + ∑∑2cij dz

dz

dV

(3)

So in the absence of an external potential, the Helmholtz free energy density of an inhomogeneous fluid, given by the gradient theory, is the sum of two contributions: the Helmholtz free energy density f0(n) of the homogeneous fluid at the local composition, n; and a corrective term, which is a function of the local density gradients. The equilibrium densities in the interface can be computed either by direct minimization of eq 3 or by solving the Euler-Lagrange equations derived from the minimum free energy criterion applied to eq 3. Assuming that the density dependence of the influence parameters can be neglected,17,40,45 the Euler-Lagrange equations are

∑j cij

d2nj dz

2

) µi0(n1, ..., nN) - µi ≡ ∆µi(n1, ..., nN)

be solved, together with the boundary conditions {ni(-∞) ) nVi ; ni(+∞) ) nLi }, where nVi and nLi are the equilibrium densities of component i in the vapor bulk phase and in the liquid bulk phase, respectively. However, using the geometric mixing rule for the influence parameters (i.e., βij ) 0 in eq 2) leads to a noticeable simplification. In this case, the system of N differential equations (eq 4) reduces to the system of (N - 1) algebraic equations.

xci∆µref(n1, ..., nN) ) xcref∆µi(n1, ..., nN)

i ) 1 ... N, i * ref (5)

These equations fix the densities of all but one component, the density of this latter one being a reference variable. Care must be taken in the selection of this reference variable. As was explained previously,41 it must be a monotonic function of z over the whole interface in order to generate the entire density profiles. In a previous article,41 we proposed physical arguments rather than mathematical ones to choose this reference variable. For mixtures of nonassociating compounds, in the case of a vapor/liquid interface, the free energy of the interfacial region can be minimized by a relative enrichment of the most volatile compounds. On the contrary, there should not be any enrichment in the liquid/vapor interface of the less volatile compound; its density would run continuously from its value in the vapor bulk phase to its value in the liquid bulk phase. This compound can, thus, be selected as the reference one. Previous applications of the gradient theory to binary and ternary mixtures has allowed us to validate this method.41 Therefore, we use it again in the present work. It is noteworthy that we have checked that the choice of the reference compound has no impact on the surface tension calculated with the gradient theory, since there is no enrichment of this compound in the interfacial region. Solving the system of algebraic equations (eq 5) allows to obtain ni(nref) for i * ref. Then, multiplying eq 4 by dni/dz, summing over i, and integrating yields

1

dni dnj

∑i ∑j 2cij dz

dz

) ∆Ω(n) ) Ω(n) - ΩB

(6)

with the grand thermodynamic potential Ω(n) ≡ f0(n) - ∑iniµi, and ΩB ) -P, P being the equilibrium pressure. From eq 6 and the chain rule of differentiation,

dni dnref dni ) dz dnref dz

(7)

one obtains

for i, j ) 1 ... N (4)

where µi0 ≡ (∂f0/∂ni)T,V,nj, µi being the chemical potential of component i in the equilibrium bulk phase and n, the number of components. To obtain the equilibrium densities through the interface, the set of differential equations (eq 4) must

dz )

x

1

dni dnj

∑i ∑j 2cij dn

ref

dnref

∆Ω(n1, ..., nN)

dnref

(8)

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Combination of eq 8 with eq 1 leads to the following expression for the surface tension.

σ)

∫nn

L ref V ref

x

dni dnj

∑i ∑j cij dn

2∆Ω(n1, ..., nN)

ref

dnref

dnref (9)

Thus, the only inputs necessary to use the gradient theory, once the vapor-liquid equilibrium (VLE) has been computed, are the Helmholtz free energy density of the homogeneous fluid f0(n1, ..., nN) (that can be derived from an equation of state) and the influence parameters of the various components. 2.2. The Equation of State. In this work, the PengRobinson equation of state46 is selected both to model the homogeneous fluid and to derive an expression for the Helmholtz free energy density required in the gradient theory. Such a combination of the gradient theory with the PR-EOS was originally presented by Carey et al.15 and since then was adopted several times in the literature.15-17,19-25,35,40,41,47 The Peng-Robinson equation of state is expressed as

P)

a(T) RT v - b v(v + b) + b(v - b)

(10)

where b is the covolume and a(T) is the energy parameter. For a mixture, these parameters are related to the ones of the pure fluids by mixing rules. In this work, the classical van der Waals mixing rules have been adopted,

∑i xibi

(11)

∑i ∑j xixj(1 - kij)xaiaj

(12)

b) a)

where xi is the mole fractions of component i in each phase and kij is the binary interaction parameters. The parameters of the pure fluid are given by

bi ) 0.07780

RTci Pci

(13)

R2Tci2 R(Tri) ai(T) ) 0.45724 Pci

(14)

with

R(Tri) ) {1 + mi(1 - xTri)}2

(15)

The coefficient mi is function of the acentric factor as

mi ) 0.37464 + 1.54226ωi - 0.26992ωi2

(16)

In 1978, Robinson and Peng48 proposed the following modification of eq 16 for components heavier than n-decane (that is to say ωi > 0.49):

mi ) 0.379642 + 1.485030ωi - 0.164423ωi2 + 0.016666ωi3 (17) Equation 16 remains valid when ωi e 0.49.

The Peng-Robinson equation of state is known to poorly represent the volumetric properties, especially the liquid volumes under saturation. As an attempt to improve the description of volumetric properties, we have used the concept of volume correction introduced by Martin et al.49 and developed by Peńeloux et al.50 (see the article of Peńeloux et al. for further details). The corrected volume vcorr is

vcorr ) v - d

(18)

where v is the molar volume computed by the original equation of state at T and P, and d is the volume correction. In this work, as for the covolumes, a linear mixing rule is applied to the volume corrections.

d)

∑i xidi

(19)

2.3. The Influence Parameters. In a previous paper,40 after having reviewed the models proposed in the literature, we derived a simple expression for the temperature dependence of the influence parameters. This expression is valid for nonpolar pure fluids when the gradient theory is used in combination with the PREOS. It describes the variation of the influence parameter with the “reduced temperature” ti ) 1 - T/Tci as follows

ci aib2/3 i

) Aiti + Bi

(20)

where bi is the covolume, and ai(T) is the energy parameter in the PR-EOS The coefficients Ai and Bi are merely correlated with the acentric factor of the component i by the relations

Ai )

- 10-16 1.2326 + 1.3757ωi

(21)

Bi )

10-16 0.9051 + 1.5410ωi

(22)

In this work, we use expressions 20-22 to compute the influence parameters of each component. 3. Application of the Gradient Theory to Multicomponent Mixtures 3.1. Numerical Resolution of the Problem. In a previous work,41 we applied the gradient theory to binary mixtures made up of a gas (carbon dioxide, methane, or nitrogen) and a hydrocarbon. We have shown that for all the gas/hydrocarbon pairs, using the geometric mixing rule for the influence parameter either has no impact on the computed surface tensions or gives the better estimation. Moreover, from previous literature results,16,41 we know that the geometric mixing rule for the influence parameters also involves the most efficient estimations for hydrocarbon/hydrocarbon pairs. Thus, for the systems under consideration in this work, only the geometric mixing rule is used for the influence parameters. In that case, the set of differential equations reduces to the following set of algebraic equations:

hi ) xci∆µref - xcref∆µi ) 0 i * ref

(23)

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The principal difficulty to apply the gradient theory lies in the resolution of this set of equations. This resolution can be performed as follows. First, the [nVref, nLref] interval is discretized with the step size

∆nref )

nLref

nVref

Nstep

(24)

The number of steps, Nstep, in the interface must be small enough to allow convergence at each step but not too small to avoid prohibitive time calculations for the whole of the N steps. For the systems of equations studied, this aim has been reached with Nstep ) 500 grid points. The resolution of eq 23 begins on the vapor side of the interface and is made step-by-step until the liquid side of the interface is reached. At each intermediate step, k, the system of algebraic equations (eq 23) is solved by the Newton-Raphson method.51 The correN ∂hi/∂nj δnj ) -hi is sponding matrix equation ∑j)1 solved by LU decomposition. The initial estimates of the densities, for i * ref are

ni(k) ) ni(k - 1) +

dni (k - 1)∆nref dnref

(25)

The (N - 1) derivatives dni/dnref are solutions of the following set of equations,

{[

∑ xci j*ref

∂µref ∂nj

- xcref

∂µi

] } dnj

∂nj dnref

∂µi ) xcref ∂nref ∂µref xci∂n i * ref (26) ref

which is solved by LU decomposition.51 The convergence criterion used at each intermediate step was 1/(N - 1)∑i*ref δni < 10-4. It proved small enough to have no impact on the value of the surface tension computed. The procedure ends when the liquid side of the interface is reached. The complete density profiles through the interface being obtained, the surface tension is computed with eq 9. The complete procedure is summarized in the algorithm of Figure. 1. 3.2. Surface Tension Estimations. The applicability of the gradient theory in combination with the PREOS is tested for mixtures consisting of hydrocarbons with a considerable amount of a volatile compound, such as nitrogen, methane, or carbon dioxide. These mixtures can be used to model the behavior of real petroleum mixtures, particularly during gas injection processes. An important condition to model surface tension is an accurate description of the phase behavior and especially the equilibrium densities in the bulk phases. As was indicated in Section 2, the volume-corrected PREOS was used to model the vapor-liquid equilibria of the mixtures considered here. For each mixture, the binary interaction parameters kij and the volume corrections di were fitted by using the experimental data to improve the representation of the equilibrium bulk properties, then the surface tension was computed with the algorithm described in Section 3.1. For comparison, the parachor method of Fanchi52 was also applied. This method, traditionally used in the

petroleum industry to model the surface tension of such complex mixtures, gives the surface tension as

(

FL FV Pai xi - yi i)1 ML MV N

σ1/4 )

∑

)

(27)

where ML and MV are the molecular weight of the equilibrium liquid and vapor phases, respectively. The parachors Pai were correlated by Fanchi52 to the critical parameters of the pure components. 3.2.1. N2/3 Hydrocarbons Mixture. Huygens et al.53 provide the experimental surface tensions and equilibrium densities at 373.15 K for pressures ranging from 330 to 400 bar for the system which contains 79% nitrogen, 6.9% methane, 8.5% n-butane, and 5.6% n-tetradecane. This synthetic quaternary mixture was used to model the behavior of the surface tension of a nitrogen/volatile oil system at typical conditions of North Sea oil reservoirs. For this mixture, we ignore both the equilibrium compositions and the bubble pressure, usually necessary to adjust the binary interaction parameters, kij. However, the PR-EOS is known to accurately describe the vapor densities of this kind of mixtures when a set of appropriate kij is used (volume corrections effect being negligible on the vapor densities). Here, with kN2-nC4 ) - 0.1778, the vapor densities can be computed with an absolute average deviation (AAD) of 0.3%. The use of volume corrections leads to an AAD of 0.4% on the saturated liquid densities. Thus, the equilibrium densities are fairly reproduced (see Figure 2), although we are not absolutely sure that the equilibrium compositions are correctly estimated because, to our knowledge, there are no experimental data for these compositions in the literature. Considering the conclusions that were drawn for binary N2/hydrocarbon mixtures in a previous work,41 the binary interaction coefficients for the influence parameters are set to zero for all the N2/hydrocarbon pairs. Moreover, from previous literature results,16,41 we know that the geometric mixing rule for the influence parameters also involves the most efficient estimations for the hydrocarbon/hydrocarbon pairs. Thus, the algorithm presented in Section 3.1 can be used to compute the surface tension with the gradient theory for this mixture. Figure 3 compares the computed surface tensions with the experimental data. Both the gradient theory and the parachor method of Fanchi underestimate the surface tension of this mixture, yet with an AAD of 14%, the gradient theory is consistently more accurate than the parachor method of Fanchi, for which the average deviation between prediction and experiment is 32%. 3.2.2. CH4/4 Hydrocarbons Mixture. To our knowledge, experimental surface tensions have never been published for mixtures of more than three components containing a high proportion of methane, yet some surface tension measurements have been performed by AEAT Reservoir Engineering on the system consisting of 80% methane, 14% n-butane, 4% n-heptane, 1.4% n-decane, and 0.6% n-tetradecane. This mixture of five components can actually be used to mimic the behavior of a real gas condensate. The ECL Technology Petroleum Engineering Laboratories (formerly of AEA Technology) has kindly provided us with the experimental


Figure 3. Surface tension versus pressure at 373.15 K for the N2/3 hydrocarbons mixture: [, experimental data;53 s, gradient theory of fluid interfaces; - - - -, parachor method of Fanchi.52

Figure 4. Pressure-density diagrams at 313.15 K for the CH4/4 hydrocarbons mixture: [, experimental data provided by AEAT Reservoir Engineering; - - - -, PR-EOS; s, PR-EOS with volume corrections. Figure 1. Schematic procedure for the calculation of the surface tension of a multicomponent mixture.

Figure 2. Pressure-density diagrams at 373.15 K for the N2/3 hydrocarbons mixture: [, experimental data;53 - - - -, PR-EOS; s, PR-EOS with volume corrections.

equilibrium bulk densities and surface tensions at 313.15 K for pressures ranging between ∼140 and 210 bar. With all binary interaction parameters kij equal to 0 and appropriate volume corrections, the equilibrium bulk densities can be reproduced with AADs of 2% in the liquid phase and 3% in the vapor phase. For comparison, Figure 4 depicts the experimental equilibrium densities together with the ones computed with the PREOS. As was mentioned for the N2/hydrocarbons mixture, previous literature results16,41 have demonstrated that

Figure 5. Surface tension versus pressure at 313.15 K for the CH4/4 hydrocarbons mixture: [, experimental data provided by AEAT Reservoir Engineering; s, gradient theory of fluid interfaces; - - - -, parachor method of Fanchi.52

the geometric mixing rule for the influence parameters is the most efficient for this kind of mixture, so we have used the algorithm described in Section 3.1 to compute the surface tension with the gradient theory. The estimated surface tensions are plotted in Figure 5 together with the experimental data and the values computed with the parachor method of Fanchi. Except for the lowest pressures (for which σ > 1 mN/m), the gradient theory provides estimations that are in excellent agreement with the experimental data. In comparison, the parachor method of Fanchi fails to predict the surface tension of this mixture, the average devia-

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Figure 6. Pressure-density diagrams at 322.04 K for the CO2/ 10 hydrocarbons mixture: [, experimental data;54 - - - -, PR-EOS; s, PR-EOS with volume corrections.

Figure 7. Surface tension versus pressure at 322.04 K for the CO2/10 hydrocarbons mixture: [, experimental data;54 s, gradient theory of fluid interfaces; - - - -, parachor method of Fanchi.52

tion between calculated and measured surface tensions being higher than 55%. 3.2.3. CO2/10 Hydrocarbons Mixture. The experimental surface tension data for this mixture are taken from Gasem et al.,54 who also provide the experimental equilibrium phase densities at 322.04 K and for pressures ranging from nearly 90 to 100 bar. This system is a synthetic mixture of 11 components containing an important amount of carbon dioxide; its composition is given in Table 1. The introduction of kCO2-alkane ) 0.15 Table 1. Composition of the CO2/10 Hydrocarbons Mixture54 component

molar composition (%)

methane carbon dioxide ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-decane n-tetradecane

2.874 91.681 0.251 0.332 0.515 0.344 0.272 0.412 0.419 2.477 0.424

for each alkane from methane to n-octane, kCO2-nC10 ) 0.11, and volume corrections allows a quite good description of the equilibrium densities, as is shown in Figure 6. Indeed, AAD is 0.5% in the liquid phase and 0.7% in the vapor phase. Vapor-liquid equilibrium being computed, the gradient theory has been applied to estimate the surface tension of this mixture. According to previous results, the binary interaction coefficients for the crossed influence parameters were set to 0 for all CO2/hydrocarbon pairs41 and for all hydrocarbon/hydrocarbon pairs.16,41 The estimated surface tensions are illustrated in Figure 7. The gradient theory allows estimations in good agreement with the experimental data (despite of a small overestimation for the lowest pressures), considering the fact that the surface tensions that have to be estimated are low (most of them lower than 1 mN/m). This range of values of surface tension is typically encountered in gas injection processes. In comparison, the systematic overestimation of the surface tension with the parachor method of Fanchi is much higher. 3.3. Density Profiles and Compositions in the Interface. The worth of the gradient theory is that it can also be used for computing interfacial properties other than surface tensions, such as density profiles,

Figure 8. Density profiles through the interface computed with eq 28 for CH4/4 hydrocarbons mixture at 313.15 K and 142 bar.

that are hardly accessible to experimental observation. Here, the gradient theory having permitted us to obtain satisfactory prediction of the surface tension of such complex mixtures, it is justified to suppose that the density profiles that were used to obtain these surface tensions are correct. Let us compute these density profiles. This information is achieved by integrating eq 8, which gives

z ) z0 +

∫nn (z) N 0 N

x

dni dnj

∑i ∑j cij dn

N

dnN

2∆Ω(n1, ..., nN)

dnN

(28)

z0 and nN0 being the coordinate and the density arbitrarily chosen as origins. This choice has no impact on the density profiles because it only leads to a translation of the coordinates. Equation 28 allows us to calculate the density profiles of each component through the interface. As an example, let us discuss the results obtained for the CH4/4 hydrocarbons mixture. Figure 8 depicts the density profiles of the components of this mixture at 313.15 K and 142 bar. Except for methane, densities run monotonically from their values in the vapor phase to their values in the liquid phase. They have the traditional hyperbolic tangent shape of ordinary interfaces, which is satisfying for the credibility of the method used. On the other hand, the density of methane does not vary monotonically: it goes through a maximum, which indicates a local enrichment of the interface


hydrocarbons mixture, the accumulation of nitrogen, whose solubility in the hydrocarbon-rich phase is very low, is located on the vapor side of the interface. 5. Conclusions

Figure 9. Density profiles of CH4 through the interface computed with eq 28 for CH4/4 hydrocarbons mixture at 313.15 K and different pressures.

The gradient theory of fluid interfaces has been applied successfully to the complex mixtures studied. It was combined with the volume-corrected PengRobinson equation of state. This application has been performed in a predictive manner, that is to say, with no adjustable parameters. It allowed very good estimates of the surface tensions of the mixtures at high pressures and high temperatures. It proved to be much more accurate than the parachor method of Fanchi. Furthermore, unlike the parachor method, the gradient theory allows one to compute the density profiles in the interface. A local enrichment of the interface in methane, nitrogen, or carbon dioxide has been found, and the high concentration of such a compound in the interface has been shown to be principally responsible for the low surface tensions encountered. The gradient theory is not very time-consuming, and it requires only easily accessible parameters, thanks to the correlation derived in a previous work for the influence parameters, so it will be advantageously used in industrial environments. Acknowledgment

Figure 10. Molar fraction of methane through the interface for CH4/4 hydrocarbons mixture at 313.15 K and different pressures.

in methane. This behavior must be linked with the minimization of the Helmholtz energy of the interface by a substantial amount of the componentsmethanes which has the lowest intrinsic free energy, that is to say, the component which is here the most volatile. This local enrichment of the interface has been analyzed with further details elsewhere for binary mixtures.39 When pressure increases, the interface thickness of the CH4/4 hydrocarbons mixture increases (Figure 9). This increase is accompanied by an increase of the methane density in the interface and a decrease of its local accumulation, which finally tends to disappear. We have plotted in Figure 10 the molar fraction of methane through the interface at several pressures. The local proportion of methane in the interface is always very important, at least higher than 60%. In the insert of Figure 10, the “average” molar fraction of methane in the whole interface (xjCH4 ) (1/Nstep)∑ixCH4(i)) is plotted as a function of pressure. Its increase with pressure reveals clearly that the concomitant decrease of the surface tension must be attributed to the increase of the quantity of methane in the interface. A similar behavior has been observed for the two other mixtures studied. For the CO2/10 hydrocarbons mixtures, a local enrichment of both methane and carbon dioxide was observed in the interface for each pressure under consideration. Unlike methane that accumulates on the vapor side of the interface, the accumulation of carbon dioxide is located nearly in the middle of the interface, which is in agreement with the higher solubility of carbon dioxide in the hydrocarbonrich phase in comparison with methane. For the N2/3

C. Miqueu gratefully acknowledges Total for the financial support of her PhD. We are very grateful to F. Montel (Total) for suggesting this project, for his helpful comments, and for the interest he has taken in this study. We gratefully acknowledge the ECL Technology Petroleum Engineering Laboratories (formerly of AEA Technology) for kindly providing us with the experimental data for the CH4/4 hydrocarbons mixture. Nomenclature a ) attractive parameter in the EOS (J‚m3/mol2) A, B ) coefficients in the correlation of the influence parameters AAD ) absolute average deviation (%) b ) covolume in the EOS (m3/mol) c ) influence parameter (J‚m5/mol2) d ) volume correction (m3/mol) F ) Helmholtz free energy (J) f ) Helmholtz free energy density (J/m3) k ) binary interaction parameter for the attractive parameter in the EOS M ) molecular weight (kg/mol) n ) number density (mol/m3) N ) number of components Nstep ) number of calculation steps P ) pressure (bar) Pa ) parachor ((N/m)1/4‚m3/mol) R ) ideal gas constant (J‚mol-1‚K-1) T ) temperature (K) t ) “reduced temperature” (1 - T/Tc) Tr ) reduced temperature (T/Tc) v ) molar volume (m3/mol) V ) volume (m3) x, y ) mole fractions z ) position in the interface (m) Greek Letters β ) binary interaction parameter for the influence parameter

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µ ) chemical potential (J/mol) F ) density (Kg/m3) σ ) surface tension (N/m) ω ) acentric factor Ω ) grand thermodynamic potential (J/m3) Subscripts 0 ) homogeneous state B ) bulk c ) critical corr ) corrected i, j ) component N ) number of component r ) reduced ref ) reference variable Superscripts L ) liquid V ) vapor 0 ) at equilibrium

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Received for review September 17, 2004 Revised manuscript received February 8, 2005 Accepted February 25, 2005 IE049086L

Modeling of the Surface Tension of Multicomponent Mixtures with the

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