Modeling Phase Equilibria of Asymmetric Mixtures Using a Group

group contribution relations inspired by the Lorentz−Berthelot combining rules. ..... Two-thirds of the experimental database is composed of met...
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Ind. Eng. Chem. Res. 2008, 47, 8847–8858

8847

GENERAL RESEARCH Modeling Phase Equilibria of Asymmetric Mixtures Using a Group-Contribution SAFT (GC-SAFT) with a kij Correlation Method Based on London’s Theory. 1. Application to CO2 + n-Alkane, Methane + n-Alkane, and Ethane + n-Alkane Systems Dong Nguyen-Huynh, Jean-Philippe Passarello,* and Pascal Tobaly Laboratoire d’Inge´nierie des Mate´riaux et des Hautes Pressions (LIMHP), CNRS UniVersite´ Paris 13, 99 aVenue J. B. Cle´ment, F-93430, Villetaneuse, France

Jean-Charles de Hemptinne Institut Franc¸ais du Pe´trole, 1 & 4 aVenue de Bois-Pre´au, 92852, Rueil-Malmaison Cedex, France

Here, a group contribution statistical associating fluid theory equation of state (SAFT EOS) (GC-SAFT) proposed earlier by our group (Tamouza et al., Fluid Phase Equilib. 2004, 222-223, 67-76) is extended to some asymmetric systems, using a method for correlating the kij binary parameters, using only pure compound parameters. The method is inspired by London’s theory of dispersive interactions and correlates the kij values to the “pseudo-ionization energies” of compounds i and j (denoted as Ji and Jj, respectively). A group contribution for the latter parameters is also proposed, in view of obtaining a more-predictive model. Correlation tests of phase equilibria are conducted on some CO2 + n-alkane systems. Using the parameters thus obtained, the phase envelopes of other CO2 + n-alkane systems, as well as methane + n-alkane and ethane + n-alkane systems, were fully predicted. Correlation and predictions are qualitatively and quantitatively satisfactory. The deviations are within 4%-5% (i.e., comparable to those obtained on previously investigated systems). 1. Introduction Recently, our research group developed a group contribution method, combined with three different versions of the statistical associating fluid theory equation of state (SAFT EOS), to predict the fluid-phase equilibria of systems of interest in the petroleum industry.1,2 Satisfactory predictions (kij ) lij ) 0) were obtained for various systems containing n-alkane, 1-alkanol, aromatic and polyaromatics compounds and alkyl esters.2-7 The approach was also applied to mixtures of hydrocarbons with small specific molecules, i.e., H2 and CO2.8 However, in the latter case, nonzero kij parameters were needed to obtain realistic computations of the phase envelope, apparently diminishing the prediction capability of our approach. However, as in the case of the EOS parameters, the kij values could also be correlated by a group contribution, using (small molecule)-(group) binary parameters, such as kH2,group k and kCO2,group k. But if this method applies well for predictions in a given series of mixtures (the CO2 + n-alkane series, for instance), it is not however easily transferable to other systems. For instance, a model for methane + n-alkane model would require knowledge of the kmethane,n-alkane value, which cannot be estimated from kCO2,methane and kCO2,n-alkane parameters (here, methane is a group). This could be done only via the regression of methane + n-alkane VLE data. It seems to be clear that this method requires the availability of a rather large amount of mixture data. For this reason, it may be only of little help when dealing with systems for which data are few or missing. * To whom correspondence should be addressed. Tel.: +33 1 49 40 34 06. Fax: +33 149 40 34 14. E-mail: [email protected].

This fact motivated the search for another way of correlating and predicting the binary parameter kij that is needed in the case of asymmetric systems. Based on London’s dispersion interaction theory, it is possible to obtain such theoretical or semi-theoretical expressions for correlating kij. As discussed below, they involve only pure compounds parameters (diameter, ionization energy) to be taken from literature, or otherwise adjusted. The total number of parameters is then dramatically decreased: a n-dimensional vector of pure parameters is needed this time, whereas the earlier group contribution kij method requires a n2-dimensional matrix of parameters. Such an approach, based on London’s theory, therefore requires less experimental data and a priori enables one to make predictions on a larger number of systems. To test the approach, it was decided, as a first step, to investigate the phase equilibria of carbon dioxide (CO2) + alkane mixtures. These systems were chosen because CO2 is the most widely used supercritical fluid, because (i) it has a relatively low critical temperature (close to 31 °C, which makes it highly suitable for processing heat-sensitive materials), (ii) it is a nontoxic, nonflammable, inexpensive compound, and (iii) it is regarded as an environmentally safe solvent. These characteristics enable a broad range of industrial applications such as petroleum production,9 polymer production,10 supercritical gas extraction, removal of wastes, processing of drugs, regeneration of poisoned catalysts, etc. For these reasons, it was studied many times, which allowed us to compare our approach to others.

10.1021/ie071643r CCC: $40.75  2008 American Chemical Society Published on Web 10/24/2008

8848 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

In a second step, the approach was tested in a fully predictive manner to methane + n-alkane and ethane + n-alkane systems. These systems were sufficiently studied experimentally to start testing the method on a solid ground. For this reason, the tests were also as exhaustive as possible. 2. GC-SAFT Equation of State The group contribution SAFT (GC-SAFT) is the combination of a SAFT EOS with group contribution relations (described below) that allow one to compute the EOS parameters. In earlier works, three different versions of SAFT EOS were used for the correlation and prediction of VLE data: the original SAFT, slightly modified;11,12 PC-SAFT, from Gross and Sadowski;13,14 and VR-SAFT.15 As shown by the previous results2,8 obtained on asymmetric systems that contain a species (such as CO2 and H2) under supercritical conditions, PC-SAFT seems to be the best EOS among those three versions. For this reason, the decision was made to use only this version in the present work. The expression of PC-SAFT EOS is not recalled here, and the interested reader is referred directly to the original paper for more details.14 However, when polar compounds such as CO2 (quadrupolar) are considered, one should use an extended version6 of this equation, which uses additional polar terms to obtain better computations of the phase diagrams.16-21 Here, one reads a ) anonpolar + aQQ

(1)

where anonpolar refers to a PC-SAFT EOS and aQQ is a term that accounts for quadrupole-quadrupole interactions. The aQQ term used here is obtained by extending the theory of Gubbins and Twu,22 which was originally developed for spherical molecules, to chain molecules. The extension here closely follows the so-called “segment approach” that was proposed by Jog and Chapman.23,24 In this approach, the polar moments are assumed to be well-localized on certain segments of the chain. From this point of view, quadrupolar interactions between chain molecules are viewed as interactions between polar spherical segments. The expression of aQQ is provided in Appendix A. In the segment approach, an additional parameter to the Q quadrupolar fraction xQ R is introduced: the quadrupolar xR fraction is defined as the fraction of quadrupolar segments in chain (species) R. This means the product xQ R mR gives the total number of quadrupole moments Q (which, here, are assumed to be all identical) in chain R. Theoretically, this product should have an integer value; however, as in the case of chain length mR, Jog and Chapman considered xRQ to be adjustable.19,21,24 We chose to follow their lead here, as in our previous work.6 In GC-SAFT, the EOS parameterssi.e., the segment parameters (energy ε, diameter σ) and chain parameter (m)sof the molecule are calculated through group contribution relations inspired by the Lorentz-Berthelot combining rules.1,2 The energy parameter ε/k is calculated by a geometric average: ε)

( ∏ ) nG

ngroups

εnk k

(2)

k)1

The segment diameter σ is calculated as an arithmetic average: ngroups

σ)

∑ k)1

nkσk nG

(3)

The chain length parameter m is calculated as a sum of the chain contributions Rk, given by the different groups:

ngroups

m)

∑ nR

(4)

k k

k)1

εk, σk, and Rk are parameters of the chemical group k, and nk is the number of k groups in the molecule that are composed of ngroups different groups. The total number of groups in the molecule is denoted as nG and is simply given by ngroups

nG )

∑n

k

k)1

Examples of parameter calculations are given in detail in the paper by Nguyen-Thi et al.4 These relations generally apply to compounds that belong to well-defined chemical families (n-alkane, 1-alkanol,...), with the exception of the first or even first two members (methane, ethane, methanol,...), which should be treated specifically. 3. Extension to Asymmetric Mixtures: Description of the kij Correlation Method PC-SAFT is applied to mixtures using the van der Waals onefluid model13,14 and modified Lorentz-Berthelot mixing rules that relate the potential parameters εij and σij for crossinteractions i-j to those of self-interactions i-i and j-j: εij ) (1 - kij)√εiiεjj

(

(5)

)

σii + σjj (6) 2 In many cases, pure predictions may be obtained by setting kij and lij each equal to zero (kij ) lij ) 0). However, in our previous works, this approximation was found out to provide reasonable results only for mixtures that contain species treated by the group contribution method previously described.2,3,5-7 In other cases, nonzero binary interaction parameters (kij and/ or lij) are necessary to improve the representation capability of the EOS. In earlier works with GC-SAFT, however, lij could be set to zero with satisfactory results. Therefore, here, we also will retain this hypothesis. The usual way of evaluating kij for a given system relies on regression of the corresponding mixture data. However, this procedure, although useful in certain situations, is only of little help for prediction. Therefore, for prediction purposes, a correlation/prediction method would be more suitable. In the context of GC-SAFT, such a kij prediction method would fall mainly into two categories: (1) The first category involves binary group-group parameters or molecule-group parameters (for a molecule treated specifically). Such a method has been used for CO2-hydrocarbons systems and H2-hydrocarbons systems,8 as explained above. (2) The second category involves pure compound parameters, such as diameters or ionization energy, which will be considered here. As will be seen below, these parameters also may be further estimated by some group contribution procedure. The latter approach seems more appealing than the former, because it uses only pure-compound parameters and, in principle, at least would not require mixture data. Therefore, this approach was selected for this work. The first attempts into that direction were made several decades ago, based on the London’s dispersive interaction theory, although more-rigorous theories were available at that time. London’s theory is indeed approximate (see the review σij ) (1 - lij)

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8849 25

of Barker ) This theory provides expressions for the crossinteraction potential uij that were then introduced, for instance, in the second virial coefficient to obtain an EOS for the mixtures. A brief account of these developments is given here, because our method is directly inspired by these considerations. London obtained an expression of the dispersive interaction potential between molecule i and molecule j, by solving the Schro¨dinger equation under simplifying assumptions and obtained the following expansion:26,27 uij(r) ) -

C6,ij 6

-

C8,ij 8

r

-

r

C10,ij r10

...

(7)

today, not incorporated in modern EOSs such as PC-SAFT. Notice, however, that a simplified form of the repulsive part of the total potential also involves ionization energy. Hence, it seems that this quantity can play the role of a pertinent parameter for correlating kij. Indeed, after a comprehensive study of binary systems of hydrogen, helium, and neon with light hydrocarbons, Hiza and Duncan30 noticed that the deviation 1 - kij is, at least in part, dependent on differences in the ionization potential of the interacting species. Therefore, to make a good correlation for kij, they suggested an empirical relation that involved only the ionization energy30 as a parameter:

()

uij(r) ≈ with C6,ij ≈

C6,ij

(8)

r6

(

)

3 hν0ihν0j RR 2 hν0i + hν0j i j

(9)

where R represents polarizability, h is Planck’s constant, and ν0 refers to the characteristic frequency of a molecule in its unperturbed state, corresponding to its “zero-point energy”.28 The parameter ν0 is related to an energy transition ∆E between electronic quantum states by the Bohr relationship ∆E ) hν0.27 The term ∆E is approximately equal to the first ionization potential I for a molecule, so that C6 can be estimated from eq 9 by replacing hν0 by I. It may easily be shown, after small rearrangements (Hildebrand et al.28), that uij

√uiiujj

)

2√IiIj Ii + Ij

(10)

giving a theoretical expression for the deviation from the geometric rule, in terms of intermolecular potential. The expression εij/(εiiεjj)1/2 ) 1-kij is dependent on the exact definition of the energy parameter εij in the potential uij. If one assumes that εij is simply proportional to C6,ij, then 2√IiIj ) 1 - kij ) I i + Ij √εiiεjj εij

(11)

However, if the interaction energy is that which lies in the Lennard-Jones (LJ) potential (usually used in the equations of state), a different expression of kij is obtained. By identifying London’s potential with the attractive part of the LJ potential, i.e., uijLJ ) εij(σij/r)6, Hudson and McCoubrey29 deduced that 1 - kij )

( )

2√IiIj 2√σiσj Ii + Ij σi + σj

6

(12)

where the diameters of the molecules are explicitly involved this time. However, as emphasized by Hiza and Duncan,30 relation 12 was observed to be not always adequate to predict realistic kij values, using experimental values of ionization energy and diameter (collision diameters for a 12:6 LJ potential). As explained by these authors, this failure is probably due partially to the fact that kij must be determined by the repulsive forces, as well as by the attractive forces. However, the corresponding theoretical expressions are very complex, and a rigorous treatment of this problem seemed formidable and is still, even

Ii Ij

kij ) 0.17(Ii - Ij)1 ⁄ 2 ln

This expression is often approximated by truncating at the first term:

(13)

However, this correlation is only valid for the specific EOS used by Hiza an Duncan30 and was tested on mixtures with smallsized species. More recently, Nishiumi and Arai31 applied the following relation to the binary interaction parameter of the Peng-Robinson equation of state (PR EOS):

[( ) ( ) ]

1 - kij ) 64

Vci Vcj

1⁄6

+

Vci Vcj

-1 ⁄ 6 -6

(14)

Relation 14 was deduced from relation 12, assuming first that 2(IiIj)1/2/(Ii + Ij) ≈ 1, arguing that eq 12 is, in fact, more sensitive to molecule diameters than to ionization energy. The diameters then were expressed in terms of critical molar volumes (Vci and Vcj), as suggested by Hudson and McCoubrey.29 The phase equilibria of more than 100 binary and ternary mixtures containing hydrocarbons, nitrogen, CO2, and H2S were investigated by Nishiumi and Arai.31 The preliminary results obtained using relation 14 clearly indicate that a realistic correlation of the kij values did require modification of eq 14. To remedy this situation, an empirical modification was proposed, introducing a dependence on the differences in acentric factors:

( ) ( )

1 - kij ) C + D

Vci Vci +E Vcj Vcj

2

(15)

where C ) c1 + c2|ωi - ωj|

(15a)

D ) d1 + d2|ωi - ωj|

(15b)

and

in which ci, di are fitted parameters, obtained by correlating the optimized kij values for each type of mixture.31 Other attempts to compute the value of kij using London’s theory or more advanced theories were performed very recently (see the works of Leonhard et al.32 and Haslam et al.33). More examples of various uses of similar theories and their modifications, as well as more references, may also be found in the work of Prausnitz et al.;34 however, the developments given here are sufficient for our purposes. From the discussion above, and after a careful examination of the literature, it seems impossible at the present level of our knowledge to propose a reliable kij correlation that is developed on purely theoretical grounds. Nevertheless, if some empirical treatment cannot be completely avoided, parts of the theoretical developments above detailed may be kept to ensure some physical soundness. For our purposes, the work of Hiza and Duncan30 is of interest because one single physical parameters namely, ionization energysis involved in their correlation (eq 13). However, because relation 13 it is too specific to the EOS

8850 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 Table 1. Parameters for the Polar PC-SAFT Model for Pure CO2

GC-PC-SAFT a

ε/k (K)

σ (Å)

139.99

2.98

Note: 1 Buckingham ) 3.335641 × 10

m 1.84 -40

Q (B)a

xpm

4.3

0.52

2

Cm.

Table 2. Deviation on Vapor Pressure and Saturated Liquid Density of CO2 (PC-SAFT) temperature, T (K)

Npt

216-304 216-302

19 18

(see section 2), a geometric average inspired by the original Lorentz-Berthelot combination rule for the energy parameter is proposed:

Average Absolute Deviation, AADa (%) vapor pressure saturated liquid volume

Jmolec )

∏

ngroups

nG

Jnk k

(17)

k)1

In conclusion, eqs 16 and 17 give the basic relations for correlating and predicting the kij values to be used in GC-PCSAFT. These relations are tested in the next two sections.

0.55 1.21

4. Determination of Parameters

AADX (%) ) (1/Np)∑Np[(Xexp - Xcal)/Xexp] × 100. Npt ) number of points. a

used by these authors, it has been discarded for this work. Instead, keeping the idea of a possible correlation of kij values to the ionization energy or to some related parameter, the simple functional expression that is decribed by eq 11, which is simple enough for engineering applications, and has, nevertheless, some physical grounds, was retained as a basis for our correlation. Let us recall at this point that, rigorously, experimental values of the zero-point energy hν0 should be used in eq 11. Yet, it is generally identified with the ionization energy. Furthermore, as shown in the review of Barker,25 one must recognize that London’s theory is only approximate. For these two reasons, it was decided to use an adjustable parameter J (which is called the pseudo-ionization energy) instead of I in eq 11 to correlate the kij values. Therefore, the proposed correlation is written as 1 - kij )

2√JiJj Ji + Jj

(16)

where the subscripts i and j refer to segments in molecules i and j. Moreover, let us recall that, indeed, unlike in the case of a cubic EOS for instance, in a SAFT EOS, the interactions are considered between segments. Equation 12 and, more generally, the dependence of kij on diameters were not considered here for reasons that deserve discussion. Equation 12, combined with PC-SAFT, should be written in terms of segment diameters and segment ionization energies. However, as a result of earlier work, in GC-PC-SAFT, the segment diameter values for different molecules seem to be generally close, so that it is expected, as a first approximation, that [2(σiσj)1/2/(σi + σj)]6 ≈ 1. Although approximate, this assumption is at least far more justified here than in the case of other EOSs (for example, cubic ones) that assume interactions between molecules (σi and σj are then the diameters of molecules i and j, which may be very different, as seen in the examples given by Hudson and McCoubrey29). Moreover, Hiza and Duncan30 have shown that, in the case of mixtures of small species (i.e., of comparable size, as in the case of our SAFT segments), the dependence of kij on the diameters σi and σj could be neglected. Finally, preliminary calculations showed that eq 11 is, in our case, superior to eq 12. To complete our prescription, in view of making useful predictions for engineering applications using relation 16, it is necessary to propose a method for computing pseudo-ionization energies. First, notice that, because the molecules are assumed to be homonuclear in PC-SAFT, a single value of the pseudoionization energy characterizes the entire chain molecule and is denoted here as Jmolec. The value of Jmolec should be dependent a priori on the chemical groups that comprise the molecule and could be computable by a group contribution method. Consistently with the group contribution used for the EOS parameters

4.1. EOS Parameters. 4.1.1. Pure n-Alkanes. These are nonpolar and nonassociative compounds. Methane and ethane, which must be treated as specific species, possess their own parameters that are determined based on the pure compound liquid density and vapor pressure. The other compounds in the series, starting from n-propane, were treated using group contributions. They are made of (CH2) and (CH3) groups. The corresponding group contribution parameters εi, σi, and Ri were determined in an earlier work2 and are reused here. The relative deviations that have been obtained on vapor pressure and saturated liquid volume data are 0.76% and 0.85%, respectively. All parameters of the GC-PC-SAFT that are used to describe the pure n-alkane VLE are recalled in Table B1 of Appendix B, for the sake of completeness. 4.1.2. Pure Carbon Dioxide. This compound is regarded as quadrupolar and is treated in a specific manner, because it does not belong to a well-defined chemical family. As noted in earlier works on polar systems,2,6,19,24 and by our own observations, the EOS parameters (and especially the energy parameter, ε) often seem to be strongly correlated to the polar parameter Q. As a consequence, numerous sets of parameters allow a good representation of the pure compound VLE, but only some of them may also provide a good prediction of the phase equilibria of the mixtures. To select an appropriate set of parameters for CO2, we proceeded as done previously.6 The quadrupolar moment of CO2 was set to the experimental value, which is 4.3 B.35 [Note: The unit B is a Buckingham: 1 Buckingham ) 3.335641 × 10-40 C m2.] All the other pure parameters of CO2 were determined by the simultaneous regression of the pure-compound data (saturation pressure and saturated liquid volume) and four series of VLE data of the CO2 + propane mixture at 230.0, 270.0, 311.05, and 361.15 K. This mixture was chosen because propane is treated by group contribution methods (this is not the case for methane and ethane) and also because the binary interaction parameter kij for CO2 + propane is expected to be small, as inferred from previous investigations of CO2 + n-alkanes mixtures.8,36 In the calculations, the value of kCO2,propane was set exactly equal to zero. Pure vapor pressures and saturated liquid densities for CO2 were taken from DIPPR.37 The references for mixture data are given in Table S1 in the Supporting Information. The regression procedure was performed as explained in ref 6, using the following objective function: FOBJ ) 0.5

∑ data

(

Pexp - Pcalc Pexp

)

2

+ 0.5

∑ data

(

liq Vliq exp - Vcalc

Vliq calc

)

2

(18)

The first term in the right-hand side (RHS) of eq 18 includes both pure and mixture data. The EOS parameters for CO2, and the deviations in vapor pressure and liquid density that are obtained, are reported in Tables 1 and 2, respectively.

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8851 Table 3. Pseudo-ionization Energy of Series of n-Alkanes

Figure 1. Relationship between the binary interaction parameter kij used to model the CO2 + n-alkane VLE and the carbon number in the n-alkane.

chemical group

pseudo-ionization energy (eV)

CH2 CH3

7.345 12.992

that very similar phase equilibria computations were obtained with this value and assuming kCO2,propane ) 0. In a third step, the pseudo-ionization energies JCH2 and JCH3 of the chemical groups (CH2) and (CH3) were obtained by fitting Jn-alkane, assuming that eq 17 holds. The values are reported in Table 3. The values of the binary interaction parameters kij used in this work for the CO2 + n-alkane systems are ∼75% lower than the results of Fu et al.40 and are ∼55% lower than those of Le Thi et al.8 This should be at least partially due to the fact that, in the former case, CO2 was regarded as quadrupolar, but not in the latter cases. As explained by Jog and Chapman,24 if CO2 is considered to be nonpolar, the self-interaction energy εCO2,CO2 and then also the cross interaction εCO2,alkane are artificially increased, to include polar interactions that do exist. 5. Results of Correlation and Prediction of Phase Equilibria

Figure 2. Comparison between the experimental energy and the pseudoionization energy obtained in this work for the n-alkanes.

4.2. Pseudo-ionization Energy. To determine the pseudoionization energies of CO2 and the chemical groups of n-alkanes, we followed a three-step procedure. In the first step, the optimum value of the interaction parameter kij for each binary system was obtained by regressing the corresponding VLE data. For this step, we have chosen only some systems, for which a large number of data is available: CO2 + C3, CO2 + C4, CO2 + C5, CO2 + C6, CO2 + C7, CO2 + C8, CO2 + C10, CO2 + C20, CO2 + C22, CO2 + C36, and CO2 + C44. Descriptions of the regressed database, together with the calculation results, are given in Table S1 in the Supporting Information. In the next step, assuming that, as a first approximation, the pseudo-ionization energy of CO2 could be set to the experimental value (13.78 eV38,39), the pseudo-ionization energies of the n-alkanes were obtained by inversion of relation 16. This choice for JCO2 may appear somewhat arbitrary, but it was considered to be acceptable here, because a good correlation and good prediction results were obtained (see the following sections and Figure 1). The pseudo-ionization energies of n-alkanes obtained from relation 16 are rather close to the true ionization energy experimental values (available up to n-decane) but are systematically lower (5%-15%). The departures between these two quantities are slightly higher than the experimental dispersion, which is up to 10% according to NIST.39 As shown in Figure 2, their trends with carbon number are comparable (i.e., decreasing with n-alkane chain length). However, the departures seem to increase with the chain length. Until now, the choice of relation 16 for correlating kij seems reasonable and seems rather physically meaningful. However, this needs confirmation and will be investigated in the next paragraph. Notice that for CO2 + propane, one finds that, at this step, kCO2,propane ) 0.007 (i.e., very close to zero). We have verified

Most of the VLE computations were made following a standard iterative bubble point algorithm (at fixed temperature and liquid phase mole fractions). Deviations in pressure and, if available, vapor mole fractions were then calculated and are given in Tables S1 and S2 in the Supporting Information. However, under conditions close to a mixture critical point, the VLE computations generally failed to converge. In those particular cases, we used a flash temperature, pressure (T,P) algorithm based on minimization of the mixing Gibbs energy, using a “brute force approach”. The search of possible phase compositions (mole fractions) in equilibrium was made in a systematic manner (by making tests on the mixing Gibbs energy at regularly sampled mole fractions, first with increments of 0.01 to localize the possible phase compositions and then with increments of 0.0001 about these compositions, to refine the search), so that this algorithm proved to be more efficient than the bubble point algorithm. However, it was also much slower and, therefore, was used only occasionally for VLE computations. It was also used for liquid-liquid phase splitting computations. Deviations are then given in mole fractions. 5.1. CO2 + Alkane Systems. A rather large database (see Table S1 in the Supporting Information) is available for these systems, ranging from methane up to C44, and mainly consisting of VLE measurements. Liquid-liquid equilibria (LLE) were measured only in the case of two systems, namely, CO2 + C15 and CO2 + C16. The database is as comprehensive as possible, except for the CO2 + CH4 system, for which only hightemperature data (>170 K) were considered. Very low temperature data are indeed of less interest in the petroleum industry. As described in section 4, some of the listed mixtures were used in the regression procedure. The other systems were then fully predicted using the adjusted parameters of CO2 and the group contribution correlation for pseudo-ionization energy (eq 17). Because the latter correlation applies to the n-alkane series, starting at n-propane, the pseudo-ionization energies of methane and ethane had to be estimated specifically. As a first approximation, they were set to the experimental values, which are 12.61 and 11.52 eV, respectively.39 Generally, a good fit and a good prediction of VLE data is observed, within 3%-4% with regard to pressure, even near VLE mixture critical points (see Figure 3a for correlation results

8852 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

Figure 3. Results of correlation for CO2 + n-alkane mixtures, using polar GC-PC-SAFT ((a) data used in the regression process and (b) data not used in the regression process); kij values for the mixture are calculated using eqs 16 and 17.

Figure 4. VLE phase envelopes of the CO2 + ethane mixture; solid lines are predicted results with polar GC-PC-SAFT EOS. The kij values for the mixture are calculated using eqs 16 and 17; the value of Jethane was set to the experimental ionization energy value. Data taken from ref 58.

of regressed data and Figure 3b for full prediction results). Furthermore, the accuracy of the model does not seem to deteriorate for very asymmetric systems, when compared to small n-alkane-containing systems. Examples of correlated and predicted phase diagrams are displayed in Figures 4-7. The model predicts liquid-liquid phase splitting for the two previously quoted systems. Although the three-phase vaporliquid-liquid equilibria (VLLE) line is rather well-located by the model, the LLE upper critical mixture pressure is overestimated. However, this result is not surprising, because, in fact, it is known that simultaneous representation of VLE and LLE is difficult. Nevertheless, the liquid-liquid phase envelope is predicted in a realistic manner in the region that is not too close to the LLE upper mixture critical point. From these results, it can be concluded that our GC-SAFT model allows a satisfactory representation and prediction of the

Figure 5. Modeling the VLE phase diagram of the CO2 + propane system, via polar GC-PC-SAFT EOS; the kij values for the mixture are calculated using eqs 16 and 17. Data taken from refs 59-61.

phase equilibria of CO2 + n-alkane systems, within an accuracy comparable to that of other systems already investigated using GC-SAFT (binary mixtures that contain n-alkane, 1-alkanols, aromatics, and esters).2,3,6,7 The method used here is a real improvement over our previous works that have been performed on this system. In Figure 8, the deviations obtained by Le Thi et al.8 and those obtained using the present method are compared based on the same database (smaller than the one considered here; see the article of Le Thi et al.8 for more details). The model used here clearly seems to be systematically and rather significantly better than that of Le Thi et al.8 As already mentioned in the Introduction, recently, several authors attempted to model the vapor-liquid phase behavior of CO2 + n-alkane mixtures with EOSs. Until a few years ago, most of these EOSs were cubic;41-45 only Cotterman and Praunitsz46 used a statistical-mechanics-based EOS (PSCT). Recently, several authors used PC-SAFT EOS for this task.8,14,40,47

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8853

Figure 6. Modeling the VLE phase diagram of the CO2 + n-decane system via polar GC-PC-SAFT EOS; the kij values of the mixture are calculated using eqs 16 and 17. Data taken from ref 62.

Figure 7. VLE phase diagram of the CO2 + n-hexadecane system; solid lines are predicted results with polar GC-PC-SAFT EOS. The kij values of the mixture are calculated using eqs 16 and 17. Data taken from refs 63-65.

Figure 8. Results of prediction for the CO2 + n-alkane mixtures, using polar GC-PC-SAFT. Comparison with a previous work by Le Thi et al.8 (using the same database as Le Thi et al.;8 see this article for more details).

However, the investigation of these systems was made systematically only in some cases. Kordas44 used the Peng-Robinson EOS (t-m PR EOS), together with a temperature-dependent kij, to obtain typical errors in the bubble point within 4%-5% for the entire series of CO2 + n-alkane mixtures. Their results compare well to those of

Figure 9. Comparison between two models, showing the VLE of the CO2 + n-hexadecane mixture ((s) predicted by polar GC-PC-SAFT (kij ) 0.037, from the present work; (- - -) correlation of VLE with PC-SAFT (kij ) 0.151), where the molecular parameters for pure CO2 and n-hexadecane are taken from the work of Gross and Sadowski14). Data taken from ref 64.

Kato et al.41 and Gasem et al.,48 although they are limited to some of these systems. Berro et al.45 applied the IUPAC EOS for CO2 and a PRtype cubic EOS for n-alkanes, combined with the group contribution method of Abdoul et al.,49 to these mixtures. Most of the CO2 + n-alkane mixture data were regressed to fit one binary group contribution parameter. The relative mean deviation in bubble pressure is 2.9%. More recently, Gross and Sadowski14 applied PC-SAFT to some of these systems. Some comparisons are provided in Figure 9. Later, Gross18 improved these results using an extended and modified version of PC-SAFT (polar PC-SAFT), but no comprehensive treatment was attempted. Garcia et al.47 investigated the pressure-temperature (PT) diagrams of several systems using PC-SAFT and a single kij value that was obtained for CO2 + tridecane systems. This value was then reused to model the other CO2 + n-alkane systems. However, this assumption does not work well for the Pxy diagram of the CO2 + ethane system. Fu et al.40 represented the VLE of the entire series using PCSAFT and a correlation between kij and the n-alkane carbon number (n). For n > 20, a constant kij was determined to be sufficient. Although their modeling seems to be good, it is difficult to compare to GC-SAFT, because the average deviations are given in terms of mole fraction, rather than bubble pressure. Comparison with the Nishiumi and Arai31 model is provided in Figure 18 (we performed the computations using their parameters and our database, as described in detail in Tables S1 and S2 in the Supporting Information). Remember from section 2, they used the original PR EOS with the original mixing rules (see the work of Peng and Robinson50) combined with a correlation for kij, in terms of critical volumes and acentric factors. This correlation is obtained by modifying the empirical Hudson and McCoubrey29 relation to fit the optimal kij values (values adjusted by regression of VLE mixtures data of each of the considered systems). However, as shown in the article of Nishiumi et al.,31 the optimal kij values does not always seem very well fitted by their correlation (see Figures 17, 19, and 22 in their article). This may probably explain why sometimes large deviations are observed between the model of Nishiumi et al.31 and the data. The deviations are comparable to those with GCSAFT, although the latter seems slightly better. Still better results

8854 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

Figure 10. Results of the prediction for mixtures with methane, ethane, and n-alkane, using GC-PC-SAFT; the kij values for the mixture are calculated using eqs 16 and 17.

Figure 11. VLE phase diagram of the methane + ethane system; symbols are experimental data, and solid line represents GC-PC-SAFT EOS with the kij values calculated using eqs 16 and 17. Data taken from refs 66-69.

Figure 13. VLE phase diagram of the methane + n-hexadecane system; symbols are experimental data, and solid line represents GC-PC-SAFT EOS, with the kij values calculated using eqs 16 and 17. Data taken from refs 72 and 73.

Figure 12. VLE phase diagram of the methane + n-decane system; symbols are experimental data, and solid line represents GC-PC-SAFT EOS, with the kij values calculated using eqs 16 and 17. Data taken from refs 70 and 71.

Figure 14. VLE phase diagram of the methane (1) + tricontane (2) system; symbols are experimental data, and solid line represents GC-PC-SAFT EOS, with the kij values calculated using eqs 16 and 17. Data taken from ref 74.

can be obtained using “simple” group contribution cubic EOS (see, for instance, Jaubert et al.51-53); however, in those cases, to our best knowledge, the kij calculation requires group-group parameters. 5.2. Methane + Alkane and Ethane + Alkane Systems. To test the prediction capability of our approach, methane + n-alkane and ethane + n-alkane VLE phase diagrams were computed using pseudo-ionization energies above determined and relation 16. The computations were then compared to the corresponding experimental values. The detailed results may

be found in Table S2 in the Supporting Information. Deviations are shown in Figure 10. Examples of predicted phase diagrams may be found in Figures 11-14. Two-thirds of the experimental database is composed of methane + n-alkane systems (up to C44). Data for the ethane + n-alkane systems, for larger n-alkanes up to C44, are relatively few. As a general comment, it seems that the deviations of GCSAFT observed here are quite comparable to those obtained for CO2 + n-alkane systems. As illustrated in Figures 11-17,

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8855

Figure 15. Effect of kij on phase equilibrium predictions of methane + n-alkane systems.

than the other ethane mixtures but remain in an acceptable range.

Figure 16. VLE phase diagram of ethane + n-pentane system; symbols are experimental data, and solid line represents GC-PC-SAFT EOS, with the kij values calculated using eqs 16 and 17. Data taken from ref 75.

Figure 17. VLE phase diagram of ethane + n-dodecane system; symbols are experimental data, and solid line represents GC-PC-SAFT EOS, with the kij values calculated using eqs 16 and 17. Data taken from refs 76 and 77.

nonzero kij values are needed for treatment of methane + n-alkane systems and are well-predicted using our approach (Figure 15). Similar results were obtained for ethane + n-alkane systems. However, GC-PC-SAFT has a tendency to overestimate the bubble-point pressure of ethane + large n-alkane systems systematically. (See Figures 16 and 17.) Predicted kij values by GC-PC-SAFT seem slightly too high. Deviations in ethane mixtures with C20 and beyond are systematically higher

Several researchers have used EOSs to model asymmetric mixtures of light and heavy n-alkane and methane or ethane, mainly using cubic EOSs. (See for example, the work of Stryjek54 and Floter et al.,55 who used PR EOS to model only some of these systems: respectively, ethane + n-alkane (up to C10) and methane + some heavy alkanes.) Gao et al.56 applied PR and SRK EOS more systematically to describe VLE of asymmetric binary mixtures of methane, ethane, and alkanes up to C44 with a deviation within 3%-4.5% in bubble pressure, i.e., within an accuracy comparable to ours. For this work, they used two binary interaction parameters correlated to the pure EOS parameter and acentric factor. As noted above in section 2, Nishiumi and Arai31 attempted to model methane and ethane + n-alkane (up to C16) series using PR EOS and a correlation for kij, in terms of critical volumes and acentric factors. Figure 18 shows a comparison between our approach and that of Nisiumi and Arai.31 The deviations are comparable. Recently, Voutsas57 compared the fitting capabilities of two versions of PR EOS, original SAFT and PC-SAFT for modeling phase behavior of asymmetric systems that contain alkanes up to C30. The PR EOSs were found out to be superior for treating these systems, but PC-SAFT nevertheless gave quite comparable deviations. However, a specific kij value was adjusted for each system. 5.3. n-Alkane + n-Alkane Systems. In earlier work,2,3 these systems were treated in a fully predictive manner, setting kij ) lij ) 0. However, strictly speaking, kij is not zero from eq 16 for these systems, because two different n-alkanes have two different pseudo-ionization energy values. Some tests have been performed to evaluate the impact of using nonzero kij and compared to the case of zero binary interactions parameters (see Figure 19). Both results are of comparable accuracy. However, a slight improvement is noticed in the case of mixtures that contain propane and n-butane. 6. Conclusion In this paper, a new method for correlating and predicting kij has been presented and successfully tested for CO2 + n-alkane, methane + n-alkane, and ethane + n-alkane binary mixtures using polar GC-PC-SAFT.

8856 Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008

Figure 18. Results of prediction for CO2 + n-alkane, methane + n-alkane, and ethane + n-alkane mixtures, using polar GC-PC-SAFT. (Made via comparison with a previous work by Nisiumi and Arai.31)

Figure 19. Modeling n-alkane + n-alkane systems, showing the effect of the kij work for the n-alkanes.

To be useful for engineering applications, this method should be tested on more systems. This will be the subject of the second part of this series of publications.78

Amult 3B (224;224;224) ) NF2 32π3 Q Q (2002π)1⁄2 xRxβxγxQpRxpβ xpγmRmβmγ × 2025 (kT)2 Rβγ

∑

(

Acknowledgment The authors are grateful to Petro Viet Nam Oil and Gas Group for financial support through a Ph.D. grant. Supporting Information Available: Detailed database used in calculation and detailed results. (PDF) This material is available free of charge via the Internet at http://pubs.acs.org. Appendix A: Quadrupolar-Quadrupolar Contribution to SAFT Equation of State The quadrupolar-quadrupolar free-energy contribution is written as a Pade´ approximant:

[

AQQ ) A2

1 A3A + A3B 1A2

]

(A1)

where A2 represents the second-order terms in the quadrupolarquadrupolar perturbation expansion of free energy and A3A and A3B represent the third-order terms in the quadrupolar-quadrupolar perturbation expansion of free energy. A2 ) -

QR2Qβ2 (10) 14 πNF Q xRxβxQpRxpβ mRmβ JRβ 5 kT Rβ d 7

mult A3A (224;224;224) )

∑

(A2)

Rβ

QR3Qβ3 (15) 144 πNF Q Q x x x x m m J 245 (kT)2 Rβ R β pR pβ R β d 12 Rβ Rβ

∑

(A3)

QR2Qβ2Qγ2 dRβ3dRγ3dβγ3

)

KRβγ(444;555) (A4)

In these expressions, F is the total number density of molecules, mR is the chain length of molecule R, and dRβ is an average hard-sphere diameter that is related to segment diameters σR and σβ (see the original papers13,14 for more details). QR represents the quadrupole moments of the polar segments in the chain compound R. Q In the above equations, the term xpR refers to the fractions of quadrupolar segments in the chain of component R; this term should not be confused with xR, which is the mole fraction of component R. Note that the theory of Gubbins and Twu has been extended by recognizing that xpRmR is simply the number of polar spherical segments in the molecule R. J and K are integrals over two- and three-body interactions. These integrals were fitted to reduced density and temperature using an empirical form. These expressions may be found in ref 22. Appendix B: Polar GC-SAFT Parameters for Hydrocarbons Treated in Previous Work6 The polar GC-SAFT parameters for hydrocarbons that have been treated in previous work are given in Table B1. Table B1. Group Contribution Parameters for the n-Alkane Series (GC-PC-SAFT EOS) group

ε/k (K)

σ (Å)

R

CH2 CH3

261.0866 189.9628

3.9308 3.4873

0.3821 0.7866

Ind. Eng. Chem. Res., Vol. 47, No. 22, 2008 8857

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ReceiVed for reView December 3, 2007 ReVised manuscript receiVed June 20, 2008 Accepted August 21, 2008 IE071643R