Capturing the Solubility Minima of n-Alkanes in Water by Soft-SAFT

J. Phys. Chem. B 2009, 113, 7621–7630

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Capturing the Solubility Minima of n-Alkanes in Water by Soft-SAFT Lourdes F. Vega,*,†,‡ Fe`lix Llovell,⊥,§ and Felipe J. Blas| MATGAS Research Center, Campus de la UAB, Bellaterra, 08193 Barcelona, Spain; Institut de Cie`ncia de Materials de Barcelona, Consejo Superior de InVestigaciones Cientı´ficas (ICMAB-CSIC), Campus de la UAB, Bellaterra, 08193 Barcelona, Spain; Departament d’Enginyeria Quı´mica, Escola Te`cnica Superior d’Enginyeria Quı´mica, UniVersitat RoVira i Virgili, Campus Sescelades, 43007 Tarragona, Spain; and Departamento de Fı´sica Aplicada, Facultad de Ciencias Experimentales, UniVersidad de HuelVa, 21071 HuelVa, Spain ReceiVed: March 1, 2009; ReVised Manuscript ReceiVed: April 7, 2009

The purpose of this work is twofold: (1) to provide an accurate molecular model for water within the softSAFT equation of state [Blas, F. J.; Vega, L. F. Mol. Phys. 1997, 92, 135; Llovell, F., et al. J. Chem. Phys. 2004, 121, 10715] and (2) to check the capability of this molecular-based equation of state for capturing the solubility minima of n-alkanes in water experimentally found at room temperature for these mixtures. Water was modeled as a Lennard-Jones sphere with four associating sites, with parameters obtained by fitting to experimental vapor-liquid equilibrium data. Special care was taken to the value of these parameters depending on the range of applicability of the equation, which turned out to be essential for accurate predictions for mixtures. A correlation available in the literature was used for the molecular parameters of the n-alkane series. The crossover soft-SAFT equation was able to accurately describe the phase behavior of water near to and far from the critical point, up to 350 K. If instead of obtaining an overall good agreement one is interested in a more precise description of the near-ambient conditions, a more refined fitting of the parameters is needed. The model was used to describe the water + methane up to water + n-decane binary mixtures. The equation was able to predict the mutual solubilities in almost quantitative agreement with experimental data, including the presence of the solubility minima at ambient temperature, with a single transferable energy binary parameter, independent of temperature and chain length. Predictions obtained from the soft-SAFT approach are clearly superior than those obtained from the Huang and Radosz version of the SAFT equation [Economou, I. G.; Tsonopoulos, C. Chem. Eng. Sci. 1997, 52, 511], due to the more refined reference term and the more accurate radial distribution function used in the chain and association terms. This is the first time a SAFT approach is able to describe this minima. 1. Introduction The world as it is conceived today needs an incredible amount of daily energy. Although many efforts are devoted to renewable and ecologically friendly sources of energy, petroleum continues to be the most important source nowadays. However, the operational costs of the extraction, transportation, and transformation are increasing while the source amount decreases. The optimization of the process passes through a better knowledge of the thermodynamic behavior of petroleum, which is, from a chemical point of view, a mixture of water and hydrocarbons. One of the main challenging aspects for this optimization process concerns the description of the mutual solubilities between these compounds. The thermodynamic behavior of aqueous solutions of hydrocarbons is extremely nonideal due to the microscopic nature of the associating interactions among water molecules. It results in some anomalous properties that are challenging to model. For example, the solubility of the hydrocarbon in water is several orders of magnitude lower than * Corresponding author. E-mail: [email protected]. † MATGAS Research Center, Campus de la UAB. ‡ Consejo Superior de Investigaciones Cientı´ficas, (ICMAB-CSIC), Campus de la UAB. § Universitat Rovira i Virgili. | Universidad de Huelva. ⊥ Present address: Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2BY, UK.

the solubility of water in the hydrocarbon-rich liquid phase. Moreover, the solubility of the hydrocarbon exhibits a minimum at room temperature, while the solubility of water is a monotonic function that increases with the temperature.1 The presence of the minimum means that, at certain conditions, the solubility of the hydrocarbon in water increases when the temperature is decreased. This phenomenon is related to the strong anisotropy of the hydrogen bonding formed among water molecules. As a consequence, the entropic effect becomes more important than the thermal molecular agitation. The features mentioned above evidence the need of an accurate microscopic model for water-hydrocarbon mixtures. As several accurate models for hydrocarbons are available, the challenge remains in finding a water model accurate enough to capture its subtle behavior in mixtures, while being tractable enough to perform several calculations within a short amount of time, for practical applications. Several models based on ab initio calculations and/or semiempirical models have been proposed in recent years and some improvements have been obtained with respect to the previous ones (see, for instance, ref 2 and references therein). Regarding the water + hydrocarbon mixtures, it is encouraging to see that some molecular simulations performed by Errington et al.3 and Boulougouris et al.4 have been able to reproduce with accuracy the mutual solubility of water with some hydrocarbons, showing that this modeling approach is able to capture this anomalous behavior.

10.1021/jp9018876 CCC: $40.75  2009 American Chemical Society Published on Web 05/05/2009

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Simulations were able to provide accurate results for the Henry constant for water + methane and water + ethane mixtures,3 and also for water + butane and water + n-hexane4 mixtures, reproducing their mutual solubility, although results deteriorated as the chain length of the hydrocarbon increased. Even though these are encouraging results, simulations are time consuming, especially if one is interested in extensive calculations. Hence, and in spite of its need, there is not a single model available yet, accurate enough to describe the thermodynamic and structural behavior of water and its mixtures over a wide range of thermodynamic conditions in a reasonable amount of time. Calculations with these models need to be fast enough to be used for engineering applications. Whatever water model is used, it has to be combined with a thermodynamic model in order to calculate this solubility behavior. Using equations of state (EoS) is an alternative approach, provided they are accurate enough for these highly nonideal systems. In fact, several different EoS have been proposed to reproduce the mutual solubilities in water + hydrocarbon binary mixtures. Given the association nature of the forces acting in these systems, only models that explicitly account for them are expected to reproduce this behavior. Early works using association models include the use of a group contribution association model (GCA),5 the original SAFT, and associated perturbed anisotropic chain theory (APACT).1 More recently, Voutsas et al.6 and Yakoumis et al.7 have compared the capabilities of original SAFT and cubic plus association (CPA) EoS for modeling the mutual solubility in the water + hydrocarbon system, showing that the CPA is superior to the original SAFT EoS, although it still presents important deviations in the waterphase hydrocarbons solubilities. Grenner et al.8 and Karakatsani et al.9 have also studied these mixtures by the use of the PCSAFT EoS. In general, all these works describe well the water content of the hydrocarbon phase (in fact, experimental data is actually used to fit the model) while the prediction of the hydrocarbon content of the water phase is less accurate. In a very recent work, Pereda et al.10 have used the group contribution plus association EoS (GCA-EoS) to model the phase equilibrium of water + hydrocarbon systems, using new and available experimental data for these systems, finding good agreement with experimental data for both phases. It should be mentioned that none of these models has been able to predict the minimum in the solubility of hydrocarbons in water at low temperatures, this still remaining a challenge from a modeling perspective. An excellent review of the most important works in the literature on the modeling mutual solubility of water + hydrocarbon systems was recently presented by Olivera et al.;11 the reader is referred to it for details. In this context, the objective of this work is dual: (1) to provide an accurate molecular model for water using the softSAFT EoS, and (2) to check the capability of this molecularbased EoS for capturing the solubility minima of n-alkanes in water experimentally found at room temperature. Soft-SAFT12,13 is a variant of the original SAFT equation14,15 that uses a Lennard-Jones (LJ) model to describe the reference fluid, in contrast to the hard-sphere model of the original version. It also uses an accurate radial distribution function of the LJ reference term, which is used in the chain and association term. The predictive power of soft-SAFT has been used to accurately calculate the solubility of several complex experimental systems, such as the solubility of hydrogen in heavy n-alkanes,16 the solubility of gases in perfluoroalkanes,17,19 and the solubility of gases in ionic liquids,20,21 among others. Since the other versions of SAFT were unable to capture the solubility minima, and there

Vega et al. are just slight differences among them based on the reference intermolecular potential, it is of interest to see how this version performs for these challenging mixtures. The rest of the paper is organized as follows: section 2 contains a brief discussion about the molecular associating models for water; sections 3 and 4 are devoted to a description of the soft-SAFT equation; results concerning the modeling of water, hydrocarbons, and their mixtures are presented and discussed in section 5; finally, section 6 provides some concluding remarks. 2. Soft-SAFT Model Within the framework of SAFT, the equation of state (EoS) of a fluid is a perturbation expansion given in terms of the residual molar Helmholtz energy, defined as the difference between the total Helmholtz energy and that of an ideal gas at the same temperature T and molar density F. SAFT implicitly assumes that there are three major contributions to the total intermolecular potential of a given molecule: (a) the repulsiondispersion contribution typical of individual segments, (b) the contribution due the fact that these segments can form longlived chains, and (c) the contribution due to the possibility that some segments form association complexes with other molecules. Furthermore, polar terms can be added, explicitly taking into account their microscopic contribution in case they exist. According to this scheme, the residual Helmholtz energy can be written as

ares ≡ a - aideal ) aref + achain + aassoc + apolar

(1)

where ares is the residual Helmholtz free energy density of the system. The superscripts ref, chain, assoc, and polar refer to the contributions from the reference term, the formation of the chain, the association, and the polar interactions, respectively, depending on the system under study. In the soft-SAFT EoS,12,13,22 the reference term is a LennardJones (LJ) spherical fluid, which accounts for both the repulsive and attractive interactions of the monomers forming the chain. The accurate EoS of Johnson et al.23 is used for the reference fluid. The chain and association terms come from Wertheim’s theory,24,27 and they are formally identical in the different versions of SAFT:

achain ) FkBT

∑ xi(1 - mi)ln gLJ i

assoc

a

) FkBT

∑ xi ∑ i

R

(

ln

XiR

)

XiR Mi + 2 2

(2)

(3)

where F is the molecular density, T is the temperature, m is the chain length, xi is the molar fraction of component i, kB the Boltzmann constant, and gLJ is the radial distribution function of a fluid of LJ spheres at density Fm ) mF, evaluated at the bond length σ. Mi is the number of associating sites of component i, and XiR is the mole fraction of molecules of component i nonbonded at site R, which accounts for the contributions of all the associating sites in each species.

XiR )

1 1 + NAF

∑ xj ∑ Xjβ∆R β j

Ri βj

(4)

i j

β

The term ∆ is related to the strength of the association bond between site R in molecule i and site β in molecule j. For

Solubility Minima of n-Alkanes in Water

J. Phys. Chem. B, Vol. 113, No. 21, 2009 7623

the square-well bonding potential and a spherical geometry of the association sites,28,29 the simplified expression is

( [ ] )

∆Riβi ) 4π exp Ri βj

Riβj

ε

T

- 1 kRiβjI

σm3 )

Ri βj

∞

a ) FkBT

) ∑ (an-1 + across n

i

∑ ∑ mimjxixj

εm )

i)1

The value of an-1 for the first iteration corresponds to the original soft-SAFT value. Details on the implementation of the crossover term can be found in the original references, refs 13 and 40. 3. Mixing Rules In order to extend the equation to mixtures, each term in the equation should be extended as a function of the composition. Since the chain and association terms are readily applicable to mixtures, no mixing rules are needed for these two terms. Only the segment term needs to be extended to mixtures. As softSAFT uses the complete pair radial distribution function of the Lennard-Jones fluid and its contact value as functions of the reduced temperature and density in the association and chain term, respectively, these terms need to be extended to mixtures as well. To obtain both extensions, the same van der Waals one-fluid mixing rules have been used (see Blas and Vega12,41 for details). In the van der Waals one-fluid theory, the size and dispersive energy of the mixture may be written, respectively, as

(7)

j

∑ ∑ mimjxixjεijσij3 i

j

∑ ∑ mimjxixjσij3 i

(8)

j

where mi is the chain length and xi is the molecular mole fraction of component i. The parameters σij and εij are obtained through the generalized Lorentz-Berthelot combining rules:

(

σij ) η

σii + σjj 2

)

and

εij ) ξ(εiiεjj)1/2

(9)

η and ξ are the size and energy binary adjustable parameters, respectively. The equation is used in a fully predictive manner from the pure component parameters when η and ξ are equal to unity, while values different from unity mean the use of one or two binary adjustable parameters, taking into account the differences in size and/or energy of the segments forming the two compounds in the mixture. In this work, only one binary parameter has been used for all mixtures considered, the one related to the dispersive energy. In the next section we compare the performance of soft-SAFT with the original SAFT as implemented by Economou and Tsonopoulos.1 In their work they use two mixing rules: the van der Waals one-fluid theory, which is the one used for soft-SAFT, and the perturbed hard chain theory-based (PHCT) mixing rules, proposed by them in the referred work. The PHCT-based mixing rules, introduced to describe asymmetric mixtures in the original SAFT EoS, allow writing the dispersive energy of a LJ mixture as following:

εm )

∑ ∑ mimjxixjεijσij3 i

j

∑ ∑ mimjxixjσjj3 i

(6)

j

i

(5)

where ε is the association energy and k represents the association volume for each association site and compound, while I is a dimensionless integral converted into a numerical function of the temperature and density.30 The soft-SAFT version uses an analytic expression for gLJ(σ) collected from extensive computer simulations of the LJ system and fitted to an empirical function of reduced temperature and density.31 It is possible to explicitly account for multipolar interactions by the addition of a perturbative multipolar term. For the quadrupolar moment of benzene and toluene we have used an expansion of the Helmholtz free energy density in terms of the perturbed quadrupole-quadrupole potential with the Pade´ approximation proposed by Stell et al.32 Details on its implementation in soft-SAFT can be found in reference.19 The classical formulation of SAFT makes the theory unable to correctly describe the scaling of thermodynamic properties as the critical point is approached. A possible solution is to splice together an equation which incorporates the fluctuation-induced scaled thermodynamic behavior of fluids asymptotically close to the critical point, but that also accounts for a crossover to classical behavior of the thermodynamic properties far away from the critical point, where the effect of fluctuations becomes negligible. This contribution is obtained when the renormalization group (RG) theory33,34 is applied. The treatment followed in the crossover soft-SAFT EoS, based on White’s work,35,39 is done by incorporating the scaling laws governing the asymptotic behavior close to the critical point, while reducing to the original EoS far from the critical point. This approach is mathematically expressed as a set of recursive equations that incorporate the fluctuations in a progressive way:

∑ ∑ mimjxixjσij3

(10)

j

4. Molecular Model for Water and Hydrocarbons Several semiempirical models for water have been developed along the years (see, for instance, refs 2, 42, and 44 for excellent reviews). When developing a water model the goal is to capture its structure on the basis that if the computed model can successfully predict the physical properties of liquid water, then the (unknown) structure of liquid water can be determined. Most of these models involve orientation electrostatic effects and LJ sites that may or may not coincide with one or more of the charged sites. The LJ interaction accounts for the size and dispersive energy of the molecules and it is a key component of the model. It is repulsive at short distances, ensuring that the structure does not completely collapse due to the electrostatic interactions. At intermediate distances it is significantly attractive but nondirectional, and it competes with the directional attractive electrostatic interactions, also essential to capture the water behavior. This competition ensures a tension between an expanded tetrahedral network and a collapsed nondirectional one. Generally, each model is developed to fit well with one particular physical structure or parameter (for example, the density anomaly, the radial distribution function, or the critical parameters) and it comes as no surprise when a model developed to fit certain parameters gives good compliance with these same

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Figure 1. Bidimensional sketch of the model used for water. (a) Ball and stick classical sketch (image from ref 45). (b) Soft-SAFT model. The big circle represents the LJ center, while the small circles stand for the square-well sites mimicking the associating sites.

parameters; the challenge is then to use this model to accurately predict the behavior of the system at conditions far away from which the parameters were fitted, or for other properties. Figure 1a depicts a bidimensional sketch of water, showing the orientation of the 2 hydrogen atoms and the 2 other pairs of electrons around the central oxygen atom of the molecule (image taken from ref 45). According to the figure, it seems that water presents four different associating sites. However, there is controversy in the literature about how many sites should be considered to accurately describe the structure and thermodynamic properties of water. Some simulations have shown that, in the clusters of liquid water, only three sites are bonded per molecule in most of the cases.46 Different site models have also been considered within different SAFT approaches; for instance, see a comparison of SAFT models with two-, three-, and fourassociating sites and their performance done by Clark et al.47 They showed that the four-site model performs superior to the other two models for the particular SAFT approach they used. This suggestion is in the same direction as conclusions from some computational chemistry studies.48,49 Following this and several modeling works done with SAFT,30,50,51 and CPA,52 we have also decided to use a four-sites model. In a search for an accurate yet simple model, the electrostatic interactions are taken into account in an effective manner through the association molecular potential parameters of water, instead of being explicitly included, an approach also taken by other authors. The accuracy of these assumptions will be assessed by comparison with the experimental data. Hence, the water molecules are modeled in this work as a single spherical LJ core, accounting for the repulsive and dispersive forces between different molecules of the fluid, with four embedded off-center square well bonding sites (see Figure 1b). These four associating sites account for the two electron lone pairs and the two hydrogen sites of the water molecule. The most important features of the water + hydrocarbon mixtures can be defined at a microscopic level: repulsive and dispersive intermolecular forces between atoms or units in the real molecules, covalent-like bonds to form the hydrocarbon chain, and the association bonds due to the formation of hydrogen bonds among the water molecules. Following previous works, hydrocarbon molecules are represented as united atoms or sites: each site is assigned parameter values to represent a group of atoms in the molecule of interest, such as CH3, CH2, or CH groups. In the soft-SAFT approach these molecules are modeled as m LJ segments of equal diameter, σ, and the same dispersive energy, ε, tangentially bonded to form the chain. 5. Results and Discussion When the original soft-SAFT equation is applied to associating chain systems, five molecular parameters are needed to describe the molecule: m, the chain length, σ, the diameter of the LJ spheres forming the chain, ε, the interaction energy

Vega et al. among them, the association volume kHB, and association energy εHB of the associating sites of the molecule. The inclusion of the crossover treatment leads to two additional parameters, the cutoff length, L, related to the maximum wavelength fluctuations that are accounted for in the uncorrected free energy, and φ, the average gradient of the wavelet function.39 These parameters are treated as adjustable when using the equation for real fluids and phase equilibrium densities and vapor pressures are employed for their optimization. Moreover, when dealing with nonideal mixtures, binary parameters η and ξ are usually required. A. Pure Components. The application of soft-SAFT to any mixture first requires values for the molecular parameters of the pure compounds. These parameters are obtained by fitting to experimental data, usually vapor-liquid equilibrium data. Water. We have followed different approaches for water in this work. First, as with any other molecule, an overall optimization of the parameters for a wide range of temperature was done. For this purpose, two versions of the equation were used, the original soft-SAFT equation12 and the crossover version of the equation,13 designed to accurately predict the phase envelope close to and far from the critical region. Results from these calculations are depicted in Figure 2a; temperature data from the critical region down to T ) 400 K was used in the optimization. As shown in the graph, both versions of the equation perform equally well in a wide range of temperatures, while, as expected, only crossover soft-SAFT also captures the critical point. Quantitative agreement with experimental data53 up to 350 K is obtained with the crossover soft-SAFT version of the equation. Figure 2b shows these two calculations, but it also includes more calculations performed with two additional sets: a set obtained by fitting experimental vapor-liquid equilibrium data over a limited range of temperatures (300-450 K) and calculations performed with parameters obtained by Mu¨ller and Gubbins30 with a LJ-SAFT EoS (also considering a LJ as a reference fluid and four associating sites, at the same level of approximation as the model used here). The values of these four sets of parameters are provided in Table 1; a foursite model for water was considered in all cases. A further test of the performance of these parameters comes from checking the extent of hydrogen-bonding association in the system, a characteristic which gives rise to many of the anomalous properties of water. Spectroscopic information on the degree of hydrogen bonding for water in the liquid phase turns out to be quite rare. Several years ago, Luck54 examined the overtone bands obtained in infrared spectra of water to obtain an estimate of the relative proportions of free and hydrogenbonded water; from this data he was able to determine the fraction of the total number of possible OH hydrogen bonds which are not bonded. This can be compared directly with the fraction of molecules not bonded at a given site, readily available from Wertheim theory of association at the core of SAFT approaches (as obtained by applying eq 4). The degree of OH hydrogen bonding obtained experimentally by Luck is shown in Figure 3. The fraction of free OH hydrogen bonds sharply increases with temperature from about 10% close to the triple point to about 75% at the critical point. The extent of hydrogen bonding predicted for the four-site models of water with softSAFT and the different sets of parameters is depicted in Figure 3. Note that these are pure predictions obtained with the parameters fitted for vapor-liquid density data. It is clear that the set of parameters obtained by adjusting the limited low range of temperature (set 1 in Table 1) provides an overall better behavior, with quantitative agreement over a wide range of



Figure 3. Degree of hydrogen bonding in water (represented by X_A, the fraction of the total possible number of OH hydrogen bonds that are free) in the coexisting vapor and liquid phases determined from the sof-SAFT EoS with the four sets of parameters given in Table 1, as compared with experimental data for the liquid phase (symbols) taken from ref 54. Lines as in Figure 2b.

optimized. With these high values, the LJ correlation used for the soft-SAFT is outside the range of applicability at temperatures below 350 K as the value of the reduced temperature T* ) T/ε falls outside of the range of the optimization of the correlation used for the LJ reference term.23 This may also explain the better results obtained by the soft-SAFT model fitted to lower temperatures (set 1, solid line) and the LJ-SAFT EoS (set 4, dashed-dotted line) for the fraction of nonbonded hydrogen bonds, as they are the ones with the lowest values of the energy parameter. Hence, we have decided to use one single set of parameters for the rest of the calculations. This corresponds to the one obtained when fitting a limited range of low-temperature data (set 1 in Table 1, the solid line in Figure 2b). This decision was based on the fact that the main interest of this work is focused on the mutual solubilities at these temperatures and in addition it is the model providing best results for the degree of association, a key issue for the water + n-alkanes mixtures under consideration in this work. Note, however, that this is not the most appropriate set if one is interested in describing the overall behavior of the phase envelope with a single set of parameters. Finally, it should be mentioned that the crossover version of the equation is not necessary at these conditions because the region of interest is far away from the critical region and, as a consequence, the performance of the equation with and without crossover will be equal. In fact, we have performed the same calculations shown in Figure 3 with the set 1 parameters, but taking into account the crossover term and the performance for

Figure 2. Vapor-liquid equilibrium of water: (a) the dotted line is the original soft-SAFT calculation while the dashed line represents the same equation with a crossover treatment; (b) dotted and dashed lines as in (a); the dashed and dotted line corresponds to the soft-SAFT performance using Mu¨ller and Gubbins water parameters30 while the solid line represents the soft-SAFT calculations optimized in the range 300-450 K; experimental data is obtained from ref 53. See text for details.

temperatures, the second more accurate set of parameters is that proposed by Mu¨ller and Gubbins (set 4 in Table 1). The poorest performance is obtained by the parameters fitted using the global optimization of the phase envelope. It is important to note that very high values of the dispersive energy ε were found when the whole phase diagram was

TABLE 1: Molecular Parameters for a Four Associating Sites Model for Water Using Soft-SAFT with and without a Crossover Terma procedure set set set set a

1: T range 300-450 K 2: global without crossover 3: global with crossoverb 4- Mu¨ller-Gubbins parameters

m

σ (Å)

ε/kB (K)

kHB (Å3)

εHB/kB (K)

1.000 1.000 1.000 1.000

3.154 3.137 3.137 3.190

365 480 458 408.54

2932 923.2 1037 2707

2388 2612 2501 2367.7

In this second case, they are optimized for the whole phase diagram and for a specific range of temperatures, from 300 to 450 K. Parameters obtained by Mu¨ller and Gubbins are also provided for comparison. See text for details. b L ) 1.00, φ ) 5.00.

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Figure 4. Vapor-liquid equilibrium of several n-alkanes. Dotted circles, hexagons, diamonds, squares, and circles represent experimental data of methane, propane, hexane, octane, and decane, respectively.53 The dashed line is the original soft-SAFT calculation while the solid line represents the same equation with a crossover treatment.

Figure 5. Vapor-liquid equilibrium of benzene. Symbols and lines as in Figure 4.

high temperatures improves only very close to the critical point, remaining identical to the classical EoS for the rest of temperatures. Alkanes. The molecular parameters of the n-alkanes investigated in this work were taken from published correlations with the molecular weight of the compounds, without22 and with the crossover term.13 The theoretical predictions of both sets of parameters are shown in Figure 4. As can be seen, the softSAFT predictions (with and without the crossover term) provide an accurate description of the whole phase diagram. In addition,

Vega et al.

Figure 6. Mutual solubilities of the system n-hexane + water. Circles represent experimental data.56 The solid lines are soft-SAFT predictions while the dashed and dotted-dashed lines represent, from bottom to top, SAFT 3-site (vdW rule), SAFT 4-site (vdW rule), SAFT 3-site (PHCT), and SAFT 4-site (PHCT) calculations.

if the renormalization group term is also included in the free energy, the critical region is accurately described. Similar results were achieved for the benzene molecule (see Figure 5), whose parameters were taken from ref 55 without crossover and were optimized by applying the renormalization group treatment for this work. They are used here in a transferable manner and their values are provided in Table 2 for completeness. B. Water + Hydrocarbon Mixtures. We now turn to the prediction of the phase behavior of several water + alkane binary mixtures. In particular, the main interest of this work is to check the ability of the soft-SAFT approach for predicting the existence of a minimum in the solubility of hydrocarbons in water at low temperatures. This minimum is observed at approximately 320 K in the LLG three-phase line of the water + n-alkane homologous mixtures. We first studied the water + n-hexane mixture, which can be considered a regular member of the homologous series. A binary energy parameter, ξ ) 0.68, was used to fit the solubility of water in n-hexane, and the same value was used to predict the solubility of n-hexane in water. Figure 6 shows the mutual solubilities of water and n-hexane, as functions of temperature, over a wide range of temperatures, from near the triple point of water to near the upper critical end point of the mixture. As can be seen, agreement between theoretical predictions and experimental data taken from the literature56 is excellent at all temperatures considered. Apart from the obvious agreement between both results in the solubility of water in n-hexane (this property was fitted during the optimization), the agreement between both data along the solubility curve of n-hexane in

TABLE 2: Soft-SAFT Molecular Parameters for Benzene and Toluene Used in This Worka benzene without crossover with crossover toluene without crossover a

M

σ (Å)

ε/kB (K)

Q (C m2)

L/σ

φ

2.333 2.343

3.754 3.754

299.3 304.7

-5.0 × 10-40 -5.0 × 10-40

1.195

7.20

2.692

3.925

296.5

-5.0 × 10-40

-

-

Those used without the crossover term are taken from ref 55 and are provided here for completeness.

Solubility Minima of n-Alkanes in Water water is remarkable. The soft-SAFT approach is able to capture not only the correct order of magnitude of these solubilities but also the minimum of the solubility of n-hexane in water at low temperature, in excellent agreement with experimental data. To our knowledge, this is the first time a SAFT approach is able to quantitatively predict the existence of this minimum. The performance of the soft-SAFT equation is also compared with predictions from other versions of SAFT and mixing rules. Results are compared here with results obtained from the original SAFT version with different water models (three- and four-site models) and two mixing rules (the traditional vdW one-fluid approximation and the PHCT mixing rules, as calculated by Economou and Tsonopoulos1). All of them are able to describe the solubility of n-hexane in water (as expected since this property has been fitted directly to obtain the value of ξ) but they are unable to predict, even qualitatively, the solubility of water in n-hexane. A plausible explanation for the different results obtained by the two versions of the equation lies in the fact that soft-SAFT uses a more realistic Helmholtz free energy expression than the original SAFT, including both attractive and dispersive interactions in the same term, which also appears in the radial distribution function used in the chain and association term. It is important to recall that these soft-SAFT calculations were done using a set of parameters optimized in the range of 300-450 K. Figure 7 shows these results for the different water parameters sets presented in Table 1. We have followed the same procedure as the one explained for optimizing the water + n-hexane mixture, optimizing the binary parameter for the water solubility in n-hexane for each set. When calculations were done with the same soft-SAFT model but with parameters optimized for the whole envelope, the minimum is still reproduced, but it appears at higher temperatures and there is an unfeasible crossing in the mutual solubilities around 300 K. The best results (apart from the ones obtained with set 1) are obtained when the Mu¨ller and Gubbins parameters for water are used for the calculations. Note that the Mu¨ller and Gubbins dispersive energy parameter is lower than for sets 2 and 3. As explained in the previous section, the 300-350 K temperature range falls outside the range of applicability of the LJ equation for high values of the dispersive energy; when these sets are used unphysical results may be obtained. Hence, only the set 1 parameters for water (corresponding to those optimized in the 300-450 K range) are used for the rest of the calculations. Figures 8 and 9 show similar calculations to those presented in Figure 6 but involving the mutual solubilities of water + n-octane and water + n-decane, respectively. Comparisons between theoretical predictions and experimental data taken from the literature are shown in both plots. Figure 9 also shows predictions obtained from the original SAFT EoS for the mutual solubilities of water and n-decane with the two mixing rules previously presented in Figure 6. Only the four-site model for water in the original SAFT was considered in this case since the three-site model provides a worse description of the solubilities of the n-alkanes in the waterrich liquid phase. As can be seen, the agreement between the theoretical predictions from soft-SAFT is superior to those obtained from the original SAFT (with all the different mixing rules), and are in excellent agreement with the experimental data taken from the literature.57,58 This is especially remarkable if we take into account that no single experimental data has been used to predict these mixtures (ξ ) 0.68 is the value obtained from the optimization of the solubility of water in the n-hexane-rich liquid phase and it is used in a transferable manner for the rest of the series).


Figure 7. Mutual solubilities of the system n-hexane + water. Circles represent experimental data.56 (a) Solid line, soft-SAFT calculations with set 2 from Table 1 (global without crossover) and ξ ) 0.60. (b) Solid line, soft-SAFT calculations with set 3 from Table 1 (global with crossover) and ξ ) 0.55 and (c) solid line, soft-SAFT calculations with set 4 from Table 1 (Mu¨ller and Gubbins parameters) and ξ ) 0.65. The rest of the lines as in Figure 6. See text for details.

Given the excellent performance of the model for these mixtures we have decided to use the same approach to check the accuracy of the model for the rest of the members of the series. The binary parameter fitted for the mixture water + n-hexane (ξ ) 0.68) was used to predict the behavior of the

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Figure 8. Mutual solubilities of the system n-octane + water. Circles represent experimental data.57 The solid lines are soft-SAFT predictions. Figure 10. Solubility of water in methane (gas phase) at different isotherms from 308.11 to 477 K. Empty squares (308.11 K), circles (313.12 K), and triangles (318.12 K) represent experimental data from ref 60. Dotted diamonds (366.5 K), circles (422.04 K), squares (466.5 K), and triangles (477.5 K) are experimental data from ref 61. The solid lines are soft-SAFT predictions.

Figure 9. Mutual solubilities of the system n-decane + water. Circles represent experimental data.58 The solid lines are soft-SAFT predictions while the dashed and dotted-dashed lines represent, from bottom to top, SAFT 4-site (vdW rule) and SAFT 4-site (PHCT) calculations.

water + n-alkanes binary mixtures, with no further fitting. The first mixture considered was the water + methane mixture. This is the only mixture of the series that does not exhibit LLG equilibria, which may be due to the small size of the methane molecule that permits to be dissolved in water. The vapor-liquid equilibrium of this mixture (at room temperature) is of crucial interest because of the hydrate formation at low temperatures, which often results in the blockage and shutdown of the gas and oil subsea pipelines.59 Figure 10 shows the solubility of water in methane (gas phase) at different temperatures, from 308.11 K (close to hydrate formation) up to 477 K. As can be observed, the equation is able to predict the behavior of the solubility, as a function of composition, at all temperatures, in excellent agreement with the experimental data taken from the literature.60,61 The first water + n-alkane binary mixture that shows LLG three-phase phase behavior is the water + propane system.

Figure 11. Mutual solubilities of the system propane + water. Circles represent solubilities in both liquid phases while the triangles are the solubility of water in the gas phase. All the experimental data are taken from ref 62. The solid lines are soft-SAFT predictions.

Figure 11 shows the theoretical predictions for the mutual solubility of water and propane over a wide range of temperatures. The theoretical calculations are able to quantitatively predict the solubility of the mixture in the gas phase as well as the solubility of water in the propane-rich liquid phase. Unfortunately, only qualitative agreement is achieved for the solubility of propane in water.62 Although the theory is able to describe at a qualitative level the existence of a minimum in the solubility of propane in water, the equation overestimates the solubility of propane in the water-rich liquid phase one order



Figure 13. Mutual solubilities of the system toluene + water with the same binary parameters used for the solubility of water in benzene (ξ ) 0.90, η ) 1.04). Circles represent experimental data63 while the full lines represent the soft-SAFT predictions.

Figure 12. Mutual solubilities of the system benzene + water (a) using the same binary parameter value employed for the alkanes-water mixtures (ξ ) 0.68) and (b) fitting two binary parameters to the solubility of water in benzene (ξ ) 0.90, η ) 1.04). Circles represent experimental data.56 Lines as in Figure 6.

of magnitude approximately in the whole range of temperatures considered. These results could be expected, as the lighter members of the n-alkanes series do not show the regular behavior obtained by longer members (the series is expected to behave regularly from butane on). It is important to understand that the binary interaction parameter was transferred from the solubility of n-hexane in water calculations along the LLG threephase line. A much better agreement for the water + n-propane mixtures can be obtained by changing the value of the binary interaction parameter, but this would result in a loss of the transferability power of the equation as it was used here. Finally, we have considered the phase behavior of water with the most important aromatic compound, benzene. Since this molecule is forming a ring and it does not have the same structure as n-alkanes, one expects that the binary parameter value optimized for the previous mixtures will not be valid in the present case either. Figure 12a shows the mutual solubilities of water and benzene, as functions of the temperature, over a wide range of temperatures. As can be seen, when the value of

the binary parameter optimized for the water + n-alkane homologous series of mixture (ξ ) 0.68) was used to predict the mutual solubilities of water and benzene, only qualitative agreement was achieved, although the solubility minima was obtained. A new calculation with an optimized value of ξ ) 0.92 and a second binary parameter η ) 1.04 is provided in Figure 12b; when these parameters are used both liquid phases are perfectly described with the soft-SAFT equation. The SAFT four-site (vdW rule) and SAFT four-site (PHCT) calculations are also shown for comparative purposes. Given the excellent performance of the equation for benzene + water, we have checked the transferability of the binary parameters for another aromatic compound, toluene. The softSAFT molecular parameters for this compound were taken from ref 55. Results from the calculations for the mutual solubilities of water and toluene as compared to experimental data63 are presented in Figure 13. As it can be observed, the predictions of the model are excellent, enhancing the accuracy of the model and the transferability of the parameters used. 6. Conclusions The soft-SAFT equation of state has been successfully applied to accurately model water and water + hydrocarbon binary mixtures, including the mutual solubility of water and hydrocarbons. The use of a renormalization-group treatment into the soft-SAFT equation has permitted an accurate evaluation of the critical region. Water was modeled using a four-site LJ sphere with different sets of parameters; the molecular parameters for the equation were obtained by fitting to vapor-liquid coexistence data taken from the literature. An overall optimization of the soft-SAFT equation provided a quantitative agreement with the experimental data over a wide range of temperatures. In particular, the crossover version of the equation provided excellent agreement with experimental data from 350 K until the critical point. However, some unrealistic features were observed at room temperature. These problems were attributed to the high value of the dispersive energy of water found during the optimization procedure of the molecular parameters; when these values are used the LJ multiparameter equation of the reference term is used outside its fitted range in temperature.

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In order to avoid this problem, and based on the fact that the present study focused on water at near-ambient conditions, an additional set of soft-SAFT parameters was fitted in a (more reduced) range of temperature, from 300 to 450 K. This choice allows a better description of the thermodynamic properties of water close to these temperatures. In particular, the new set of parameters provides accurate predictions of the degree of hydrogen bonding in water, in quantitative agreement with experimental data over a wide range of temperatures, a key property related to the unusual behavior of water in mixtures. Soft-SAFT, with the set of parameters fitted to the limited range of temperatures, was used to predict the behavior of water + n-alkanes mixtures as well as the water + benzene and water + toluene mixtures. The equation was able to capture the main features of mixtures of water + n-hexane, n-octane, and n-decane, including the solubility minima of hydrocarbons in water at low temperatures, in excellent agreement with experimental data taken from the literature. Comparison with calculations performed with the original SAFT using different model sites for water and different mixing rules show that soft-SAFT is clearly superior to all of them. Excellent results for the water + aromatic compounds were also obtained. The model also gave an excellent description of the water with methane, propane, benzene, and toluene mixtures. A single energy parameter, independent of temperature, and with the same value for the whole series was used in this work. To our knowledge, this is the first time that a theoretical treatment based on the SAFT approach is able to capture the solubility minima of hydrocarbons in water. This successful result is ascribed to the accurate and realistic reference term and radial distribution function used in the soft-SAFT equation of state versus other versions of SAFT. Acknowledgment. We are indebted to Ioannis Economou for several discussions, for encouraging us to perform this work several years ago, and for providing data for comparison (Figures 6-9 and 12). Helpful discussions with Carlos Vega are also gratefully acknowledged. This research has been possible thanks to the financial support received from the Spanish Government (projects CTQ2005-00296/PPQ, CTQ200805370/PPQ, as well as NANOSELECT and CEN2008-01027, a Consolider and a CENIT project, respectively, both belonging to the Programa Ingenio 2010). Additional support from the Catalan Government is also acknowledged (SGR2005-00288). F.J.B. also acknowledges the financial support from the Spanish Government project FIS2007-66079-C02-02. References and Notes (1) Economou, I. G.; Tsonopoulos, C. Chem. Eng. Sci. 1997, 52, 511. (2) Vega, C.; Abascal, J. L. F.; Conde, M. M.; Aragones, J. L. Faraday Discuss. 2009, 141, 251. (3) Errington, J. R.; Boulougouris, G. C.; Economou, I. G.; Panagiotopoulos, A. Z.; Theodorou, D. N. J. Phys. Chem. B 1998, 102, 8865. (4) Boulougouris, G. C.; Errington, J. R.; Economou, I. G.; Panagiotopoulos, A. Z.; Theodorou, D. N. J. Phys. Chem. B 2000, 104, 4958. (5) Zabaloy, M. S.; Mabe, G. D. B.; Bottini, S. B.; Brignole, E. A. Fluid Phase Equilib. 1993, 83, 159. (6) Voutsas, E. C.; Boulougouris, G. C.; Economou, I. G.; Tassios, D. P. Ind. Eng. Chem. Res. 2000, 39, 797. (7) Yakoumis, I. V.; Kontogeorgis, G. M.; Voutsas, E. C.; Hendriks, E. M.; Tassios, D. P. Ind. Eng. Chem. Res. 1998, 37, 4175. (8) Grenner, A.; Schmelzer, J.; von Solms, N.; Kontogeorgis, G. M. Ind. Eng. Chem. Res. 2006, 45, 8170. (9) Karakatsani, E. K.; Kontogeorgis, G. M.; Economou, I. G. Ind. Eng. Chem. Res. 2006, 45, 6063.

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Capturing the Solubility Minima of n-Alkanes in Water by Soft-SAFT

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