Prediction of Thermodynamic Properties and Phase Behavior of Fluids

Article pubs.acs.org/jced

Prediction of Thermodynamic Properties and Phase Behavior of Fluids and Mixtures with the SAFT‑γ Mie Group-Contribution Equation of State Simon Dufal, Vasileios Papaioannou, Majid Sadeqzadeh, Thomas Pogiatzis, Alexandros Chremos, Claire S. Adjiman, George Jackson, and Amparo Galindo* Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom ABSTRACT: Group contribution (GC) approaches are based on the premise that the properties of a molecule or a mixture can be determined from the appropriate contributions of the functional chemical groups present in the system of interest. Although this is clearly an approximation, GC methods can provide accurate estimates of the properties of many systems and are often used as predictive tools when experimental data are scarce or not available. Our focus is on the SAFT-γ Mie approach [Papaioannou, V.; Lafitte, T.; Avendaño, C.; Adjiman, C. S.; Jackson, G.; Müller, E. A.; Galindo, A. Group contribution methodology based on the statistical associating fluid theory for heteronuclear molecules formed from Mie segments. J. Chem. Phys. 2014, 140, 054107−29] which incorporates a detailed heteronuclear molecular model specifically designed for use as a GC thermodynamic platform. It is based on a formulation of the recent statistical associating fluid theory for Mie potentials of variable range, where a formal statistical−mechanical perturbation theory is used to maintain a firm link between the molecular model and the macroscopic thermodynamic properties. Here we summarize the current status of the SAFT-γ Mie approach, presenting a compilation of the parameters for all functional groups developed to date and a number of new groups. Examples of the capability of the GC method in describing experimental data accurately are provided, both as a correlative and as a predictive tool for the phase behavior and the thermodynamic properties of a broad range of complex fluids.

■

INTRODUCTION Thermodynamic models play a crucial role in the design and optimization of processes across different applications of chemical engineering and physical sciences. They are employed for the calculation and prediction of the thermodynamic properties of the materials associated with the process and can greatly affect the design, cost, and in some cases even determine the feasibility of a given unit operation.1 The need for an accurate thermodynamic description of an increasing variety of compounds to support the design of processes places a strong emphasis on the development of techniques with predictive capabilities that can be used for a wide range of process conditions and mixtures with limited reliance on experimental data beyond that used at the early stages of the process development. A specific class of predictive models that has drawn much attention over the past 40 years are group-contribution (GC) methods. In group-contribution approaches molecules are modeled in terms of the functional groups that they comprise, and it is proposed that the properties of any given system of interest can be obtained from appropriate contributions of the corresponding groups to the thermodynamic properties of the system. Functional groups are typically characterized by assessing a large amount of experimental data, but once parameters that describe a given set of groups have been determined, these can then be used in a straightforward © XXXX American Chemical Society

manner for the prediction of the properties of any compound or mixture that features the functional groups. The initial formulation of GC methods was intended for the prediction of thermodynamic properties of pure components, e.g., the methods of van Krevelen,2 Joback and Reid,3 and the more recent work of Gani and co-workers.4,5 GC approaches have also been developed for the study of binary and multicomponent mixtures, initially developed from the contribution that each group makes to the activity coefficients in liquid mixtures. The most prominent example of approaches of activity-coefficient-based methods is the well-known universal functional activity coefficient (UNIFAC) approach and its modifications.6−9 To overcome some of the limitations inherent to activitycoefficient approaches, such as the treatment of the liquid phase alone, the GC concept has also been applied within equations of state (EoSs), which provide a consistent platform for both liquid and vapor phases. This line of work has led to the development of, among others, the group-contribution EoS,10,11 Special Issue: Modeling and Simulation of Real Systems Received: March 19, 2014 Accepted: July 24, 2014

A

dx.doi.org/10.1021/je500248h | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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the group-contribution associating EoS,12,13 the predictive Soave−Redlich−Kwong EoS,14 the volume-translated Peng− Robinson EoS,15,16 and the predictive Peng−Robinson approach.17 In view of the success of the statistical associating fluid theory (SAFT)18,19 and its many incarnations20−22 in the modeling of a wide range of challenging and complex systems, the GC concept has also been applied in recent years within the framework of SAFT. Early work on GC-based SAFT-type approaches focused on developing methodologies for the calculation of the intermolecular parameters for use within SAFT from the appropriate contributions of the specified functional groups, while the underlying molecular model and theory remained unchanged. These so-called homonuclear approaches are based on the original molecular description proposed by Chapman et al.,18,19 where molecules are represented as associating chains of bonded identical spherical segments. Several functional forms of the segment−segment intermolecular potential (square-well and Lennard−Jones) have been considered; see the work of Vijande and co-workers23−25 and Tobaly and co-workers26−29 as illustrative examples. As an alternative to these homonuclear approaches, the GC concept has been employed within a SAFT formalism based on a more detailed heteronuclear molecular model, with different types of monomeric segments used to describe the different chemical functional groups constituting a given molecule. Such a heteronuclear molecular model is formally implemented in the SAFT-γ30−32 and SAFT-VR-GC33 approaches, and has the key advantage of allowing for the prediction of thermodynamic properties of mixtures based on pure component data alone.31,33−35 The unlike EoS parameters which characterize the interactions between the different groups present in a mixture are determined from selected pure component data that contain the relevant unlike interactions or from appropriate combining rules (e.g., see ref 36). The most recent variant of SAFT-based group-contribution approaches is the SAFT-γ Mie EoS,32 where a heteronuclear model is implemented and a Mie (generalized Lennard−Jonesium) potential of variable repulsive and attractive ranges is used to represent the segment−segment interactions. In our current work we examine the performance of the SAFT-γ Mie approach for the description of a broad variety of properties of pure fluids and binary mixtures including phase behavior (vapor−liquid, liquid−liquid, vapor−liquid−liquid, solid−liquid), single−phase derivative properties (compressibility, heat capacity, and speed of sound), and excess properties of mixing (enthalpy, volume, heat capacity, and speed of sound). Comparisons of the calculations are made with the experimental data used in the development of the group parameters and for predictions of properties not used in the characterization of the groups. A brief summary of the SAFT-γ Mie theoretical formalism is first presented, followed by a short discussion of the parameter estimation scheme applied to obtain the parameters that describe each functional group within the context of the theory. The performance of the theory in the description of the thermodynamic properties of selected pure components and binary systems is then illustrated, followed by a discussion of the main conclusions from our work.

Figure 1. Example of the decomposition of a molecule into functional groups: pentanoic acid is composed of one CH3 group (shaded blue), three CH2 groups (shaded gray), and one COOH group (shaded red, with an association site in black to mediate the dimerization in the system).

Each chemical functional group k (CH3, CH2, ...) is represented as a fused spherical segment or number of segments vk*. Two segments k and l are assumed to interact via a Mie potential of variable range:37 ⎡⎛ ⎞ λklr ⎛ ⎞ λkla ⎤ σ σ ΦklMie(rkl) = *klϵkl ⎢⎜ kl ⎟ − ⎜ kl ⎟ ⎥ ⎢⎝ rkl ⎠ ⎝ rkl ⎠ ⎥⎦ ⎣

(1)

where rkl is the distance between the centers of the segments, σkl the segment diameter, ϵkl the depth of the potential well, and λrkl and λakl the repulsive and attractive exponents of the segment−segment interactions, respectively. The prefactor * kl is a function of these exponents and ensures that the minimum of the interaction is −ϵkl: a

r

a

λkl /(λkl − λkl) λklr ⎛ λklr ⎞ *kl = r ⎜ ⎟ λkl − λkla ⎝ λkla ⎠

(2)

The Helmholtz free energy of this fluid can be obtained from the appropriate contributions of the different groups, noting that the implementation of this type of united-atom model of fused segments requires the additional use of a shape factor Sk, which reflects the proportion which a given segment contributes to the total free energy. It has been shown30,31 that, while the SAFT-γ equation provides a very accurate representation of a fluid of model heteronuclear chains of tangentially bonded spherical segments, such a model is not appropriate to treat real fluids at a molecular level; we should note that tangential models have been used reliably in the development of coarse-grained force fields for use in molecular dynamics simulation38−40 with the SAFT-γ Mie approach, but we do not consider this type of coarser treatment here. The precise form of the SAFT-γ free energy stems from the first-order thermodynamic perturbation theory (TPT1) of Wertheim,41−45 who showed that an expression for the free energy of chains of tangentially bonded segments can be obtained when the free energy and structure of a corresponding reference monomeric fluid are known. As in other SAFT approaches, hydrogen bonding or strongly polar interactions can be treated through the incorporation of a number of additional short-range square-well association sites, which are placed on any given segments as required. The association interaction between two square-well association sites of type a in segment k and b in segment l is given by

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SAFT-γ MIE MODEL AND THEORY In the SAFT-γ Mie32 approach molecules are represented as associating heteronuclear chains of fused spherical segments. As an example, a SAFT-γ Mie representation of pentanoic acid is given in Figure 1.

ΦklHB, ab(rkl , ab) B

HB c ⎧ ⎪− ϵkl , ab if rkl , ab ≤ rkl , ab =⎨ ⎪ if rkl , ab > rklc , ab ⎩ 0

(3)



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where rkl, ab is the center−center distance between sites a and c b, −ϵHB kl, ab is the association energy, and rkl, ab the cutoff range of the interaction between sites a and b on groups k and l, respectively the cutoff distance rckl,ab can equivalently be described in terms of a bonding volume Kkl,ab.46 Each site is positioned at a distance rdkk, aa from the center of the segment on which it is placed; at the TPT1 level the relative positions of the various sites on a segment are not taken into account explicitly. In summary, a functional group k is fully described by the number v*k of identical spherical segments forming the group, the shape factor of the segments Sk, the diameter of the segments σkk, the segment energy of interaction ϵkk, and the exponents of the Mie potential λrkk and λakk, as well as the parameters characterizing any site−site association interactions as described previously. In the case of associating groups several additional parameters are required: the number NST,k of different site types, the number of sites of each type, e.g., nk,a, nk,b, ..., nk,NST,k, together with the energy ϵHB kk,ab and the bonding volume parameter Kkk,ab for the association between sites of the same or of different type. The parameters rdkk,ab and rckk,ab are both fixed to a value of 0.4σkk in our version of the theory.47 By extension, the interactions between groups of different types k and l are characterized by the corresponding unlike parameters σkl, ϵkl, λrkl, λakl, ϵHB kl,ab, and Kkl,ab. The relevant parameters need to be determined in order to evaluate the thermodynamic properties of a given pure fluid or mixture. As is common in a formal statistical−mechanical treatment within the canonical ensemble the link between the molecular model and the macroscopic thermodynamic properties is given in terms of the Helmholtz free energy; in our case the expressions are those of the SAFT-γ Mie EoS.32 The total Helmholtz free energy A of a mixture of associating heteronuclear chains of fused spherical segments that interact via Mie potentials can be written in the usual SAFT form as the sum of four separate contributions: A = Aideal + Amono + Achain + Aassoc

characterized by the Mie potential is obtained following a Barker−Henderson49,50 high-temperature perturbation expansion up to third order,51 which can be expressed as A3 A1 A2 Amono AHS = + + + NkBT NkBT NkBT NkBT NkBT

where the repulsive term A is the free energy of a hard-sphere reference system of diameter dkk, which is temperature-dependent.50 In the SAFT-γ group-contribution treatment each of the segment’s contribution to the free energy is considered following30 NG ⎛ NC ⎞ AHS = ⎜⎜∑ xi ∑ νk , i νk*Sk ⎟⎟a HS NkBT ⎝ i=1 k=1 ⎠

(7)

to obtain the total hard-sphere Helmholtz free energy of the mixture of segments. Here NG is the number of types of groups present, νk,i the number of occurrences of a group of type k on component i, and aHS is the dimensionless contribution to the hard-sphere free energy per segment, obtained using the expression of Boublík52 and Mansoori et al.,53 which is a function of the density ρs, the mole fractions, and diameters, of the segments in the fluid. In SAFT-γ the segment density is related to the molecular number density by ⎛ NC NG ⎞ ρs = ρ⎜⎜∑ xi ∑ νk , i νk*Sk ⎟⎟ ⎝ i=1 k=1 ⎠

(8)

and the appropriate moment densities are given by πρs

ζm =

6

NG

∑ xs,kdkkm

m = 0, 1, 2, 3 (9)

k=1

where the Barker-Henderson hard-sphere diameter dkk of the reference fluid is used, and xs,k is the fraction of segments of a group of type k in the mixture:

(4)

N

ideal

where A is the free energy of an ideal gas mixture of the corresponding molecules, Amono accounts for the Mie segment− segment interactions (repulsion and dispersion) of the reference monomeric system, Achain accounts for the change in the free energy due to the formation of molecules from Mie segments, and Aassoc is the term accounting for the association interactions. The detailed expressions for each of these terms are given in the original publication32 where the association term has been replaced by the Lennard−Jones based associative contribution described in ref 47; here we provide only a summary of the key aspects of the approach. Ideal Term. The free energy corresponding to an ideal mixture of molecules is given by48 ⎛ NC ⎞ Aideal = ⎜⎜∑ xi ln(ρi Λi3)⎟⎟ − 1 NkBT ⎝ i=1 ⎠

(6)

HS

xs, k =

∑i =C1 xi νk , i νk*Sk N N ∑ j =C1 xj∑l =G1 νl , j νl*Sl

(10)

After substituting this expression for the group fraction xs,k in the definition of the moment densities (eq 9) and expressing aHS as a function of the molecular density ρ, one obtains the usual form of the Helmholtz free energy of a hard-sphere mixture.51 Subsequent terms in the expansion are obtained following similar summations over the free-energy contributions per segment, each with their corresponding power of inverse temperature, so that the mean-attractive energy, the energy fluctuation, and the third-order terms are given by ⎞ ⎛ 1 ⎞q ⎛ NC NG =⎜ ⎟ ⎜⎜∑ xi ∑ νk , i νk*Sk ⎟⎟aq NkBT ⎝ kBT ⎠ ⎝ i = 1 k = 1 ⎠ Aq

q = 1, 2, 3 (11)

(5)

The free-energy contributions per segment aq are obtained, as for the hard-sphere reference term, by summing the pairwise interactions aq,kl between groups k and l over all pairs of functional groups present in the system:

where xi is the mole fraction, ρi = Ni/V the number density, with Ni the number of molecules of component i in the mixture, V the total volume, N the total number of molecules, kB the Boltzmann constant, and T the absolute temperature. The summation is over N all of the components NC of the mixture (i.e., N = ∑i CNi). The ideal free energy incorporates the effects of the translational, rotational and vibrational contributions to the kinetic energy implicitly in the thermal de Broglie volume Λ3i . Monomer Term. The free-energy contribution due to repulsive and attractive interactions for the monomeric fluid

NG NG

aq =

∑ ∑ xs ,kxs ,l aq,kl k=1 l=1

q = 1, 2, 3 (12)

explicit expressions for these contributions are given in ref 51. The calculation of the first perturbation term a1 requires a knowledge of the radial distribution function of the reference C



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hard-sphere fluid for a range of separations and subsequent numerical integration. In order to obtain an analytical expression for this contribution the mean-value theorem can be employed, following the original SAFT-VR54 methodology, and a mapping of the real density of the fluid to an effective density is carried out. This mapping has been undertaken for ranges of exponents of the Mie potentials 5 < λkl ≤ 100. The fluctuation term a2 is calculated following the improved macroscopic compressibility approximation (MCA) proposed by Zhang et al.55 combined with a correction in the same spirit as that proposed by Paricaud56 for soft potentials, using a number of coefficients obtained from the analysis of Monte Carlo molecular simulation data of selected Mie fluids. An empirical expression is used for the third-order term a3; it is worth noting that the coefficients in the expression were obtained from phase equilibrium and critical data of selected Mie fluids, so that higher-order terms of the expansion (in essence the entire series) are taken into account in an effective manner. This allows one to obtain an excellent representation of the of real fluids even for near-critical regions of the thermodynamic space.51 Chain Term. In the SAFT-γ approach the change in free energy associated with the formation of a molecule from its constituting segments is obtained using average molecular parameters (σ̅ii, d̅ii, ϵi̅ i, and λ̅ii) for each molecular species i,30,31 so that the formal TPT1 expression45,57 can be retained. The averaging of the molecular size and energy parameters is independent of the composition of the mixture; it requires the molecular fraction zk,i of a given group k in a molecule i: zk , i =

to represent this function:51 giiMie(σii̅ ; ζx ) = gdHS(σii̅ ; ζx ) exp[βϵii̅ g1(σii̅ )/gdHS(σii̅ ; ζx ) + (βϵii̅ )2 g2(σii̅ )/gdHS(σii̅ ; ζx )]

Boublík’s58 expression is employed for the evaluation of gHS d (σi̅ i; ζx), which represents the radial distribution function of a system of hard spheres of diameter d̅ii evaluated at distance σi̅ i and packing fraction ζx. The first and second-order terms are approximated by their value at the contact distance d̅ii, i.e., gq(σii̅ ) ≈ gq(dii̅ )

N

(13)

The average molecular segment diameter σ̅ii and the reference hard-sphere diameter d̅ii are then defined as

Aassoc = NkBT

NG NG

σii̅ 3 =

∑ ∑ zk ,i zl , i σkl3 k=1 l=1

3

NG NG

∑ ∑ zk ,i zl , i dkl3 k=1 l=1

(15)

Xi , k , a

respectively. The other molecular parameters are obtained in the same way, so that the average interaction energy ϵi̅ i and exponents which characterize the range of the potential λ̅ii are obtained as

∑ ∑ zk ,i zl , i ϵkl k=1 l=1

(16)

and NG NG

λii̅ =

∑ ∑ zk ,i zl , i λkl k=1 l=1

NC

NG

∑ xi ∑ νk , i i=1

k=1

NST, k

⎛

∑ nk , a⎜⎝ln Xi , k , a + a=1

1 − Xi , k , a ⎞ ⎟ ⎠ 2

(21)

⎡ NC NG = ⎢1 + ρ ∑ xj ∑ νl , j ⎢⎣ j=1 l=1

NST, l

∑ b=1

⎤−1 nl , bXj , l , bΔij , kl , ab⎥ ⎥⎦

(22)

where the integral Δij,kl,ab characterizes the overall strength of the association between a site of type a on a group of type k of component i and a site of type b on a group of type l of component j. It is approximated as

NG NG

ϵii̅ =

(20)

where nk,a is the number of sites of type a on group k, and Xi,k,a is the fraction of molecules of component i that are not bonded at a site of type a on group k. Xi,k,a is obtained from the solution of the mass action equations as given in refs 30, 43, and 46,

(14)

and dii̅ =

q = 1, 2

The first term in the expansion g1(σi̅ i; ζx) can be obtained by means of a self-consistent method for the calculation of pressure from the virial and the free-energy routes,51,54,59 while an expression based on the MCA with a correction term is used for the second perturbation term g2(σi̅ i; ζx). All of the parameters and free-energy terms needed are evaluated using the effective molecular parameters given in eqs 14 to 17. We refer the reader to the original SAFT-γ Mie paper for a more detailed description of the development of all of the expressions;32 note that there was a typographical error (misplaced bracket) in the expression for the chain term (eq 46 in ref 32), which is now correctly given by eq 18. Association Term. The contribution to the Helmholtz free energy due to the association of molecules via short-range bonding sites follows from the original TPT1 expressions of Wertheim41−44,46,57 by summing over the number of species NC, the number of groups NG, and the number of site types on each group NST,k so that

νk , i νk*Sk ∑l =G1 νl , i νl*Sl

(19)

Δij , kl , ab = Fkl , abKkl , abIij , kl , ab

(23)

where Fkl,ab = exp(ϵHB kl,ab/kBT) − 1, Kkl,ab is a bonding-volume parameter, and Iij,kl,ab is a temperature-density polynomial correlation of the association integral for a Lennard−Jones monomer, expressed as

(17)

respectively, for both the repulsive λ̅iir and the attractive λ̅iia exponents. The resulting contribution to the free energy of the mixture due to the formation of chains of segments using the effective molecular parameters is given by NC ⎛ NG ⎞ Achain = −∑ xi⎜⎜∑ νk , i νk*Sk − 1⎟⎟ln giiMie(σii̅ ; ζx ) NkBT ⎠ (18) i=1 ⎝k=1

10 10 − p

Iij , kl , ab =

⎛

⎞q

kT cpq(ρσx3) p ⎜⎜ B ⎟⎟ ⎝ ϵij̅ ⎠ p=0 q=0

∑∑

(24)

where the coefficients cpq are given in ref 47; σ3x is obtained as

where gMie ii (σi̅ i; ζx) is the value of the radial distribution function (RDF) evaluated at a distance σi̅ i in a hypothetical fluid of packing N N fraction ζx , defined as ζx = (π/6)ρs∑k=1G ∑l=1G x s,k x s,l d kl3 . A second-order expansion of the logarithm of the RDF is used

σx3

NG NG

=

∑ ∑ xs,kxs,l σkl3 k=1 l=1

D

(25)



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Table 1. Groups Developed for Use within the SAFT-γ Mie Approacha

a

The green shading indicates the group−group interaction parameters estimated from experimental data, while the grey shading indicates the unlike interaction parameters predicted using the combining rules given in eqs 28 to 31.

and ϵi̅ j is given by

ϵij̅ =

σii̅ 3σ ̅jj3 σij̅ 3

The exponents of the unlike segment−segment interaction λrkl and λakl are obtained as λkl = 3 +

ϵii̅ ϵjj̅ (26)

where σ̅ii is given by eq 14 and σii̅ + σjj̅ σij̅ = 2

σkk + σll 2

(27)

ϵklHB , ab =

Kkl , ab

(29)

Although a more rigorous approach would be to calculate the corresponding Barker-Henderson reference diameter by numerical integration using the segment diameter, as for the like group interactions, this simple rule has been shown to be of comparable accuracy, at a fraction of the computational cost.51 The unlike dispersion energy ϵkl between groups k and l is obtained by applying an augmented geometric mean (Berthelot-like rule), which also accounts for asymmetries in size:36 ϵkl =

σkk3 σll3 σkl3

ϵkkϵll

(32)

⎛ 3 Kkk , aa + = ⎜⎜ 2 ⎝

3

3 Kll , bb ⎞ ⎟ ⎟ ⎠

(33)

These combining rules provide a good first estimate of the values of the required unlike group parameters; however, it is best to use experimental data when available to estimate these parameters especially in the case of the unlike attractive interactions (dispersion and association energies). An advantage of the heteronuclear model used in our approach is that the use of pure component data is sufficient in many cases to obtain an accurate estimate of the values of these unlike energetic parameters. Once the functional form of the Helmholtz free energy is fully specified, other properties can be easily determined using standard thermodynamic relations.61 In particular, the pressure P = (−∂A/∂V)T,N and chemical potential μi = (∂A/∂Ni)T,V,Nj of each species i, with the total Gibbs free energy G = A + PV = ∑Ni CNi μi, are used to determine the fluid-phase equilibria of the systems of interest using the HELD algorithm.62

(28)

dkk + dll 2

ϵkkHB, aaϵllHB , bb

while the unlike bonding volume Kkl,ab is obtained as

The same combining rule is applied for the calculation of the unlike reference hard-sphere diameter dkl:

dkl =

(31)

which results from the imposition of the geometric mean of the integrated van der Waals energy (Berthelot rule) for a Sutherland fluid of range λkl.32 In associating mixtures, the unlike association energy ϵHB kl,ab can be obtained by using a simple geometric mean

Combining Rules. The expressions for the Helmholtz free energy presented in the previous section require the prescription of a number of unlike group parameters. These are typically determined using combining rules and refined by estimation from experimental data when required. As a consequence of the use of a heteronuclear model in the SAFT-γ EOS, the unlike group interactions are needed not only to treat mixtures, but also for pure component calculations. For example, the unlike interaction between the CH3 and CH2 groups required to represent pure n-alkanes has to be characterized in order to calculate their thermodynamic properties. The unlike segment diameter σkl is obtained using the Lorentz-like arithmetic mean of the like diameters:60 σkl =

(λkk − 3)(λll − 3)

■

GROUP PARAMETERS Wherever possible, the parameters that characterize the like interactions of each functional group are estimated by regression to the experimental data of a series of compounds that contain that group. In many cases, unlike interaction parameters that are not derived from combining rules are obtained by estimation to pure component data. Where a group constitutes a whole molecule (e.g., water), or when additional data are needed to increase the statistical significance of the parameters, mixture data are also used. The task of populating

(30) E



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Table 2. Like Group Parameters for Use within the SAFT-γ Mie Group-Contribution Approacha k

group k

v*k

Sk

λrkk

λakk

σkk/Å

(ϵkk/kB)/K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CH3 CH2 CH C aCH aCCH2 aCCH CH2 CH cCH2 COOH CH3COCH3 COO H2O CH3OH

1 1 1 1 1 1 1 1 1 1 1 3 1 1 2

0.57255 0.22932 0.07210 0.04072 0.32184 0.20859 0.20650 0.44887 0.20037 0.24751 0.55593 0.72135 0.65264 1.00000 0.83517

15.050 19.871 8.0000 8.0000 14.756 8.5433 8.0000 20.271 15.974 20.386 8.0000 17.433 31.189 17.020 19.235

6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000 6.0000

4.0773 4.8801 5.2950 5.6571 4.0578 5.2648 4.3128 4.3175 4.7488 4.7852 4.3331 3.5981 3.9939 3.0063 3.2462

256.77 473.39 95.621 50.020 371.53 591.56 61.325 300.90 952.54 477.36 405.78 286.02 868.92 266.68 307.69

NST,k

nk,H

nk,e1

nk,e2

1 1 1

0 0 0

1 1 1

0 0 0

3 3

1 1

2 1

2 1

2 2

2 1

2 2

0 0

v*k , Sk, and σkk are the number of segment constituting group k, the shape factor, and the segment diameter of group k, respectively; λrkk and λakk are the repulsive and attractive exponents, and ϵkk is the dispersion energy of the Mie potential characterizing the interaction of two k groups; NST,k represents the number of association site types on group k, with nk,H, nk,e1, and nk,e2 denoting the number of association sites of type H, e1, and e2 respectively. a

Table 3. Group Dispersion Interaction Energies ϵkl and Repulsive Exponent λrkl for Use within the SAFT-γ Mie GroupContribution Approacha k

l

group k

group l

(ϵkl/kB)/K

λrkl

k

l

group k

group l

(ϵkl/kB)/K

λrkl

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3 5 6 7 8 9

CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH3 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH2 CH CH CH CH CH CH

CH3 CH2 CH C aCH aCCH2 aCCH CH2 CH cCH2 COOH CH3COCH3 COO H2O CH3OH CH2 CH C aCH aCCH2 aCCH CH2 CH cCH2 COOH CH3COCH3 COO H2O CH3OH CH aCH aCCH2 aCCH CH2 CH

256.77 350.77 387.48 339.91 305.81 396.91 455.85 333.48 252.41 355.95 255.99 233.48 402.75 274.80 275.76 473.39 506.21 300.07 415.64 454.16 345.80 386.80 459.40 469.67 413.74 299.48 498.86 284.53 341.41 95.621 441.43 65.410 67.510 426.76 502.99

15.050 CR ″ ″ ″ ″ ″ ″ ″ ″ ″ 14.449 CR ″ 15.537 19.871 CR ″ ″ ″ ″ ″ ″ ″ ″ 11.594 CR ″ 17.050 8.0000 CR ″ ″ ″ ″

3 3 3 3 4 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 9 9 10 10 11 11 11 12 12 13 14 14 15

10 11 12 14 4 5 6 7 11 12 14 6 7 11 12 14 7 11 12 14 8 9 9 10 10 13 11 12 14 12 14 13 14 15 15

CH CH CH CH C aCH aCH aCH aCH aCH aCH aCCH2 aCCH2 aCCH2 aCCH2 aCCH2 aCCH aCCH aCCH aCCH CH2 CH2 CH CH cCH2 cCH2 COOH COOH COOH CH3COCH3 CH3COCH3 COO H2O H2O CH3OH

cCH2 COOH CH3COCH3 H2O C aCH aCCH2 aCCH COOH CH3COCH3 H2O aCCH2 aCCH COOH CH3COCH3 H2O aCCH COOH CH3COCH3 H2O CH2 CH CH cCH2 cCH2 COO COOH CH3COCH3 H2O CH3COCH3 H2O COO H2O CH3OH CH3OH

570.45 504.99 637.29 101.89 50.020 371.53 416.69 429.16 331.61 333.11 357.78 591.56 462.04 473.66 394.83 220.00 61.325 599.28 459.22 190.00 300.90 275.75 952.54 398.35 477.36 498.60 405.78 393.71 399.00 286.02 287.26 868.92 266.68 278.45 307.69

CR ″ ″ ″ 8.0000 14.756 CR ″ 9.0687 CR 38.640 8.5433 CR ″ ″ ″ 8.0000 CR ″ ″ 20.271 CR 15.974 CR 20.386 CR 8.0000 CR ″ 17.433 CR 31.189 17.020 CR 19.235

a In all cases the unlike diameters σkl and dkl as well as the unlike attractive exponent of the Mie potential λakl are obtained from eqs 28, 29, and 31, respectively. CR indicates that the unlike repulsive exponent λrkl is obtained from the combining rule given by eq 31. Only the interaction parameters that have been estimated from experimental data are reported in this table.

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HB Table 4. Group Association Energies ϵkl,ab and Bonding Volume Parameters Kkl,ab for Use within the SAFT-γ Mie GroupContribution Approach

k

l

group k

site a of group k

group l

site b of group l

(ϵHB kl,ab/kB)/K

Kkl,ab/Å3

5 6 7 11 11 11 11 12 12 12 12 14 14 14 15

14 14 14 11 14 14 14 12 14 14 14 14 15 15 15

aCH aCCH2 aCCH COOH COOH COOH COOH CH3COCH3 CH3COCH3 CH3COCH3 CH3COCH3 H2O H2O H2O CH3OH

e1 e1 e1 H H e1 e2 H H e1 e2 H H e1 H

H2O H2O H2O COOH H2O H2O H2O CH3COCH3 H2O H2O H2O H2O CH3OH CH3OH CH3OH

H H H H e1 H H e1 e1 H H e1 e1 H e1

563.56 563.56 563.56 6427.9 188.61 2450.1 214.55 980.20 1386.8 1588.7 417.24 1985.4 1993.5 1993.5 2062.1

339.61 339.61 339.61 0.8062 3075.5 11.141 1159.4 2865.2 188.83 772.77 1304.3 101.69 104.11 104.11 106.57

liquid densities (for a temperature range spanning from the triple point to approximately 90% of the experimental critical temperature) together with single-phase densities in the compressed liquid region (corresponding to high pressure). Limited mixture data for the excess thermodynamic properties (such as volumes and/or enthalpies of mixing) and fluid-phase behavior (vapor−liquid or liquid−liquid equilibria) are used where necessary. The parameter estimation functionality of the commercial software package gPROMS63 is used as the regression tool: the parameter estimation problem is described with a maximum likelihood formulation, and the resulting nonconvex optimization problem is solved using the built-in gradient-based nonlinear solver. A multistart gradient-based optimization algorithm is also used when mixture fluid-phase equilibrium data are included in the estimation. In such a case, the HELD algorithm62 is used to calculate the equilibrium phases, thereby ensuring that phase stability is achieved. In order to make direct comparisons with the experimental data, the accuracy of the theoretical description of a generic property R consisting of nR separate data points is quantified in terms of the percentage average absolute deviation (%AAD) defined as

Table 5. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) for Selected Families of Compounds with the SAFT-γ Mie Group-Contribution Approacha family

% AAD Pvap(T)

% AAD ρsat(T)

nC

n-alkanes n-alkenes branched alkanes (CH group) branched alkanes (C group) n-alkylbenzenes n-alkyl esters n-alkyl carboxylic acids

1.55 2.70 2.78 9.76 2.64 0.83 2.74

0.59 0.77 0.85 6.30 0.66 0.24 0.69

9 15 9 9 9 6 7

further details Table Table Table Table Table Table Table

6 7 8 9 10 11 12

a nC indicates the number of compounds included in the parameter estimation procedure. For full details of the compounds and data considered the reader is referred to Tables 6 to 12, as indicated in the last column.

the parameter table is undertaken in a sequential manner, whereby the parameters for the methyl CH3 and methylene CH2 groups are determined first by using experimental data of the n-alkane series. These group parameters are then transferred to represent other chemical families and subsequently allow for the estimation of other group parameters, such as, for example, the methine CH group of the branched alkanes, or the carboxyl COO group of the n-alkyl esters, etc.32 When the parameters are estimated using experimental data of pure components, the data include vapor pressures and saturated

%AADR =

1 nR

nR

∑ i=1

R iexp − R icalc × 100 R iexp

(34)

where the exp and calc superscripts denote measured and calculated quantities, respectively. In this work we present the

Table 6. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for the n-Alkanes Considered in the Parameter Estimation compound

T range/K

n

%AAD Pvap(T)

ref

T range/K

n

%AAD ρsat(T)

ref

ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane

125−275 147−332 170−385 187−422 201−456 216−486 227−512 237−532 245−555

31 38 44 48 52 55 58 60 63

2.24 2.22 1.27 1.90 1.68 1.01 1.22 0.69 1.75

104 104 104 104 104 104 104 104 104

125−275 147−332 170−385 187−422 201−456 216−486 227−512 237−532 245−555

31 38 44 48 52 55 58 60 63

1.48 0.74 0.37 0.36 0.27 0.46 0.54 0.59 0.52

104 104 104 104 104 104 104 104 104

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Table 7. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for Selected n-Alkenes compound

T range/K

n

%AAD Pvap(T)

ethene propene but-1-ene pent-1-ene hex-1-ene trans-but-2-ene hept-1-ene oct-1-ene trans-pent-2-ene trans-hex-2-ene dec-1-ene cis-but-2-ene cis-pent-2-ene cis-hept-2-ene buta-1,3-diene

104−257 120−320 213−371 213−413 213−445 206−293 223−433 243−453 213−383 292−341 360−445 195−302 281−427 314−371 164−373

31 22 26 35 48 25 39 40 19 13 18 33 20 31 23

2.06 2.07 0.92 1.39 1.97 9.90 1.61 1.06 2.35 2.71 0.23 6.44 0.56 1.70 5.57

ref 105 105 106, 108, 108, 106 108, 108, 108 120 116 121 121 121 122

107 110, 111 113−115 116 116

T range/K

n

%AAD ρsat(T)

104−257 120−320 153−371 153−413 193−448 231−291 213−483 233−503 203−403 193−463 223−538 215−342 150−427 288−299 164−373

31 22 24 23 23 26 21 31 18 11 10 39 37 7 23

0.72 0.88 0.89 1.04 0.71 0.84 0.45 0.33 1.25 0.92 0.54 1.26 0.68 0.10 0.91

ref 105 105 107−109 108, 109, 108, 109, 106 108, 109, 108, 109, 109, 119 109 109 121 121 121 122

111, 112 115 117 118

Table 8. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for Selected Branched Alkanes Comprising the Methine CH Group compound

T range/K

n

%AAD Pvap(T)

ref

T range/K

n

%AAD ρsat(T)

ref

2-methylpropane 2-methylbutane 2-methylpentane 2-methylhexane 2-methylheptane 2-methyldecane 2,4-dimethylpentane 2,5-dimethylhexane 2,7-dimethyloctane

122−408 190−410 289−445 273−477 233−500 273−462 289−463 246−495 293−432

45 30 19 36 37 24 32 37 28

1.19 6.44 2.64 0.56 2.01 5.32 1.37 2.45 3.08

121 121 121 121 121 123 121 121 121

213−369 145−410 158−445 273−477 277−500 253−553 273−463 273−495 253−523

25 36 34 36 31 12 35 33 11

1.53 0.51 0.59 0.63 0.80 0.62 1.14 1.07 0.76

121 121 121 121 121 109 121 121 121

Table 9. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for Selected Branched Alkanes Comprising the Quaternary Carbon C Group compound

T range/K

n

%AAD Pvap(T)

ref

T range/K

n

%AAD ρsat(T)

ref

2,2-dimethylpropane 2,2-dimethylbutane 2,2-dimethylpentane 2,2-dimethylhexane 2,2-dimethylheptane 2,2-dimethyloctane 2,2-dimethylnonane 2,2-dimethyldecane 2,2-dimethylundecane

261−431 233−378 150−500 200−540 165−575 248−563 343−553 343−553 363−573

35 30 36 75 84 11 8 8 8

15.44 6.58 15.88 15.21 13.45 8.41 4.35 4.08 4.47

104 124 104 104 104 109 109 109 109

261−431 233−378 150−500 200−540 165−575 213−563 253−553 253−553 273−573

35 30 36 75 84 12 12 12 12

5.80 7.90 7.63 6.86 6.17 5.96 5.90 5.23 5.27

104 125 104 104 104 109 109 109 109

current group parameter table for use with the SAFT-γ Mie approach, which features the functional groups required to calculate the properties of a wide variety of chemical families including, but not limited to, linear and branched alkanes (CH3, CH2, CH, and C groups), linear and branched alkylbenzenes (aCH, aCCH2, and aCCH groups), alkenes (CH2, and CH groups), esters (COO group), carboxylic acids (COOH group), as well as the molecular groups for the modeling of cyclohexane (cCH2), water (H2O), methanol (CH3OH), and acetone (CH3COCH3) as individual species. The parameters characterizing the CH3, CH2 and COO groups have already been presented in the original SAFT-γ Mie paper.32

A list of the current groups developed for use with the SAFT-γ Mie approach is given in Table 1: like group parameters that describe each functional group are given in Table 2; the values of the unlike group parameters for the Mie potential interactions are given in Table 3; and those for the association interactions in Table 4. The SAFT-γ Mie approach using the parameters presented in Tables 2 to 4 provides an excellent description of the vapor−liquid equiliria of pure components, both for the vapor pressures and the saturated liquid densities, as evidenced by the % AADs presented in Table 5, where the deviations for a number of chemical families are shown. More details of the temperature ranges and the specific compounds for each of the families considered can be found in Tables 6 to 12. H



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Table 10. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for the n-Alkylbenzenes Considered in the Parameter Estimation compound

T range/K

n

%AAD Pvap(T)

ethylbenzene n-propylbenzene n-butylbenzene n-pentylbenzene n-hexylbenzene n-heptylbenzene n-octylbenzene n-nonylbenzene n-decylbenzene

424−555 283−433 313−523 293−477 264−463 309−513 293−463 332−466 343−571

29 25 43 18 27 20 25 29 25

1.10 1.77 0.99 2.63 1.57 5.30 5.24 3.30 1.82

ref 126 128, 108, 128, 139 136, 132, 139, 132,

129 132−134 136, 137 137 141 141, 142 136, 143

T range/K

n

%AAD ρsat(T)

ref

183−490 223−543 223−583 233−633 253−613 281−368 243−648 282−368 273−678

38 20 22 14 53 11 12 9 11

0.49 0.43 0.44 1.46 0.37 0.25 1.00 0.14 1.35

108, 127 109, 130, 131 95, 108, 109, 130, 133, 135 109, 138 109, 140 137, 138 109 137 109

Table 11. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for the n-Alkyl Acetates Considered in the Parameter Estimation compound

T range/K

n

%AAD Pvap(T)

ref

T range/K

n

%AAD ρsat(T)

ref

ethyl acetate n-propyl acetate n-butyl acetate n-pentyl acetate n-hexyl acetate n-heptyl acetate

307−473 303−493 334−399 321−462 274−459 274−478

29 20 27 23 33 30

0.37 0.49 0.73 0.85 1.86 0.67

108, 144 145 146 148 149 149, 150

273−473 303−493 298−523 298−393 273−431 273−428

21 20 25 20 18 19

0.25 0.17 0.42 0.18 0.24 0.20

108 145 109, 147 147 150 150

Table 12. Percentage Average Absolute Deviations (%AAD) for the Vapor Pressures Pvap(T) and the Saturated Liquid Densities ρsat(T) Obtained with the SAFT-γ Mie Group-Contribution Approach (Where n Is the Number of Data Points) for Selected Carboxylic Acids compound

T range/K

n

%AAD Pvap(T)

n-butanoic acid n-pentanoic acid n-hexanoic acid n-heptanoic acid n-octanoic acid n-nonanoic acid n-decanoic acid

343−452 303−460 356−479 348−494 403−513 372−528 350−543

34 19 38 40 35 11 13

1.98 3.32 2.56 2.24 2.50 2.59 3.98

■

ref 151, 154, 151, 157, 154, 157 157,

RESULTS AND DISCUSSION The group parameters summarized in Table 1 can be used for the prediction of the thermodynamic properties of a wide variety of systems (pure components and multicomponent mixtures) in a straightforward manner, without the need for additional parameters. The transferability of the group parameters lends a great predictive capability to the SAFT-γ Mie approach, as exemplified here for representative systems. We consider first the adequacy of the description of the fluidphase behavior of a broad selection of binary mixtures, including apolar, polar, and associating components. In Figure 2a the experimental data for the isobaric vapor−liquid equilibria (VLE) of the binary mixture benzene + isopropylbenzene (cumene) is compared with the corresponding SAFT-γ Mie calculations. The chemical affinity of the two components results in an almost ideal fluid-phase behavior, which is accurately reproduced by the theory. A similar level of agreement is obtained for binary mixtures with a fluid-phase behavior which departs markedly from ideality due to the presence of polar or associating components. The description of the isothermal VLE of the acetone + isobutane and n-heptane + pentanoic acid binary mixtures with the SAFT-γ approach are shown in Figure 2b and c, respectively. The quality of the description over a broad range pressures is apparent.

152 155 156, 157 158 156 162

T range/K

n

%AAD ρsat(T)

273−563 233−553 273−588 263−583 293−573 288−323 313−573

17 25 27 16 22 7 14

0.75 0.88 0.59 1.00 0.58 0.80 0.23

ref 109, 109, 109, 109, 153, 160, 153

153 153 153 159 159 161

The theoretical description of the isothermal VLE of the binary mixtures of cyclohexane + ethyl acetate, cyclohexane + n-propyl acetate, and cyclohexane + n-butyl acetate is compared with the corresponding experimental measurements in Figure 2d. It is interesting to observe that the theory accurately captures the change in thermodynamic behavior caused by the addition of a methylene group to the ester chain; the azeotropic composition is displaced from a near-equimolar value in the case of cyclohexane + ethyl acetate to a higher content of cyclohexane in the case of cyclohexane + n-propyl acetate, while the azeotrope eventually disappears for cyclohexane + n-butyl acetate. The SAFT-γ Mie approach allows one to predict this behavior in remarkable agreement with the available experimental data. It is also important to consider the accuracy and predictive ability of the methodology in mixtures that exhibit demixing in the liquid phase. The binary mixture of benzene + water is chosen as a challenging case study in this context. The mixing of water and of a hydrophobic component such as benzene leads to phase separation into two liquid phases, one rich in benzene and the other rich in water. The compositions of the minority component in each phase are very low, with the mole fraction of water in the benzene-rich phase ∼10−1 and the mole fraction of benzene in the water-rich phase ∼10−3. Such systems are notoriously difficult to treat, as has been discussed in a previous I



Article

Figure 2. Vapor−liquid equilibria of selected binary mixtures. The symbols represent the experimental data and the continuous curves the description with the SAFT-γ Mie approach: (a) isobaric temperature−mole fraction (T−x) phase diagram of benzene + isopropylbenzene (cumene) at pressures of P = 40 kPa (circles),78 P = 53.33 kPa (squares),79 and P = 101 kPa (triangles);80 (b) isothermal pressure−mole fraction (P−x) phase diagram of acetone + iso-butane at temperatures of T = 318.6 K (circles)81 and T = 364.5 K (squares);82 (c) isothermal (P−x) phase diagram of n-heptane + n-pentanoic acid83 at temperatures of T = 323.15 K (circles), T = 348.15 K (squares), and T = 373.15 K (triangles); (d) isothermal (P−x) phase diagram of cyclohexane + ethyl acetate (circles), cyclohexane + n-propyl acetate (squares), and cyclohexane + n-butyl acetate (triangles) at a temperature of T = 335 K.84

work.64−67 The fluid-phase behavior of benzene + water, along the orthobaric line of three-phase coexistence (vapor−liquid−liquid equilibria), as calculated with the SAFT-γ Mie EoS is shown in Figure 3a. As can be seen a very reliable simultaneous description of the low mutual solubilities is obtained. This type of phase behavior (liquid−liquid split with a minimum of the solubility of a hydrophobic molecule in water) is also observed in mixtures of water + n-alkanes, which have been investigated using different association-based EoSs.34,64−66,68 The SAFT-γ Mie approach provides a very satisfactory representation of the phase equilibria of these systems, as exemplified in Figure 3b for the case of water + n-hexane; though the minimum in the solubility of n-hexane in water is not reproduced, it is remarkable to find that a group contribution methodology of this kind can capture the very low solubilities for this system (of the order of 10−5 to 10−6) considering that the corresponding data was not used in the parameter estimation. The application of the SAFT-γ Mie approach is not limited to the study of fluid phases; the methodology can be easily extended to the description of solid−liquid equilibria, i.e., the determination of solubility of a solid component in the fluid. This is typically undertaken by means of a standard thermodynamic cycle which leads to the following compact working expression69 (where the temperature dependence of the heat capacity has been neglected): ⎛ T ⎞ ΔHfus, j ln xj = ⎜⎜ − 1⎟⎟ − ln γj(T , P , xj) ⎝ Tfus, j ⎠ NkBT

Figure 3. Coexistence compositions of the minority component x in the water-rich and hydrocarbon-rich liquid phases for increasing temperature T along the three-phase (orthobaric) line for (a) water + benzene and (b) water + n-hexane. The symbols represent the experimental data for the water-rich phase (circles,85 squares86,87) and the hydrocarbon-rich phase (triangles,85 diamonds86,87), and the continuous curves the description with the SAFT-γ Mie approach.

(35)

Here, the maximum amount of solute j that can be dissolved in the liquid phase, expressed as the mole fraction xj, is obtained given the enthalpy of melting ΔHfus,j at its (normal) melting J



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Figure 4. Isobaric temperature−mole fraction (T−x) phase diagram of the solid−liquid equilibria of n-butyl acetate + n-hexadecane (circles),88 n-butyl acetate + n-octadecane (squares),89 and n-butyl acetate + n-eicosane (triangles)89 at a pressure of P = 101 kPa. The symbols represent the experimental data and the continuous curves the description with the SAFT-γ Mie approach. A logarithmic scale representation of the x axis is provided in the inset to highlight the low-concentration region.

Figure 6. Second-derivative properties for pure components. The symbols represent the experimental data and the continuous curves the description with the SAFT-γ Mie approach: (a) isobaric specific heat capacity of selected alkenes96 in the liquid phase (open symbols) and the gaseous phase (filled symbols) for but-1-ene (circles), hex-1-ene (squares), oct-1-ene (triangles), and dec-1-ene (diamonds) at a pressure of P = X kPa; (b) speed of sound of pure carboxylic acids: nbutanoic acid (circles),97,98 n-pentanoic acid (squares),97,98 n-hexanoic acid (triangle),97,98 n-heptanoic acid (diamonds),97 and n-octanoic acid (asterisks)97 at a pressure of P = 101 kPa.

hexadecane,70 n-octadecane,70 and n-eicosane,70 respectively, while the enthalpies of melting are 53.332 kJ mol−1, 61.306 kJ mol−1, and 69.73 kJ mol−1 for n-hexadecane,70 n-octadecane,70 and n-eicosane,70 respectively. These results are particularly encouraging considering that the group parameters have been estimated using only the fluid-phase behavior data of pure components and excess enthalpy data of selected binary mixtures. We further examine the performance of the SAFT-γ Mie approach in the description of excess thermodynamic properties of selected binary mixtures (see Figure 5). Excess properties of mixing are very sensitive to both the like and unlike intermolecular interactions71,72 and constitute a stringent test for any model; these properties have previously been studied, e.g., by dos Ramos et al.73−75 In Figure 5a the theoretical description of the excess enthalpy is compared with the experimental data for two binary mixtures: hexanoic acid + n-hexane and hexanoic acid + n-heptane. The predictions are seen to follow the trend of the experimental data as the chain length of the n-alkane in the mixture is increased, and quantitative agreement for both mixtures is observed. In Figure 5b the SAFT-γ Mie description of the excess volume is shown for the binary mixtures: propylbenzene + n-octane, propylbenzene + nnonane, and butylbenzene + n-heptane. The proposed GC approach provides a good semiquantitative description of the experimental data: a positive excess volume is correctly predicted for propylbenzene + n-octane and propylbenzene + n-nonane, while a negative excess volume is correctly predicted for butylbenzene + n-heptane. The overall good performance of

Figure 5. Isothermal−isobaric excess properties of mixing for selected binary mixtures at ambient conditions (T = 298.15 K and P = 101 kPa). The symbols represent the experimental data and the continuous curves the description with the SAFT-γ Mie approach: (a) excess molar enthalpy of mixing for n-hexane + n-hexanoic acid (circles)90 and n-heptane + n-hexanoic acid (squares);91 (b) excess molar volume of mixing for npropylbenzene + n-octane (circles),92,93 n-propylbenzene + n-nonane (squares),93 and n-butylbenzene + n-heptane (triangles).94,95

temperature Tfus,j. The SAFT-γ Mie approach is used to calculate the activity coefficient of the solute γj(T,P,xj) at the specified temperature T, pressure P, and composition xj. We use this expression to predict the solubility of the relatively long-chain linear alkanes n-hexadecane, n-octadecane, and n-eicosane in n-butyl acetate. The solid−liquid coexistence curves are displayed in Figure 4, where the predicted solubility is shown to be in very good agreement with the experimental data even for very low compositions of the solute (see the inset of Figure 4). The values of the melting temperatures of the long-chain linear alkanes considered are 291.2 K, 301.2 K, and 309.5 K for nK



Article

Figure 7. Isothermal−isobaric second-derivative properties for selected binary mixtures. The symbols represent the experimental data and the continuous curves the description with the SAFT-γ Mie approach: (a) isothermal compressibility99 of benzene + n-hexane (circles), benzene + noctane (squares), benzene + n-decane (triangles), and benzene + n-hexadecane (diamonds) at ambient conditions (T = 298.15 K and P = 101 kPa); (b) excess speed of sound100 of cyclohexane + n-hexane (circles), cyclohexane + n-heptane (squares), and cyclohexane + n-octane (triangles) at a temperature of T = 303.15 K and a pressure of P = 101 kPa; (c) excess isobaric heat capacity of acetone + water101 at a pressure of P = 101 kPa and temperatures of T = 283 K (circles), T = 298 K (squares), and T = 313 K (triangles); (d) excess isobaric heat capacity of acetone + n-hexane (circles)102 and acetone + n-heptane (triangles)103 at ambient conditions (T = 298.15 K and P = 101 kPa).

even for sophisticated thermodynamic models and equations of state. The performance of the SAFT-γ Mie approach in representing second-order thermodynamic derivative properties is illustrated in Figure 6a and b for selected pure components: 1-alkenes and n-alkyl carboxylic acids. In Figure 6a, the SAFT-γ Mie predictions of the isobaric heat capacity for but-1-ene, hex-1-ene, oct-1-ene, and dec-1-ene are compared with the corresponding experimental data. The theoretical predictions are found to be in very good agreement with the measured data for both the liquid and the vapor phases. The theory also provides a reliable description of the speed of sound of selected carboxylic acids as a function of temperature, as can be seen in Figure 6b. Unfortunately a higher deviation is observed at the lower temperatures, which might be due to the increased degree of hydrogen bonding in the vicinity of the freezing point. As a final example, we examine the performance of the SAFT-γ Mie approach in the description of the derivative properties for selected binary mixtures including: the isothermal compressibility for binary mixtures of benzene with n-alkanes of varying chain length (Figure 7a); the excess speed of sound for binary mixtures of cyclohexane with n-hexane, n-heptane, and n-octane (Figure 7b); the excess isobaric heat capacity for acetone + water at different pressures (Figure 7c); and the excess isobaric heat capacity for acetone + n-hexane and acetone + n-heptane at ambient pressure and temperature (Figure 7d). Overall, a very satisfactory level of agreement is observed between the predictions of the SAFT-γ Mie approach and the experimental data. It is important to reiterate that the parameters used in the calculation of the derivative properties are obtained solely from

the approach in the simultaneous description of the vapor− liquid and liquid−liquid phase equilibria and excess thermodynamic properties highlights the versatility of the theory. The parameters developed to describe the various functional groups clearly allow one to capture the fine features of the interactions between the constituent groups of the mixtures studied. As indicated in previous work,32 the SAFT-γ Mie approach, while providing a good description of the excess volume and the excess speed of sound for binary mixtures of n-alkanes, does not provide a quantitative description of the very small values of the excess enthalpy of asymmetric alkane mixtures, as the EoS does not account for conformational effects, which are important in this instance.76 Group-contribution methods based on activity coefficients cannot be used directly to obtain volumetric thermodynamic properties, as they are based on a fully occupied lattice-fluid model. The enhanced capability of the SAFT-γ approach in describing volumetric properties emphasizes the distinct advantage that a GC EoS provides over traditional activitycoefficient GC approaches such as UNIFAC. The SAFT-γ Mie approach has also been shown to provide an accurate description of the second-order thermodynamic derivative properties for pure components and mixtures.32 This is a consequence of the use of the versatile Mie intermolecular potential employed for the description of the intersegment interactions, which allows for a variable attractive and repulsive range, and the use of a third-order expansion in the treatment of the monomer free energy. An accurate description of derivative properties, such as the isobaric heat capacity or the speed of sound, is considered to be a great challenge L



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fluid-phase equilibrium of pure components and selected fluid-phase behavior and excess properties of binary mixtures. The results presented in Figures 6 and 7 are obtained in a fully predictive manner in the sense that neither heat capacity nor speed of sound data were used in estimating the parameters for the relevant groups. An accurate description of these challenging properties is achieved while retaining an excellent description of the fluid-phase behavior of the corresponding pure components and binary mixtures employed to estimate the various group parameters.

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CONCLUDING REMARKS An enhanced set of functional groups have been developed for the SAFT-γ Mie EoS presented previously,32 extending the range of application of this approach to a broader selection of chemical families. The group parameters are estimated using pure-component data mainly, providing an excellent description of the VLE of many compounds for different chemical families. An accurate description of the VLE of binary mixtures for a wide range of conditions and systems is obtained with the same group parameters. The excellent reliability of the methodology is further exemplified by the capability that the SAFT-γ Mie approach provides in reproducing the LLE and SLE for binary mixtures, enabling the study of the solubility of a variety of components. One of the key features of the approach lies in the description not only of phase equilibria, but also of single-phase thermodynamic properties, such as excess properties of mixing and second-derivative properties of both pure-component and mixtures. The models developed for the different chemical groups thus provide an accurate platform for the calculation of a broad range of properties, systems, and conditions, making it an ideal tool for the prediction of thermodynamic properties.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

The authors acknowledge financial support from the Technology Strategy Board (TSB) of the United Kingdom (project CADSEP-101326), the Commission of the European Union (project FP7-ENERGY-2011-282789), and Pfizer, Inc. V.P. is thankful to the Engineering and Physical Sciences Research Council (EPSRC) of the UK for the award of a PhD studentship and a Doctoral Prize Fellowship 2013. C.S.A. is grateful to the EPSRC for the award of a Leadership Fellowship (EP/J003840/ 1). Additional funding to the Molecular Systems Engineering Group from the EPSRC (grants GR/T17595, GR/N35991, EP/ E016340 and EP/J014958), the Joint Research Equipment Initiative (JREI) (GR/M94426), and the Royal Society-Wolfson Foundation refurbishment scheme is gratefully acknowledged. Notes

The authors declare no competing financial interest.

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REFERENCES

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