Heterosegmented Perturbed-Chain Statistical Associating Fluid

Article pubs.acs.org/IECR

Heterosegmented Perturbed-Chain Statistical Associating Fluid Theory as a Robust and Accurate Tool for Modeling of Various Alkanes. 1. Pure Fluids Kamil Paduszyński* and Urszula Domańska Department of Physical Chemistry, Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warsaw, Poland S Supporting Information *

ABSTRACT: Heterosegmented molecular models based on statistical associating fluid theory (SAFT) seem to be very promising and robust tools for modeling thermodynamic properties of fluid mixtures. They differ from conventional SAFT-based methodologies as they take into account varying sizes and interactions of segments constituting chain molecules. Those different types of segments can be assigned to functional groups, and hence, the group contribution (GC) method is incorporated in a straightforward manner into the SAFT approach. In this contribution, we applied a heterosegmented version of perturbed-chain SAFT (hs-PC-SAFT) for modeling thermodynamic behavior of a great variety of pure saturated hydrocarbons, including nalkanes, branched alkanes, and alkyl-monosubstituted cyclohexanes and cyclopentanes. All the investigated compounds were assumed to be composed of 11 distinct functional groups defined within the GC model. The properties under consideration were saturated liquid and vapor density, vapor pressure, enthalpy of vaporization, surface tension, isobaric heat capacity, speed of sound, the Joule−Thomson coefficient, the Joule−Thomson inversion curve, and the second virial coefficient. The respective model parameters (segments number mi, segment diameter σi) for each defined functional group i as well as self- and/or crossinteraction parameters (uij/kB) for each pair of groups i−j were determined by fitting experimental saturated liquid densities and vapor pressures of some selected alkanes over a wide range of temperature. Then, the optimized parameters were used to predict the properties of other compounds, and finally, the resulting predictions were compared to those obtained by using different similar methods described in the literature. It was shown that an overall predictive capacity of the hs-PC-SAFT approach is comparable to other similar methods based on “variable-range” SAFT (SAFT-VR) and superior over the conventional homosegmented SAFT models involving groups contributions.

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INTRODUCTION It is beyond a shadow of a doubt that the knowledge of experimental thermodynamics, accurate phase equilibrium data, and thermophysical properties is essential from the point of view of both fundamental and applied sciences, particularly in design/operation/simulation of separation units for petroleum and chemical industry. It is well-established that the experimental data alone will never allow one to capture the effects of working conditions (such as temperature, pressure, density, composition) and molecular characteristics on fluids’ thermodynamic behavior. This is due to the fact that the measurements are usually very expensive and time-consuming, while the number of industrially significant chemicals continues to grow rapidly. This growth leads to an unfeasible number of experiments that would be required to carry out a complete and systematic study covering at least a part of all possible systems and properties. Therefore, developing and testing versatile and robust thermodynamic models remains a research area of great significance, as concluded recently by European Federation of Chemical Engineers (EFCE).1 Such models should enable chemical engineers to simultaneously describe various characteristics of both pure fluids and mixtures, including different types of phase equilibrium diagrams and volumetric/transport/ surface/thermal/acoustic properties. In particular, theoretically established (i.e., rooted in statistical mechanics) equations of state (EoSs) combined with group contribution (GC) methods © 2012 American Chemical Society

give more physical insight into mutual relationship between the macroscopic properties of a given system and its molecularscale features. Predictive and/or correlative capabilities of such models and the range of their possible applications (usually given in terms of complexity and size of molecules) are expected to be much broader than for other models, e.g., empirical correlations. Furthermore, the outcomes provided by such models are expected to be thermodynamically consistent and physically meaningful in the entire thermodynamic phase space. Naturally, a detailed and critical overview of the state of the art in approaches for thermodynamic modeling is beyond the scope of this paper. In fact, there are so many models described in the open literature that it is impossible to detail all the references here. For a comprehensive review, we direct the reader to the very extensive compilation by Folas and Kontogeorgis2 and references therein. Other monographic books are available for the interested reader as well.3−5 In this paper, we will provide a brief survey on GC approaches combined with statistical associating fluid theory (SAFT). Received: Revised: Accepted: Published: 12967

July 26, 2012 August 30, 2012 September 6, 2012 September 7, 2012 dx.doi.org/10.1021/ie301998j | Ind. Eng. Chem. Res. 2012, 51, 12967−12983

Industrial & Engineering Chemistry Research

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by Peters et al.40 Another GC method based on a simplified version of PC-SAFT (sPC-SAFT41) was elaborated by Tihic and co-workers and applied for a great variety of hydrocarbons, polar compounds, and polymers.42−45 The essential dissimilarity between the GC-sPC-SAFT model and the GC-SAFT approaches of Tamouza et al.30−37 is in the way of averaging the group parameters. Furthermore, in the method of Tihic et al.,43−45 a much greater number of groups is defined, including first- and second-order building blocks. However, treatment of associating compounds with the GC-sPC-SAFT cannot be carried out, contrary to the methods of Tamouza and coworkers.30−37 The major shortcoming of the described homosegmented methods is that they disregard connections between the groups. To overcome this limitation, Vijande et al.46 established another GC-based methodology for the SAFT parameters in which the topology of the molecule and the “proximity effects” were taken into account explicitly through mutual perturbation coefficients describing to what extent each group present in the molecule affects the others. The method was shown to be capable of accurately reproducing the fitted PC-SAFT parameters reported in the literature for some linear and branched hydrocarbons, ethers, and esters.46 Apart from the homosegmented models, the GC-SAFT approaches have been proposed in which a heterosegmented representation of the model chains is adopted. In this case, the theory is constructed at the level of the distinct segments making up the molecule rather than at the level of the molecular parameters. In spite of the universality of this methodology, the study on its applications was initially focused mainly on copolymer systems. The paper of Banaszak et al.47 concerned with generalization of Huang−Radosz SAFT48 as well as contributions by Gross et al.49 and Dominik et al.50 regarding PC-SAFT are pioneering in the field. In those heterosegmented SAFT approaches, copolymers chains are assumed to be made up of backbone and branch segments, whereas the heterogeneity of the polymer chain is described in terms of distinct structural units rather than small and relatively simple functional groups (like CH3, CH2, and so on), and hence, the GC idea is not completely applied in the SAFT framework. Thus far, only the SAFT-VR has been used within a full GC formalism. In 2007, Galindo and co-workers proposed a heteronuclear version of SAFT-VR, referred to as SAFT-γ, to treat alkanes, α-olefins, aromatics, and alcohols assuming their molecules to be formed from fused (i.e., not tangent) spherical segments differing in diameter, shape factor, and interactions potential.51 In the very first version of SAFT-γ, each segment corresponds to a respective functional group. In the subsequent work,52 the method was generalized to multiple spherical segments, and finally, the parameter matrix was extended to compounds consisting of polar and self-associating molecules, including water.52,53 Very recently the SAFT-γ approach has been very extended to ionic liquids.54 It should be emphasized that SAFT-γ is not an entirely heterosegmented approach, since it describes heterosegmented monomer fluid only. Then, effective molecular parameters are obtained from segmental ones by means of averaging, and in the final state the system is formed from homosegmented fused chains. A similar heterosegmented version of SAFT-VR has been developed and systematically extended in the group of McCabe.55−57 Contrary to the SAFT-γ approach, the effects of molecular topology and connectivity of the groups are

The SAFT approach, derived from the thermodynamic perturbation theory of Wertheim,6−9 is nowadays the most important, the most representative, and the most successful example of analytical molecular-based EoS. Thus far, several refined versions of the original SAFT of Chapman et al.10,11 (denoted by SAFT-0) have been proposed to account for different types of intermolecular forces and, hence, to model thermodynamic behavior of different families of chemicals, including nonpolar, polar, quadrupolar, cross-associating compounds, polymers/copolymers, and electrolytes solutions.12−15 From among the SAFT-based approaches, the original SAFT-0,11 perturbed-chain SAFT (PC-SAFT),16−19 SAFT for potentials of variable range (SAFT-VR),20,21 softSAFT,22 truncated PC-SAFT (tPC-SAFT),23 and (SAFT + Cubic)24 are seen to be the most versatile and promising tools for thermodynamic modeling of different classes of fluids. In particular, some of them have been very recently adopted to treat phase equilibria and related properties of complex crossassociating binary systems involving ionic liquids.25−27 Finally, it should be stressed that, apart from the advantages of the SAFT family of models, some of them may exhibit practically unrealistic and even nonphysical predictions, as pointed out recently by Polishuk.28,29 A few procedures for employing GC methodology into the SAFT approach have been proposed to combine the predictive capabilities of the GC idea with the accuracy of the SAFT-type equations of state.2 In such so-called GC-SAFT approaches, a system is characterized by specifying the functional groups that represent the molecules, assuming that the behavior of the molecular system can be obtained from the sets of SAFT parameters assigned to each group separately. Most frequently, the homosegmented (homonuclear) GC model is incorporated within the SAFT formalism, where the group parameters are averaged over the whole molecule, and finally, all the segments are described by using the same set of parameters representing their averaged size and interactions potential. It should be emphasized that the EoS adopted in these types of GC approaches is used without further modifications. Therefore, the outlined idea should be classified as a GC-based mixing rule for molecular parameters rather than underlying theory explicitly capturing and explaining thermodynamic behavior in terms of the different groups making up the molecules. Besides, the most important drawback of this kind of GC-SAFT method is that, unlike group interaction, parameters cannot be determined on the basis of pure component data alone. Thus far, several versions of SAFT based on groupparameters averaging have been reported. Tamouza et al.30−37 proposed and consistently developed GC-SAFT methods applied to SAFT-0, SAFT-VR, and PC-SAFT EoSs for modeling of pure-fluid properties of homologous series of various chemicals, including hydrocarbons,30,31 esters,32,33 alcohols,30,34 amines,35 ethers, aldehydes, ketones,36 and long chain multifunctional molecules like diamines, alkanolamines, and alkanediols.37 Moreover, those methods have been complemented by the GC method for binary corrections to Lorentz−Berthelot combining rules, resulting in an entirely predictive model for multicomponent mixtures.38,39 However, it should be stressed that the only properties investigated in terms of those GC-SAFT approaches were, thus far, vapor pressure and liquid density for pure fluids and vapor−liquid equilibrium (for the most part) and liquid−liquid equilibrium for binary mixtures. Applicability of the GC-SAFT methodology (with PC-SAFT EoS) to polymer systems was recently demonstrated 12968

dx.doi.org/10.1021/ie301998j | Ind. Eng. Chem. Res. 2012, 51, 12967−12983


Article

Ares ≡ a ̃res = a ̃hc + a disp + a ̃polar ̃ + a assoc ̃ NkBT

considered in chain formation contribution to the residual free energy of the system. In particular, GC-SAFT-VR of McCabe et al.55−57 requires a definition of the numbers of connections (bonds) between respective functional groups. This may be viewed as an additional advantage over SAFT-γ, as it allows differentiating between some skeletal and positional isomers. The predictive capabilities of the GC-SAFT-VR method was examined by calculating vapor−liquid equilibrium diagrams of binary mixtures composed of hydrocarbons, ketones, esters,55 ethers,56 alcohols, aldehydes, amines, and carboxylic acids.57 Moreover, the method has been used to model liquid−liquid equilibrium in polymeric systems.56 Being encouraged and motivated by the very good performance of the conventional (i.e., homosegmented) PCSAFT model, we decided to incorporate the GC formalism into the heterosegmented version of this EoS proposed by Gross et al.49 In further text we will abbreviate this approach as hs-PCSAFT. In particular, the main objective of this work was to treat different series of saturated hydrocarbons in terms of several simple functional groups defined within the hs-PC-SAFT framework. The investigated systems comprise pure n-alkanes, branched alkanes, and alkyl-monosubstituted cyclohexanes and cyclopentanes. To estimate the performance and versatility of the hs-PC-SAFT method, diverse thermodynamic properties of saturated and compressed fluids were investiagted, including liquid and vapor density, vapor pressure, enthalpy of vaporization, surface tension, isobaric heat capacity, speed of sound, the Joule−Thomson coefficient, the Joule−Thomson inversion curve, and the second virial coefficient. In total, 99 compounds were considered in this study. It should be stressed that this is the very first contribution where calculations for such a wide spectrum of compounds and properties are reported. As mentioned above, the previous papers concerned with SAFTγ51−53 and GC-SAFT-VR55−57 focus mainly on some basic properties of a pure saturated fluid and binary vapor−liquid equilibria. Finally, we stress that this contribution is concerned only with pure components while a comprehensive study on applications of the hs-PC-SAFT approach for mixtures will be demonstrated in the subsequent paper.

(1)

where kB is the Boltzmann constant. Superscripts hc, disp, assoc, and polar refer to contributions due to chain formation (the reference term), dispersive interactions, association, and multipolar (dipolar, quadrupolar, etc.) interactions, respectively. Since this work is focused on the treatment of hydrocarbons, the first two terms are relevant. Thus, only they will be described in detail in the following paragraphs. Explicit chain heterogeneity implemented in the hs-PCSAFT approach allows straightforward incorporation of the GC methodology into the theory. In the first place, a set of chemical functional groups has to be defined. Let NG be number of distinct types of groups that can be found in the molecules forming the fluid. Then, each group of type i is represented as a homo-segmented building block formed by mi spherical segments. It should be emphasized that the parameter mi means, originally, the number of tangent spheres forming a molecular chain. However, in real chainlike molecules, the segmental units overlap so that the bond lengths can be much less than the segment diameters. Thus, mi should be interpreted as the effective number of segments that can have values that are not necessarily integers.58 In turn, each segment making up the group i is characterized by two standard SAFT parameters, i.e., the hard-sphere diameter, σi, and the depth of the potential well for dispersive interactions, ui/kB. The group assignment is given in the traditional manner, namely, by specifying the number of occurrences of functional group of type i in a molecule. We will denote that number by νi (i = 1, ..., NG). In particular, νi = 0, if group i does not occur in the molecules of the considered fluid. Moreover, connectivity of the groups is included in the hs-PC-SAFT model. It is done by specifying the number of bonds between groups i and j (i.e., not segments), denoted by Bij (where, conventionally, i ≥ j, because Bij = Bji). Having the hs-PC-SAFT parameters defined, we can describe all the important terms from eq 1. The hard-chain contribution is given by49

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THEORY In the following section, we recall all the essential hs-PC-SAFT working equations for a pure fluid and present them in a slightly different form compared to the expressions given in the original paper of Gross et al.49 We believe that the equations presented herein are much easier to be implemented in commonly used programming languages. In our investigations we use MATLAB numerical computing environment (MathWorks, Inc.; version R2009a). Finally, functional groups definitions, hs-PC-SAFT parameters estimation procedures, and the schemes for treatment of different families of hydrocarbons are described below. Heterosegmented PC-SAFT Model (hs-PC-SAFT). As with the other SAFT formulations, hs-PC-SAFT formulates thermodynamics of a pure fluid consisting of N molecules in terms of residual Helmholtz free energy, Ares, expressed as a function of temperature T and number density ρ. Thermodynamic potential, usually represented in dimensionless form (denoted by ãres), is written as the sum of contributions regarding different types of molecular interactions, namely,

hs a ̃hc = ma ̅ ̃ −

NG

i

∑ ∑ βij ln gijhs i=1 j=1

(2)

where m̅ is the total number of segments per molecule calculated as NG

m̅ =

∑ νimi i=1

(3)

The Helmholtz free energy of the hard-sphere fluid expressed on a per-segment basis, ãhs, and its radial distribution function, 59 and Mansouri et al.60 ghs ij , are adopted from Boublik a ̃hs =

⎞ ⎛ζ 3 ζ2 3 1 ⎡ 3ζ1ζ2 ⎢ + + ⎜ 2 3 − ζ0⎟ 2 ζ0 ⎢⎣ 1 − ζ3 ζ3(1 − ζ3) ⎠ ⎝ ζ2 ⎤ ln(1 − ζ3)⎥ ⎥⎦

12969

(4)


Industrial & Engineering Chemistry Research gijhs

Article

⎛ didj ⎞ 3ζ ⎛ didj ⎞2 1 2 ⎜ ⎟ ⎟⎟ = +⎜ + ⎜⎜ ⎟ 2 1 − ζ3 ⎝ di + dj ⎠ (1 − ζ3) ⎝ di + dj ⎠

C = 1 + m̅

20η − 27η2 + 12η3 − 2η 4 [(1 − η)(2 − η)]2

2ζ2 2 (1 − ζ3)3

(5)

I1(m̅ , η) = (n = 0, 1, 2, 3)

⎡ ⎛ u ⎞⎤ di = σi⎢1 − 0.12 exp⎜ −3 i ⎟⎥ ⎢⎣ ⎝ kBT ⎠⎥⎦

I2(m̅ , η) =

m̅ − 1 m − 1 m̅ − 2 a1n + ̅ a 2n (17) m̅ m̅ m̅ m−1 m − 1 m̅ − 2 bn(m̅ ) = b0n + ̅ b1n + ̅ b 2n (18) m̅ m̅ m̅ The terms uij/kB = uji/kB and σij = σji in eqs 12 and 13 represent potential parameters for cross-dispersive interactions between segments forming groups i and j. By default, uii/kB ≡ ui/kB and σii ≡ σi. In order to calculate σij, we adopt the following combining rule

(7)

σij = (σi + σj)/2

(9)

Combining eqs 8 and 9 we obtain the general formula for βij (10)

where δij is the Kronecker delta. The same procedure for calculating chain contribution was adopted by McCabe et al. in GC-SAFT-VR.55 However, those authors explained how to properly get βij by employing some examples instead of the generalized relation given in eq 10. The dispersion (disp) term from eq 1 is given by49 a disp ̃ = −2πρI1(m̅ , η)m2

u 3 σ − πρC −1I2(m̅ , η)m2 kBT

⎛ u ⎞2 3 ⎜ ⎟σ ⎝ kBT ⎠

(11)

where η = ζ3 is the packing fraction and the remaining quantities are calculated as follows: NG NG

∑ ∑ mimj i=1 j=1

⎛ u ⎞2 3 m2⎜ ⎟σ = ⎝ kBT ⎠

NG

NG

uij kBT

σij 3

⎛ uij ⎞2 3 ⎟ σij ⎝ kBT ⎠

(12)

∑ ∑ mimj⎜ i=1 j=1

(19)

whereas the cross-interaction energies, uij/kB, are treated as adjustable parameters of the theory. Finally, the coefficients a0n, a1n, a2n, b0n, b1n, and b2n from eqs 17 and 18 are the universal model constants, which can be found elsewhere.16 Having defined the Helmholtz free energy of the fluid, the other thermodynamic properties (pressure, fugacity coefficient, enthalpy, heat capacity, speed of sound, Joule−Thomson coefficient, second virial coefficient) follow from the usual thermodynamic relations, which can be found elsewhere.62 In particular, calculations of surface tension with an EoS can be performed by using density gradient theory (DGT) proposed by Cahn and Hilliard.63 Functional Groups Definition. In this work, NG = 11 distinct groups have been defined. The functional groups examined in the context of n-alkanes are CH4 group (i = 1) used to represent methane and CH3 (i = 2) and CH2 (i = 3) groups used to represent the remaining members of the series. The introduced building blocks are the same as in SAFT-γ51 and GC-SAFT-VR.55 Nevertheless, it should be mentioned that methane was not considered in those methods. In the case of ethane, the hs-PC-SAFT approach is equivalent to homosegmented verion of PC-SAFT.16 This is due to the fact that molecules of ethane are modeled as composed of two CH3 groups (i.e., ν2 = 2) with single CH3−CH3 bond (i.e., B22 = 1). Finally, the group assignment for the remaining linear alkanes, i.e., CnH2n+2 with n ≥ 3, is 2 × CH3 (ν2 = 2) and (n − 2) × CH2 (ν3 = n − 2), while the corresponding bonding assignment is 2 × CH3−CH2 (B32 = 2) and (n − 3) × CH2−CH2 (B33 = n − 3). Determination of a suitable set of functional groups for an accurate representation of branched alkanes is not so trivial a problem as in the case of n-alkanes. McCabe et al.55 introduced a lone CH group in the GC-SAFT-VR method55 to treat methylalkanes, dimethylalkanes, and ethylalkanes. In turn, bulkier CH3CH group was incorporated into SAFT-γ by Lymperiadis et al.51 in order to capture methylalkanes and dimethylalkanes. Tihic et al.43 in their GC-sPC-SAFT model

On the other hand, in summation of the number of bonds between like segments i one must consider Bii external bonds between the groups of the same type i and (mi − 1) internal bonds per group between segments making up the groups of type i. Thus,

βij = Bij + δijνi(mi − 1)

(16)

an(m̅ ) = a0n +

(8)

βii = Bii + νi(mi − 1)

∑ bn(m̅ )ηn n=0

The symbol βij (i ≥ j) in eq 2 denotes total number of bonds between segments (in contrast to Bij) making up the groups of types i and j. As can be easily noticed, if i ≠ j, then the number of bonds between segments i and j equals the number of bonds between the respective groups i and j when i ≠ j

(15)

6

(6)

and temperature-dependent segment diameter di calculated from Baker−Henderson’s perturbation theory61 and dispersive interactions potential incorporated into the original PC-SAFT model16

u 3 σ = kBT

∑ an(m̅ )ηn n=0

N

G π ζn = ρ ∑ νimidi n 6 i=1

m2

(14)

6

with ζn defined by

βij = Bij

8η − 2η2 + (1 − m̅ ) (1 − η)4

(13) 12970



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compound i at temperature Tj with the parameters given as vector x. The numerical minimization of F(x) was performed by using the Levenberg−Marquardt nonlinear least-squares algorithm implemented in the lsqnonlin function of the MATLAB Optimization Toolbox with the default optimization options structure. It should be emphasized that, beside fitting, the hs-PC-SAFT parameters of some functional groups can be obtained directly from pure-fluid parameters of the corresponding original homosegmented model. Indeed, as can be easily noted, both approaches are equivalent for molecules composed only of functional groups of the same type. In such a case, the segment potential well width and depth of the group (i.e., σ and u/kB parameters, respectively) in hs-PC-SAFT have the same values as those characterizing segments making up the respective molecules in the PC-SAFT:

defined a greater number of groups dividing all of them into first- and second-order groups. Such an approach permits more accurate representation of branched hydrocarbons, including those containing quaternary carbon atoms. In this work, we introduce four groups that can be assigned to branched alkanes: (CH3)2CH (isopropyl; i = 4), CH3CH (i = 5), CH2CH (i = 6), and (CH3)3C (tert-butyl; i = 7). It should be mentioned that the idea of incorporation of simpler CH and C groups for tertiary and quaternary alkanes, respectively, was also considered and tested. However, it was discarded because of the unsatisfactory accuracy of correlations and predictions generated by the model. Finally, cyclic structures treated in this paper are alkylmonosubstituted cyclohexanes and cyclopentanes. Two functional groups, namely, C6H11 (cyclohexyl; i = 9) and C5H9 (cyclopentyl; i = 11), are introduced in order to represent cyclohexane and cyclopentane derivatives, respectively. Additionally, cyclohexane (C6H12; i = 8) and cyclopentane (C5H10; i = 10) are considered as distinct structural units. It should be stressed that the proposed definition of functional groups may lead to ambiguous assignments, particularly in the case of branched structures. In order to avoid incorrect descriptions, we assume that the groups defined are the “first-order” groups with respect to the classification given by Marrero and Gani.64 Thus, we adopt the following rules developed by those authors: (1) the groups should describe the entire molecule; (2) no group is allowed to overlap any other group; and (3) if the same fragment of a given compound is related to more than one group, the heavier group must be chosen to represent it instead of the lighter groups. For instance, 2-methylpentane can be represented either as 2 × CH3 (ν2 = 2), 2 × CH2 (ν3 = 2), 1 × CH3CH (ν5 = 1) or as 1 × CH3 (ν2 = 1), 2 × CH2 (ν3 = 2), 1 × (CH3)2CH (ν4 = 1). According to the third rule, the latter assignment is correct. The corresponding bond assignment for 2-methylpentane is then 1 × (CH3)2CH−CH2 (B43 = 1), 1 × CH3−CH2 (B32 = 1), 1 × CH2−CH2 (B33 = 1). In order to avoid ambiguities in group and bond assignments,l a detailed list of the compounds studied in this paper and the final values of νi and Bij for each alkane are summarized in the Supporting Information (Table S1). hs-PC-SAFT Parameters. Parameters Estimation Procedure. The hs-PC-SAFT model parameters were obtained by means of fitting hs-PC-SAFT predictions to experimental data sat for saturated liquid density data (ρsat l ) and vapor pressure (P ) for several selected compounds that contain the treated groups. In this work, we employed least-squares optimization with the following objective function:

σ(group; hs‐PC‐SAFT) = σ (molecule; PC‐SAFT)

u(group; hs‐PC‐SAFT)/ kB = u(molecule; PC‐SAFT)/ kB (22)

Furthermore, the following relation between the number of segments composing the group and molecule must hold: m(group; hs‐PC‐SAFT) =

m(molecule; PC‐SAFT) number of groups (23)

In this way, the hs-PC-SAFT parameters for CH3, (CH3)2CH, and (CH3)3C groups were obtained on the basis of the PCSAFT parameters for ethane,16 2,3-dimethylbutane,16 and 2,2,3,3-tetramethylbutane,42 respectively. The parameters for methylene group, CH2, including “puregroup” parameters and those related to its interactions with CH3 group, have been fitted to saturated properties data for short-chain n-alkanes (from n-propane to n-decane). The heavier n-alkanes were used to check the predictive capacity of the model under consideration. Besides, they were not included in the optimization process, as the vapor pressures of these compounds are very low, and thus large relative deviations are generated, introducing a bias in the objective function F(x). Finally, the parameters obtained for CH3 and CH2 groups have been transferred to branched and cyclic hydrocarbons, where the parameters of each new group were estimated in a sequential manner. Experimental Data Used. The experimental liquid densities and vapor pressures used in the optimization, see eq 20, were taken from Design Institute for Physical Property Research Project 801 (DIPPR 801) correlations.65 For each hydrocarbon, 20 equidistant data points in a reduced temperature range from T/Tc = 0.5 to T/Tc = 0.9 (where Tc stands for critical temperature) have been generated and then employed in the fitting. Critical constants, enthalpies of vaporization, surface tensions, and ideal-gas and liquid phase heat capacities as a function of temperature were taken from DIPPR database as well. Thermodynamic data on the remaining properties investigated in this work, namely, vapor density, speed of sound, isobaric heat capacity, and Joule−Thomson coefficient, were taken from the NIST Chemistry WebBook.66 Unfortunately, the data collected in this database were available only for some selected hydrocarbons falling inside the bounds of this study. The ranges of temperature and pressure studied in this work comprise both subcritical and supercritical conditions.

⎡⎛ sat ⎞2 ⎢⎜ ρl, i (Tj ; x) − 1⎟⎟ min F(x) = ∑ ∑ ⎢⎜ x ρl,sat i j ⎣⎝ ⎠ i,j ⎛ Pisat(Tj ; x) ⎞2 ⎤ ⎥ + ⎜⎜ − 1⎟⎟ ⎥ sat P ⎝ ⎠⎦ i,j

(21)

(20)

The first summation in eq 20 runs over the selected compounds, whereas the second is over experimental data sat points. The symbols ρsat l,i,j and Pi,j denote the jth experimental saturated liquid density and vapor pressure data point for compound i at temperature Tj, respectively, whereas ρsat l,i (Tj;x) and Psat i (Tj;x) are the hs-PC-SAFT-calculated quantities for 12971



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Table 1. List of the Compounds Considered in Optimization of the hs-PC-SAFT Parameters i

group i

1

CH4

2

CH3

3

CH2

4

(CH3)2CH

5

CH3CH

6

CH2CH

7

(CH3)3C

8

C6H12

9

C6H11

j

group j

1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 4 5 6 7 8 9 10 11 5 6 7 8 9 10 11 6 7 8 9 10 11 7 8 9 10 11 8 9 10 11 9

CH4 CH3 CH2 (CH3)2CH CH3CH CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9 CH3 CH2 (CH3)2CH CH3CH CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9 CH2 (CH3)2CH CH3CH CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9 (CH3)2CH CH3CH CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9 CH3CH CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9 CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9 (CH3)3C C6H12 C6H11 C5H10 C5H9 C6H12 C6H11 C5H10 C5H9 C6H11

parameters mi, σi, uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB uij/kB mi, σi, uij/kB uij/kB uij/kB mi, σi,

ui/kB

ui/kB

ui/kB

ui/kB

ui/kB

ui/kB

ui/kB

ui/kB

ui/kB

compounds methane (Gross and Sadowski16) LBa LB LB LB LB LB LB LB LB LB ethane (Gross and Sadowski16) propane, n-butane, n-pentane, n-hexane, n-heptane, n-octane, n-nonane, n-decane 2-methylbutane, 2-methylpentane, 2-methylhexane 3-methylpentane, 3-methylhexane 3-ethylpentane, 3-ethylhexane 2,2-dimethylbutane, 2,2-dimethylpentane, 2,2-dimethylhexane LB methylcyclohexane, ethylcyclohexane, n-propylcyclohexane LB methylcyclopentane, ethylcyclopentane, n-propylcyclopentane as for CH3 group as for CH3 group as for CH3 group as for CH3 group as for CH3 group as for CH3 group as for CH3 group as for CH3 group as for CH3 group 2,3-dimethylbutane (Gross and Sadowski16) 2,3-dimethylpentane 2,4-dimethyl-3-ethylpentane 2,2,3-trimethylbutane LB isopropylcyclohexane LB isopropylcyclopentane 3-methylpentane, 3-methylhexane LB 2,2,3-trimethylpentane LB sec-butylcyclohexane, isopropylcyclohexane LB LB 3-ethylpentane, 3-ethylhexane 2,2-dimethyl-3-ethylpentane LB LB LB LB 2,2,3,3-tetramethylbutane; Tihic et al.42 LB tert-butylcyclohexane LB LB cyclohexane (Gross and Sadowski16) LB LB LB bicyclohexyl 12972



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Table 1. continued i

a

group i

10

C5H10

11

C5H9

j 10 11 10 11 11

group j C5H10 C5H9 C5H10 C5H9 C5H9

parameters

compounds

uij/kB uij/kB mi, σi, ui/kB uij/kB mi, σi, ui/kB

LB LB cyclopentane (Gross and Sadowski16) LB methylcyclopentane, ethylcyclopentane, n-propylcyclopentane

Calculated with the Lorentz−Berthelot combining rule; see eq 26.

Finally, it should be remarked that the used DIPPR 80165 and NIST Chemistry WebBook66 compilations are actually only pseudoexperimental in nature, with some uncertainties. In fact, for some compounds they were established on the basis of either real experiments and/or results generated by some empirical correlations. In particular, for a great majority of hydrocarbons considered in this work, a valid temperature range for the DIPPR 801 correlations is much broader than the range of temperature for real experimental data used to obtain the equations coefficients. One must carefully bear it in mind when interpreting the performance of the hs-PC-SAFT and the deviations from the DIPPR 801/NIST values referred in this paper as the “experimental” ones.

It should be stressed that some values of the group interaction parameters (uij/kB) were assumed to be calculable by using conventional Lorentz−Berthelot combining rule for crossdispersive energy, namely, uij /kB =

RESULTS AND DISCUSSION In order to express the accuracy of hs-PC-SAFT and compare its performance with similar methods proposed previously, the following measures of accuracy of correlation and prediction are introduced: the average absolute relative deviation (AARD) 1 NP

∑ i

Xicalcd − Xiexptl Xiexptl

× 100% (24)

and the average absolute deviation (AAD) AAD(X ) =

1 NP

∑ |Xicalcd − Xiexptl| i

(26)

The need for such approximation arises either from the lack of appropriate experimental data or the physical impossibility of molecular structures involving the corresponding groups i and j. In particular, there is no molecule that contains CH4, C6H12, C5H10, and any other group because these groups themselves are individual molecules. Of course, one could fit the respective cross-interaction parameters to thermodynamic binary data for mixtures containing methane or cyclohexane or cyclopentane. The main goal of this work was, however, the development and evaluation of the entirely predictive method employing parameters obtained on the basis of pure-fluid data only, and thus, we decided to exclude the binary data from the parameter estimation process. Moreover, the considered functional groups are similar in chemical nature, and hence, the approximation given by eq 26 can be seen as reliable. The optimized “pure-group” parameters, i.e., mi, σi, and ui/kB, are shown in Table 2, whereas the matrix of cross-interaction

■

AARD(X ) =

(ui /kB)(uj /kB)

Table 2. Optimized Parameters for Chemical Functional Groups Treated in This Work in the Framework of the hsPC-SAFT Approach

(25)

Superscripts exptl and calcd refer to the calculated and experimental values, respectively. The symbols NP and X in eqs 24 and 25 denote the number of data points and a quantity of interest, respectively. The properties under consideration considered are density (ρ), saturated vapor pressure (Psat), enthalpy of vaporization (Δvl H), isobaric heat capacity (Cp), surface tension (γ), speed of sound (u), the Joule−Thomson coefficient (μJT), and the second virial coefficient (B). The remaining part of this section is organized as follows. First, the results of parameter fitting process and the optimized values for the model parameters are presented and discussed. Then, correlation and prediction results for density and vapor pressure are shown followed by comparison to the results obtained by using similar heterosegmented and homosegmented GC-SAFT models. Finally, the predictive capabilities and versatility of the hs-PC-SAFT approach in calculating other important thermodynamic properties are highlighted and presented in detail for a few selected n-alkanes for which comprehensive experimental data were available in the NIST Chemistry WebBook.66 Determination of hs-PC-SAFT Parameters. Table 1 shows all the compounds used in the parameter estimation and the corresponding hs-PC-SAFT parameters fitted. As can be seen, for each pair of groups a representative set of compounds was selected to be included in the fitting process. The remaining compounds were used to test the model parameters.

i

group

mi

σi (Å)

ui/kB (K)

1 2 3 4 5 6 7 8 9 10 11

CH4a CH3b CH2 (CH3)2CHc CH3CH CH2CH (CH3)3Cd C6H12a C6H11 C5H10a C5H9

1.0000 0.8035 0.3940 1.3426 0.5302 0.3043 1.4621 2.5303 1.8357 2.3655 1.8095

3.7039 3.5206 3.9455 3.9545 4.4323 4.8124 4.1622 3.8499 4.1787 3.7114 3.9455

150.030 191.420 257.581 246.070 317.327 378.124 266.480 278.110 318.351 265.830 297.846

a Literature value.16 bEstimated from PC-SAFT parameters for ethane16 (eqs 21−23). cEstimated from PC-SAFT parameters for 2,3-dimethylbutane16 (eqs 21−23). dEstimated from PC-SAFT parameters for 2,2,3,3-tetramethylbutane42 (eqs 21−23).

energies between segments forming unlike groups, i.e., uij/kB, is given in Table 3. It is worth noticing that for the CH2 group the obtained values are quite similar to the respective values used in SAFT-γ of Lymeriadis et al.51 and GC-SAFT-VR of McCabe et al.,55 even though those models are based on different versions of SAFT. Interestingly, the values of σi and ui/kB for similar groups are higher for the groups having a higher degree of alkyl substitution. On the other hand, corresponding values of mi 12973



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Table 3. Optimized hs-PC-SAFT Cross-Dispersive Interaction Parameters for the Chemical Functional Groups Treated in This Work uij/kB (K) i/j

group

CH4

CH3

CH2

(CH3)2CH

CH3CH

CH2CH

(CH3)3C

C6H12

C6H11

C5H10

C5H9

1 2 3 4 5 6 7 8 9 10 11

CH4 CH3 CH2 (CH3)2CH CH3CH CH2CH (CH3)3C C6H12 C6H11 C5H10 C5H9

150.030a 169.466b 196.583b 192.140b 218.194b 238.180b 199.950b 204.267b 218.546b 199.706b 211.390b

191.420c 230.799 211.628 258.857 290.129 214.967 230.729b 247.895 225.577b 233.403

257.581 250.677 255.049 258.001 249.417 267.649b 288.535 261.673b 275.847

246.070d 287.149 299.329 254.305 261.600b 280.922 255.759b 264.632

317.327 346.394b 297.919 297.072b 313.421 290.439b 307.432b

378.124 332.922 324.284b 346.953b 317.044b 335.593b

266.480e 272.233b 287.418 266.155b 281.727b

278.110a 297.551b 271.901b 287.809b

318.351 290.908b 307.928b

265.830a 281.383b

297.846

a

Literature value.16 bCalculated with the Lorentz−Berthelot combining rule (see eq 26). cEstimated from PC-SAFT parameters for ethane16 (eqs 21−23). dEstimated from PC-SAFT parameters for 2,3-dimethylbutane16 (eqs 21−23). eEstimated from PC-SAFT parameters for 2,2,3,3tetramethylbutane42 (eqs 21−23).

Table 4. Average Absolute Relative Deviations (AARD) and Average Absolute Deviations (AAD) between Experimental Data65 and hs-PC-SAFT Correlations or Predictions (Values in Parentheses) for the Saturated Liquid Densities (ρsat 1 ) and the Vapor Pressures (Psat) of the Hydrocarbons Studied in This Work X = ρsat l

X = Psat

family

Nca

AARD(X)b (%)

AAD(X)c (kg·m−3)

AAD(X) (kmol·m−3)

AARD(X)b (%)

AAD(X)c (kPa)

n-alkanes

10 (22) 16 (34) 13 (4) 39 (60) 99

0.56 (0.67) 0.74 (1.4) 0.83 (1.3) 0.72 (1.1) 0.98

3.1 (4.1) 4.4 (8.3) 5.3 (7.7) 4.4 (6.7) 5.8

0.054 (0.013) 0.043 (0.073) 0.051 (0.051) 0.048 (0.049) 0.049

0.89 (15) 2.0 (11) 1.4 (8.6) 1.5 (12) 7.8

3.6 (3.2) 5.8 (20) 6.0 (16) 5.3 (14) 10

branched cyclic altogetherd in totale a

Number of compounds falling within a given category. bSee eq 24. cSee eq 25. dRegardless of a family of compounds. eIncluding correlation and prediction set.

decreases. These trends coupled together result finally in an increase of the “combined” parameters miσi3 and miui/kB (corresponding to hard-core volume and dispersive energy per group, respectively) as the bulkiness of a functional group increases. This observation is in quantitative agreement with that noted previously for all the mentioned GC-SAFT models. Nevertheless, a regular variation of mi, miσi3, and miui/kB with molecular mass was not observed as it was in the contribution of McCabe et al.55 Correlation and Prediction of Density and Vapor Pressure. hs-PC-SAFT Calculations. In the first place, the obtained parameters have been examined by calculating saturated liquid density (ρsat l ) and saturated vapor pressure (Psat) of hydrocarbons not included in the fitting (see Table 1). In particular, the parameters of CH3 and CH2 groups obtained from short-chain n-alkanes have been used to predict the data for heavier compounds belonging to this series, whereas the parameters representing (CH3)2CH, CH3CH, CH2CH, and (CH3)3C functional groups have been used to predict the data for branched alkanes consisting of two or more tertiary or quaternary carbon atoms. Extensive evaluation of the hs-PCSAFT parameters of cyclohexyl/cyclopentyl (C6H11/C5H9) groups was not possible because of the limited amount of available experimental data for alkyl-monosubstituted cyclohexanes and cyclopentanes.

sat correlation A brief overview of the accuracy of ρsat l and P and prediction is presented in Table 4 for linear, branched, and cyclic hydrocarbons separately. For a detailed summary presenting all the studied compounds individually, the reader is referred to the Supporting Information (Table S2). As can be seen from Table 4, in total 99 compounds collected in the DIPPR 80165 database and NIST Chemistry WebBook66 were taken into consideration within the framework of the current study, including 39 compounds for which the experimental data have been used in the parameter optimization process (correlation set). The other compounds were used to check the predictive abilities of the hs-PC-SAFT model (prediction set). As expected, the deviations observed in the prediction set are higher than the corresponding values for the correlation set. Moreover, we find that the AARD and AAD between the model predictions and experimental data for the ρsat l are in general much lower than in the case of Psat, indicating that the vapor pressure is a more sensitive quantity with respect to the hs-PCSAFT parameters. In particular, the overall correlation and prediction AARD for the ρsat l are 0.72% and 1.1%, respectively, while for Psat the respective values are 1.5% and 12%. Significant difference in AARD(Psat) can be explained by very low vapor pressures of heavier compounds included in the prediction set (see Table 1). In fact, even small absolute deviations in Psat

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generate large relative deviations as the vapor pressure decreases. It should be noted that the AAD(Psat) for the linear alkanes is about 3.5 kPa and the values resulting from correlation and prediction are closely the same. On the basis of these observations, we conclude that the hs-PC-SAFT with the parameters optimized in this work is capable of predicting sat simultaneously ρsat over a wide range of temperature. l and P However, the critical point is overestimated by about 2% in critical temperature (Tc) and, thus, by about 20% in critical pressure (Pc). The same was observed in the SAFT-γ51 and the GC-SAFT-VR55 approaches. It was shown that a “crossover” treatment of the critical region has to be made to circumvent this limitation of different SAFT-based models.67−72 However, this is beyond the scope of the present work. Some representative correlations and predictions of ρsat l and sat P with the hs-PC-SAFT are illustrated by Figures 1−3. Vapor

Figure 2. Experimental data65 vs hs-PC-SAFT correlations/predictions for the saturated fluid properties for representative branched alkanes: sat (a) saturated liquid and vapor density (ρsat l and ρv ) and (b) saturated sat vapor pressure (P ).

AARD(ρ lsat ) = 1.1% and AARD(P sat ) = 38% for nhexatriacontane (C36H74). The predictions for branched alkanes are also in good agreement with the DIPPR 801 data,65 including compounds containing in their backbone chains different numbers of tertiary or quaternary carbon atoms (see Figure 2 and Table S2 in Supporting Information). In particular, AARD for heavy and highly branched hydrocarbon squalane (2,6,10,15,19,23-hexamethyltetracosane) is 1.9% for ρsat and 38% for Psat. The l accuracy of density calculation is satisfactory considering the purely predictive character of calculations. In turn, description of vapor pressure is poor, which can be partially assigned to large uncertainties of experimental reference data. Indeed, the declared error of the DIPPR 801 correlation for squalane is

Heterosegmented Perturbed-Chain Statistical Associating Fluid

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