Optimized Binary Interaction Parameters for VLE Calculations of

Article pubs.acs.org/IECR

Optimized Binary Interaction Parameters for VLE Calculations of Natural Gas Mixtures via Cubic and Molecular-Based Equations of State Mert Atilhan* Department of Chemical Engineering, Qatar University, 2713 Doha, Qatar

Santiago Aparicio* Department of Chemistry, University of Burgos, 09001 Burgos, Spain

Kenneth R. Hall Department of Chemical Engineering, Texas A&M University, College Station, Texas, United States S Supporting Information *

ABSTRACT: The objective of this study is to analyze the performance of a selected collection of equations of state (EOS) for the prediction of vapor−liquid equilibrium of the components of a typical natural gas mixture. In this work, 13 biparametric, triparametric, and tetraparametric, widely used simple-cubic-type EOS and three complex, but state-of-the-art, molecular-based EOS are used. A simple, one-fluid-based mixing rule was applied for the extension to mixtures of the models. Binary interaction parameters for the different equations were calculated by correlation of available binary vapor−liquid equilibria data in the experimental literature, in wide temperature−pressure ranges, for the key binary systems relevant to natural gases. The results allow one to infer the performance of the different EOS commonly used for phase equilibria analysis in the natural gas industry, and to study how the different complexity of the studied EOS does (or does not) lead to an improvement in the quality of the predictions.

1. INTRODUCTION Natural gas plays a very significant role in regard to the global energy supply;1 it is projected to be the fastest-growing source of energy in the next several years.2 Moreover, its clean-burning properties makes natural gas the most favorable fossil fuel,3 producing less greenhouse gases than coal or oil. Prediction of thermodynamic properties of natural gas mixtures is important from both industrial and economic viewpoints, because they are required for the efficient design of the operations involved in the natural gas extraction, equipment design and selection, production, transportation, and processing. Moreover, from a theoretical perspective, compels structure of natural-gas-like systems are suitable as strong test systems, for available and developing models. Natural gases are extremely complex multicomponent mixtures whose thermodynamic behavior is not easy to predict; thus, reliable and sufficiently accurate models should be developed and tested against high-quality experimental data. Among all the properties, phase envelopes are one of the most important characteristics that shall be calculated accurately in the custody transfer business. The temperature and pressures ranges in which a natural gas, with a certain composition, is in the homogeneous one-phase region is drawn in a pressure−temperature diagram with a line called a phase envelope. In order to avoid retrograde condensation, accurate knowledge of the phase envelope (or loop) is necessary.4 © 2012 American Chemical Society

Equations of state (EOS) play a pivotal role in the thermodynamic property modeling of complex mixtures and in chemical engineering design, and they are the most common and convenient models for the study of phase equilibria of fluid multicomponent mixtures. EOS have different advantages over other models: they can be applied over wide ranges of pressure and temperature, from low pressure to supercritical conditions; they may be extended straightforward from pure fluids to multicomponent mixtures through adequate mixing and combining rules; and they can be used to model phase equilibria between different phases without any conceptual difficulty.5 Cubic EOS are the most common option in the chemical and gas industries to model complex phase behavior and to design chemical processes,6,7 because of their simplicity. They are widely used for the prediction of phase envelopes8 in academia and industry via commercial software available for process design. These types of EOS have showed good capabilities for correlating properties of complex reservoirs fluids.9,10 In contrast, modern molecular-based EOS, whose theoretical background is stronger by applying principles of statistical Received: Revised: Accepted: Published: 9687

April 18, 2012 June 20, 2012 June 25, 2012 June 25, 2012 dx.doi.org/10.1021/ie301012q | Ind. Eng. Chem. Res. 2012, 51, 9687−9699

Industrial & Engineering Chemistry Research

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practical interest, therefore, the studied EOS should be selected according to certain criteria. The EOS models studied in this work could be classified into two main groups: cubic and molecular-based.10 Cubic-type EOS were selected because they are thoroughly used in the industry. On the other hand, despite the fact that molecular-based EOS have strong theoretical foundations, they are scarcely considered for industrial purposes, because of their complex analytical nature. Molecular-based EOS do not have the almost-semiempirical character of most of the common cubic EOS, which makes studying their performance necessary. The main strength of cubic EOS is their ability to describe the vapor−liquid equilibria (VLE) of mixtures, because it is known that, when density is required, cubic EOS are not the most suitable models.12 Cubic Equations of State. Cubic EOS are modifications of the original basic van der Waals EOS, proposed more than a century ago, and they are termed as cubic because they take a cubic form, in terms of compressibility. In these EOS, two contributions are considered to the total pressure: a repulsive term (usually hard spheres) and an attractive term (to describe the intermolecular interactions). Most of the different modifications to the original EOS have centered on the attractive term, while less attention have been paid to the repulsive term, because most of the proposed variations of the repulsive term give rise to noncubic EOS, with a subsequent loss of simplicity. Between all of the modifications of the attractive term of the original equation, the proposal by Redlich−Kwong (RK)13 (eq 1) is probably the most important, because it was the starting point of many modified EOS. The RK EOS did not have a strong theoretical background, but their inspired EOS gave good results for many different systems. In the RK EOS, the temperature dependence of the attractive term was recognized.

mechanics, are not considered in most of the software tools and no systematic studies have been developed with regard to their ability for phase envelope predictions for complex mixtures, especially for natural-gas-like mixtures.11 Thus, the scarce industrial use of these EOS may be produced, because it is not known if their additional complexity and numerical efforts are justified or not by more-accurate results than those obtained with the relatively simple-form cubic EOS. The extension of EOS to mixtures requires of adequate mixing and combining rules to obtain the mixture parameters from the involved pure compounds. Usually, only binary intermolecular interactions are considered and, hence, only binary interaction parameters should be available for multicomponent mixtures. Natural gas mixtures are mainly composed of hydrocarbons and other simple substances such as nitrogen and carbon dioxide, for which specific intermolecular interactions are absent for many EOS; therefore, these systems does not show strong deviations from ideality. Consequently, we may infer that simple mixing rules, such as those based on the one-fluid model, will give good results. In this work, we have studied the performance of 13 selected cubic EOS, biparametric/triparametric/tetraparametric, and three molecular-based equations for vapor−liquid equilibria predictions. The simple monoparametric van der Waals onefluid mixing rule was applied for the cubic EOS whereas the simple Berthelot−Lorentz mixing rule, which also is based on the one-fluid model, was applied for the noncubic EOS. The binary interaction parameters (BIPs) of all the binary systems involved are required for the phase envelope predictions in multicomponent mixtures; most of them are not available in the literature or they were reported for narrow temperature ranges. In this work, we have determined the binary interaction parameters for all of the EOS considered by correlation of experimental binary vapor−liquid equilibria data available in the literature over wide pressure−temperature ranges for the key binary systems involved in usual natural gas mixtures. The aims of this work are (i) obtain a database of BIPs for widely used EOS via reliable literature sources in wide pressure−temperature ranges of binary systems obtained from the open literature, (ii) check the EOS accuracy for phase equilibria predictions for natural-gas-like mixtures, and (iii) test if the introduction of additional complexity in the modeling, through more-complex cubic EOS or by molecular-based EOS, is justified by an improvement of the results. The BIPs available in the open literature are obtained using different experimental vapor−liquid equilibria data, in different pressure−temperature ranges, and using different mathematical correlating procedures. Therefore, a fair comparison between the performance of the different EOS is not straightforward, using the available data. Similarly, the BIPs used for many process design software programs are proprietary parameters and are not easily available. Moreover, BIPs for all the studied EOS for the required binary systems are not available in the literature.8 Therefore, the results reported in this work would lead to a collection of BIPs for the most relevant EOS, applied to the key binary systems, determined using the same database of vapor−liquid equilibria data.

P=

RT − v−b

a T v(v + b)

(1)

In this equation, the force constant a, the so-called “copressure”, takes account of the attractive interactions whereas the b constant, the so-called “co-volume”, takes account of the repulsive forces. The a term was considered in the RK EOS as temperature-independent. Soave14 replaced the temperature dependency of the attractive term in the RK EOS by the moregeneral α(T,ω) function, in his widely known Soave−Redlich− Kwong (SRK) EOS to improve the accuracy of the EOS to predict vapor pressure (see eqs 2and 3). P=

a(T ) RT − v−b v(v + b)

(2)

a(T ) = aCα(T , ω)

(3)

The α(T,ω) function that developed (eq 4) was correlated by matching the predicted vapor pressure at a reduced temperature (Tr) of 0.7 to the experimental values. α(T , ω) = ⎡⎣1 + m(1 −

2

Tr )⎤⎦

(4)

The α(T,ω) function of the SRK EOS causes this EOS to predict anomalous behaviors under extreme conditions. Thus, Peng−Robinson15 (PR) was developed as a new EOS (eq 5), by modifying the attractive term in an attempt to solve the weakness of SRK EOS in the critical region and its inaccurate liquid predictions. In PR EOS, a α(T,ω) function similar to that

2. EQUATIONS OF STATE A detailed analysis of the literature shows the enormous amount of different EOS available for pressure−volume− temperature (PVT) and phase equilibria calculations.5,6,10 The evaluation of all developed EOS is neither possible nor of 9688

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The Adachie−Lu−Sugie (ALS) EOS25 is the only fourparametric cubic EOS considered in this work (eq 12); the larger number of parameters of this EOS confers it with a moreflexible character. Nevertheless, its use is not very common for industrial purposes.

via the SRK EOS is considered, but they used a different reduced temperature range to correlate it (Tr = 0.7−1). P=

a(T ) RT − v−b v(v + b) + b(v − b)

(5)

Because of the correlation procedure used to obtain the α(T,ω) function, the vapor pressures obtained by the SRK EOS are less accurate for Tr < 0.7; thus, Twu et al.16 modified the α(T,ω) function in their EOS (TRK) (see eqs 6 and 7) that works well over a wide temperature range. α(i) = TrN(M − 1) exp⎡⎣L(1 − TrNM )⎤⎦

(6)

α = α(0) + ω(α(1) − α(0))

(7)

P=

a(T ) RT − 2 v−b v + (1 + 3ω)bv − 3ωb2

P=

(8)

The Patel−Teja (PT) EOS, and the Valderrama modification of the Patel−Teja (PT) EOS (PTV),21 are also three-parametric EOS (see eq 9), in which a more-flexible third parameter (c) is included. Both EOS use the same α(T,ω) function (eqs 3 and 4), but different ways of calculating the EOS parameters. The PT EOS may be considered a general form of the SRK and PR EOS, which will reduce to either of them at their prevailing constant critical compressibility factors.9 a(T ) RT − v−b v(v + b) + c(v − b)

(9)

a(T ) RT − v−b v(v + c) + c(v − b)

seg

a(T ) RT − v−b (v + c)2

(13)

chain

hs

(14)

disp

where A , A , A , and A are, respectively, the segment, chain, hard-sphere, and dispersive contributions to the total Helmholtz free energy, and m̅ is the segment number. In the Huang and Radosz version of the SAFT EOS, the hard-sphere term is calculated according to the Carnahan−Starling model,36 and the dispersion term by Chen and Kreglewski is applied in the framework of SAFT theory.37 Another successful version of the original SAFT model has been developed by Gross and Sadowski,38 in which the perturbed chain modification of SAFT (PC-SAFT) is developed by extending the perturbation theory of Barker and Henderson39 to a hard-chain reference. The general expression for the residual Helmholtz energy of a

(10)

Another three-parametric cubic EOS was proposed by Kubic (KU)23 (eq 11), in which a virial interpretation of the Martin24 equation was developed to infer the temperature dependence of the parameters. P=

a(T ) T v(v + NMεb)

⎛ A hs A disp ⎞ A Aseg Achain Achain = + = m̅ ⎜⎜ 0 + 0 ⎟⎟ + RT RT RT RT ⎠ RT ⎝ RT

Guo and Du22 developed a three-parametric EOS (GD) similar to the PT EOS (eq 10), with an α(T,ω) function similar to that of eqs 3 and 4. It was developed specially for hydrocarbon mixtures and reservoir fluids. P=

RT ⎛ v + εb ⎞ ⎜ ⎟ − v ⎝v−b⎠

Molecular-Based Equations of State. The advances in statistical mechanics and computational capabilities have allowed the development of EOS based on molecular principlesthe so-called “molecular-based EOS”with a strong theoretical basis. They are accurate for pure fluids as well as multicomponent mixtures and have shown appreciable performance in predicting volumetric properties and phase equilibria in wide temperature and pressure ranges. These EOS allow separation and quantification of the effects of molecular structure and interactions on bulk properties and phase behavior. Despite the advances in this field, impact and practical use of these EOS for industrial purposes has been very limited, because of their intrinsic algebraic complexity, which causes demanding computational power and time.27 Wertheim’s first-order thermodynamic perturbation theory28−31 constitutes one of the most successful starting points to develop molecular-based EOS. Based on this approach, Chapman et al.32,33 developed the so−called Statistical Associating Fluid Theory (SAFT). One of the most successful modifications to the original SAFT model is the parametrization by Huang and Radosz;34,35 this version is the one that has been applied in this work. SAFT describes the residual Helmholtz free energy (A) of a mixture of nonassociating fluids as a sum of different contributions (see eq 14):

20

P=

(12)

All the considered EOS, until now, have used the same expressions for the repulsive term of the EOS; the Mohsen− Nia et al. EOS (MMM)26 is the only cubic EOS considered in this work in which the repulsive term of the equation is modified (see eq 13). In this EOS, a more-accurate empirical repulsive term obtained from a molecular simulation of hard spheres is used; therefore, it has been used in an attempt to improve the accuracy of the EOS results by considering the importance of repulsive interactions in the fluid behavior. The α(T,ω) function is expressed the same way as in eqs 3 and 4:

They also modified, in the same way, the α(T,ω) function for the PR EOS (TPR)17 for a similar reason. Another improvement of the PR EOS was developed by Stryjek and Vera (PRSV),18 through a modification of the temperature and acentric factor dependence of m in eq 4. The aforementioned EOS are two-parametric and these types of EOS predict the same critical compressibility factor for all of the substances; hence, inaccurate vapor pressure and volume predictions are possible via the above-mentioned EOS. Thus, some authors have tried to solve this limitation through the inclusion of a third parameter in the EOS. Schmidt and Wenzel developed an EOS (SW)19 that may be included in this group; they incorporated the acentric factor as a third parameter in the EOS (eq 8), the temperature dependency of attractive form takes the same form as that of the previously defined EOS (see eqs 3 and 4). This EOS accurately predicts the liquid density and vapor pressures of light and moderate hydrocarbons. P=

a(T ) RT − v−b (v − c)(v + c)

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Figure 1. Scheme of the equation of state (EOS) study proposed in this work.

Thus, mixture parameters are obtained from the pure fluid ones from eqs 16 and 17:

nonassociating fluid is the same as for the SAFT model (eq 14). The Mansoori et al.40 model is used for the hard-sphere reference term. A different approach to obtain a molecular-based EOS was developed by Muller et al.41,42 in the so-called “BACKONE EOS”. In this EOS, different contributions are considered to comprise the total residual Helmholtz energy (see eq 15): H

A

a=

i

b=

(15)

j

∑ xibi i

POL

A A A A = + + RT RT RT RT

∑ ∑ xixjaij= ∑ ∑ xixj(aiaj)0.5 (1 − kij) i

j

(16)

(17)

For EOS with more than two parameters, a mixing rule, such as eq 17, is applied for the other parameters. The binary interaction parameter, kij, is obtained by fitting EOS predictions to experimental binary phase equilibria data. Hence, this parameter should be considered as a fitting parameter rather than a rigorous physical term. Therefore, the interaction parameters obtained for particular EOS should be applied only for the same EOS. The inclusion of binary interaction parameters improves the results obtained from an EOS; however, sometimes, this procedure is just a mere exercise of curvefitting and the extension to multicomponent mixtures without any additional parameters. Thus, some EOS may provide better results without binary interaction parameters (pure predictions) than other EOS with them. For SAFT and PC-SAFT molecular-based EOS, the one-fluid model was also applied to obtain the mixture parameters with just one binary interaction parameter to take account of the nonideality of mixtures.35,38 For the BACKONE EOS, the mixing rules designed by Muller et al. were applied.42 General hydrocarbon mixtures of reservoir fluids (i.e., light natural gases) were considered in this work for binary interaction parameter calculations. The one-fluid mixing rule should provide accurate enough results; therefore, other complex mixing rules available in the literature were not explored.

where AH, AA, and APOL are the hardcore, dispersive, and polar contributions, respectively. The hard convex body (HCB) equation of Boublik43 is used for AH. The functional form of the AA term is constructed using data from methane, oxygen, and ethane. The APOL term contains the dipolar and quadrupolar contributions (in this work, we do not consider compounds with a permanent dipole moment, and, thus, only quadrupolar contributions need to be considered; hence, this EOS may be termed as QUABACKONE. The quadrupolar term was obtained from extensive molecular dynamics simulations. The BACKONE EOS may be included in the group of generalized semi-empiric multiparametric EOS.44 A scheme of the EOS studied in this work is reported in Figure 1.

3. MIXING RULES Equations of state (EOS) are applied to multicomponent fluids by employing mixing rules to determine their parameters for mixtures. The mixing rules should describe the prevailing intermolecular forces between the different molecules forming the mixture.9 Extension of EOS to mixtures is, at this moment, empirical, unless semi-empirical in nature, because there is neither an exact statistical mechanical treatment relating the properties of dense fluids with their intermolecular potentials, nor detailed information about these potentials.45 The first successful method for the extension of cubic EOS to mixtures was the one-fluid model proposed by van der Waals.46 This model considers that the same EOS used for pure fluids may be applied to a mixture if a way is found to obtain the mixture parameters.45 This is done considering that the mixing rule in any EOS should attain the same form as that of the virial equation under conditions where both equations are valid.

4. RESULTS AND DISCUSSION Pure Compounds. The required EOS parameters for the pure compounds involved in the mixtures were obtained from the literature.35,38,47 Several generalized correlations have appeared in the literature for temperature-dependent BIPs. Natural gas mixtures are composed mainly of alkanes up to C5; therefore, we will limit our analysis up to pentane alkanes. 9690



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Figure 2. Relative deviation between experimental (methane,48 n-pentane,44 nitrogen,49 and carbon dioxide50) and EOS calculated vapor pressure for selected compounds. Legend: (−) PR, (- - -) PT, (− − −) SAFT, (− · −) and PC-SAFT.

Figure 3. Relative deviation between experiment and EOS calculated saturated vapor and liquid density for selected compounds. Experimental values and symbols as in Figure 2.

Any EOS shall represent accurate property prediction of the pure components that forms up the multicomponent mixtures. Any systematic deviation in the pure property determination will propagate into mixture calculations, which would result in unrealistic correlations and/or predictions. In Figure 2, vapor pressures for different compounds and EOS comparisons with the experimental ones are shown. The quality of the correlations is good for the EOS and selected compounds. As being the major component of natural gas, PC-SAFT shows the best results for methane, despite the problematic predications at low temperatures. As the n-alkane chain length increases, correlations are worse for all the EOS (i.e., see results for npentane) mainly at the low temperature region; but the results from PC-SAFT are clearly superior to those from the other EOS studied. For nitrogen and carbon dioxide, the correlations with PC-SAFT are clearly better than for any other EOS. The only problem that appears for PC−SAFT is that it overpredicts the critical properties.

In Figure 3, calculated vapor−liquid equilibria of pure compounds are presented with comparisons of the experimental saturated densities. From this figure, we may deduce that significant deviations are presented by most of the EOS, mainly in the low-temperature region and in the critical region, but, again, PC-SAFT is clearly superior to the other ones; only the deviations close to the critical point for this equation are remarkable, but, generally, PC-SAFT accurately correlates the equilibrium vapor and liquid densities of the main compounds of natural gases. PC-SAFT seems to be the most adequate EOS for the description of properties of pure compounds involved in natural gas mixtures; it is remarkable that PC-SAFT is clearly superior not only to cubic ones but also to among the other molecular-based EOS. Binary Interaction Parameter (BIP) Optimization. The prediction of phase envelopes in multicomponent mixtures requires the previous knowledge of binary interaction parameters for different EOS. Although some BIP might be found in the literature for some EOS,9 most of the parameters 9691



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where Nn represents the number of experimental VLE data points of the n binary system. In order to analyze the global performance of each studied EOS, a weighed value of AAD% was defined for each EOS, according to eq 21:

are not available for all the EOS. Therefore, we have correlated available literature binary VLE data, in order to obtain BIPs for all the binary mixtures commonly involved in multicomponent natural gas mixtures. As mentioned in the previous section, we considered alkanes only up to C5 for BIPs correlation, and thus, BIPs for any mixture containing alkanes higher than C5 and binary systems for which no experimental VLE data are available were set as zero indicating no interaction. In Table S1 in the Supporting Information, we reported the literature sources of experimental data together with the temperature range and number of data points considered in the correlation of each binary system. In this work, we considered the binary interaction parameters as temperature-independent; therefore, single BIP is reported for each binary system and EOS. Despite the simplified approach considered for BIPs in this work, we should remark that the temperature variation of BIPs is wellknown in the literature, and thus, several generalized correlations have been proposed with variable degrees of success.45 Similarly, some authors have proposed the dependence of BIPs with pressure and mixture composition.9 Nevertheless, the objective of the work is to analyze the performance of several families of EOS, and thus, to allow a fair comparison between them, we considered the simplified, temperature-independent BIPs, approach for all the studied systems. Similarly, this approach is common for the prediction of natural gas properties using EOS and has led to acceptable results for VLE predictions.8,9 The fitting of experimental VLE data to all the EOS to obtain BIPs was carried out using our own MATLAB codes. The BIPs were obtained by minimizing the objective function, obj, reported in eq 18, which was previously used in the literature for the same purposes:38,51

r

AAD% weighed =

Figure 4. Percentage weighted absolute average deviations obtained from the correlation of experimental vapor−liquid equilibria data reported in Table S1 in the Supporting Information, and the EOS with BIPs from Tables S1−S17 in the Supporting Information. Arrows indicate EOS leading to lower deviations for biparametric, triparametric, and molecular-type EOS. Horizontal dotted lines show the value of lower deviation, obtained for the PC-SAFT EOS; vertical dashed lines divide the EOS according to their nature: biparametric (2P), triparametric (3-P), tetra-parametric (4-P), and molecular-type EOS.

(18)

yi , j xi , j

(19)

where the subscript i stands for the number of experimental VLE data points and the subscripts 1 and 2 represent the two components of the binary mixture. Optimizations were performed using different initial guesses and different algorithms. After trying different algorithms and initial guesses, consistent results were obtained confirming that local minima are not found. Algorithms used were Nelder−Mead simplex search method52 (f minsearch function in MATLAB), Levenberg−Marquardt,53 and Trust−Region−Reflective optimization54 (these last two ones algorithms in the lsqnonlin function in MATLAB). Results of the correlations of VLE data, for the 36 binary systems and 16 EOS, are reported in the Supporting Information (Tables S2−S17) . Discussion. Results reported in the Supporting Information (Tables S2−S17) show the percentage absolute average deviation (AAD%n) (see eq 20), between experimental and calculated vapor−liquid equilibria data, for each of the n (= 36) studied binary systems, for the temperature ranges reported in the Supporting Information (Table S1) and the studied EOS. EOS ⎛ 100 ⎞ k (K iexp ,j − Ki,j ) AAD%n = ⎜ ⎟∑ K iexp ⎝ Nn ⎠ i = 1 ,j

(21)

where N is the total number of experimental VLE data points used in the correlation, for all the considered binary systems (N = 1714). The values of AAD% weighed show the global performance of each EOS for the studied binary systems that characterize lean natural gas mixtures, and they are plotted in Figure 4. According to Figure 4, PC-SAFT EOS leads to the

⎡ (K exp − K EOS) ⎤2 ⎡ (K exp − K EOS) ⎤2 i ,1 i,1 ⎥ + ∑ ⎢ i ,2 exp i ,2 ⎥ obj = ∑ ⎢ exp K K i ,2 ⎢ ⎥⎦ ⎢ ⎥⎦ i ,1 i ⎣ i ⎣

Ki,j =

⎛ Nn ⎞ ⎟AAD% n ⎠ N n=1

∑ ⎜⎝

lower deviations between the 16 studied EOS. Moreover, PR and ALS show the best performance between the studied cubic EOS. Yet RK, in the cubic EOS group, and SAFT, in the molecular-based EOS group, lead to the larger deviations with experimental results. The number of parameters in the studied cubic EOS has a very mild effect on the performance of the models, and in some cases (PT, SW, KU, MMM, and GD), the evolution from biparametric to triparametric cubic EOS leads to worse correlative ability. Only for ALS, the evolution from biparametric to tetraparametric cubic EOS leads to a slight improvement of the model performance, although the additional complexity of the EOS is not justified by a remarkable improvement oversimpler biparametric EOS such as PR. The modification of the attractive term in the cubic EOS does not lead to a remarkable improvement of the EOS performances, we may say that α terms of PR and ALS type give rise to slightly lower deviations. In contrast, modifications of the repulsive term of the EOS, studied in this work through the MMM EOS, decreases the correlative ability of the model, and the van der Waals type repulsive term is more suitable for the studied systems.

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Figure 5. Behavior of the calculated BIPs along homologous series for the reported EOS; the parameter m represents the number of carbon atoms of the n-alkane. Solid circles (●) represent BIPs from tables in the Supporting Information, and lines () represent linear fits for guiding purposes.

For the molecular-based EOS family, PC-SAFT showed remarkably lower deviations than the original SAFT model and to a better performance than BACKONE for the studied binary mixtures. Therefore, within the group of the 16 studied EOS, PR and PC-SAFT EOS should be recommended for the study of the considered binary systems, although similar correlations may be obtained with other biparametric cubic EOS such as TRK or TPR. Obtained BIPs reported in this work will be used in a future manuscript for phase equilibria predictions in the multicomponent mixtures in which they are involved; therefore, it is essential to analyze their behavior along homologous series. For this purpose, we report, in Figure 5, the behavior of BIPs for four selected EOS (PR (biparametric), PT (triparametric), ALS (tetraparametric), and PC-SAFT (molecular-based)) along the series methane + n-alkane, CO2 + n-alkane and N2 + nalkane, as a function of the number of carbon atoms (m) of the n-alkane). The plots in Figure 5 show an almost linear behavior for the reported EOS and systems, with increasing BIPs as m increases. Such smooth variation along the three main homologous series for the modeling of natural gas mixtures points to the reliability of the obtained BIPs.

The performance of the PC-SAFT, PR, and PT EOS for the correlation of VLE is reported in Figures 6−8. In Figure 6, the VLE for methane + ethane and methane + n-pentane are plotted as they are the representative examples of mixtures

Figure 6. Vapor−liquid equilibria (VLE) for x methane + (a) (1 − x) ethane and (b) (1 − x) n-pentane at different temperatures. Solid circles (●) represent experimental data,55−58 and solid lines () represent values calculated with reported EOS and binary interaction parameters from the Supporting Information. 9693



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temperature increases. Quadrupolar character of CO2 molecules leads to larger deviations of ideality for these systems. Results reported in Figures 6−8 show that the predictions of the studied EOS diverges as the n-alkane chain length increases, as the temperature increases, and as the critical region is approached. These three factors should lead to remarkably different predictions of the VLE for multicomponent mixtures when the reported BIPs/EOS are applied with predictive purposes. The behavior of the EOS in the vicinity of the critical region is best analyzed if the global phase behavior of the mixtures is calculated. Global phase behavior of binary mixtures involved in the description of natural gas mixtures was previously analyzed in an extensive way using PC-SAFT in a series of works reported by our group.64−66 Thus, we report in Figures 9 and 10 the P−T projection of the global phase diagrams for methane + n-alkane, CO2 + n-alkane, and N2 + nalkane mixtures for PC-SAFT, PR, and PT EOS. Analogous conclusions may be inferred for the remaining EOS but they are not included in these figures, for the sake of better visibility. For methane + n-alkane (from ethane to n-pentane) binary mixtures show Type I behavior, according to the van Konynenburg classification.67 All the studied EOS predict Type I behavior; nevertheless, as the n-alkane chain length increases, the predictions of the vapor−liquid critical loci are worse by the three reported EOS, leading to an overestimation of the critical loci maxima, which is more remarkable for PCSAFT EOS. On the other hand, for N2 + n-alkane, it may be concluded that, although EOS predict correctly as Type I behavior for methane mixtures and Type III for those with longer n-alkanes; predictions of the critical loci is worse for the longer n-alkanes. Nevertheless, for these systems, results from PC-SAFT are closer to the experimental values. An extended plot of the behavior in the vicinity of nitrogen critical point is reported in Figure 10a; the vapor−liquid−liquid equilibrium (VLLE) threephase line is properly predicted by the three EOS with similar quality (although slightly better results are obtained from PCSAFT). It should be remarked that BIPs are obtained from the correlation of only VLE data and they are temperatureindependent; therefore, the performance of the models for the prediction of VLLE may be considered as successful considering the simplicity of the models, especially for reported simple cubic EOS. Moreover, the predicted upper critical end

Figure 7. VLE for x N2 + (a) (1 − x) methane and (b) (1 − x) npentane at different temperatures. Solid circles (●) represent experimental data,59,60 and solid lines () represented values calculated with reported EOS and binary interaction parameters from the Supporting Information.

containing two short alkanes (and the main constituents of natural gases), and thus with a highly symmetric character and moderate deviations from the ideality, and highly asymmetric mixtures (short + long n-alkanes), respectively. The performance of the three selected EOS for methane + ethane mixtures is very similar, with a slightly better behavior of PC-SAFT for the higher temperatures. For methane + n-pentane mixtures, the results from PC-SAFT are remarkably different from those obtained from the PR and PT EOS, mainly in the vicinity of the critical region and, more remarkably, in the high-temperature region. These subtle differences between PC-SAFT EOS and PR/PT behavior is mainly for short + long n-alkane mixtures. This may have remarkable effects on the EOS performances for VLE in multicomponent mixtures, because of the well-known behavior of long n-alkanes on the condensation behavior of natural gas multicomponent mixtures at the cricondentherm and cricondenbar values. Mixtures of N2 + methane or + n-pentane are reported in Figure 7. The systems CO2 + (methane, ethane, or n-pentane) are analyzed in Figure 8. The binary system CO2 + ethane shows azeotropic behavior, which is properly predicted by PCSAFT and PR; yet, the PT triparametric EOS has larger deviations. The mixtures of CO2 + n-pentane shows different predictions of the critical region for the PC-SAFT as the

Figure 8. VLE for (a) x methane + (a) (1 − x) CO2, (b) x CO2 + (1 − x) ethane and (c) x CO2 + (1 − x) n-pentane at different temperatures. Solid circles (●) represent experimental data,61−63 and solid lines () represent values calculated with reported EOS and binary interaction parameters from the Supporting Information. 9694



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Figure 9. P−T projection of the global phase diagrams for the reported binary systems. Open circles (○) represent experimental gas−liquid critical points,68−95 solid lines () represent calculated pure−compound vapor pressures, and dashed lines (- - -) represent calculated critical loci using PC−SAFT (black), PR (blue), and PT (red). k12 values have been taken from the tables given in the Supporting Information. Pure-compound vapor pressures are calculated with PC-SAFT, PR, and PT. It should be remarked that critical properties for pure compounds are overestimated by the PCSAFT model. The numbers within each panel shows the number of carbon atoms for each n-alkane.

Although we have mentioned in the Introduction section, the problems with the BIPs available in the open literature, we report in Figure 11 a comparison between the BIPs obtained in this work and those from the literature, for SRK, PR, and PT EOS (the most common EOS for process design purposes).9 The literature comparison shows acceptable agreement with values reported in this work; the discrepancies may be attributed to the different experimental VLE data used for the BIPs correlation, and, thus, the different pressure−temperature ranges considered, together with the mathematical fitting procedure applied (different objective functions along the correlation). Finally, although the objective of this work is to analyze the performance of the studied EOS for binary mixtures relevant for the description of natural gas mixtures, it is also important to analyze the performance of the reported EOS/BIPs for the prediction of phase equilibria in multicomponent mixtures. The VLE of natural gas mixtures is commonly described for industrial purposes using the pressure−temperature projection, the so-called “phase envelope”, in which a curve separating the two-phase (VLE) and single-phase (vapor or liquid) regions is shown. Although a systematic study on the EOS performance for the prediction of phase envelopes is underway and will be published in future work, we report the initial results here using our recently published experimental phase equilibria data for natural-gas-like mixtures.96 The results reported in Figure 12 for the comparison with a model multicomponent natural-gaslike mixture show that the parameters reported in this work lead to an improvement in the prediction of phase envelopes, especially in the region close to cricondenbar. Nevertheless, these initial results will be confirmed in future works using different types of multicomponent mixtures.

Figure 10. P−T projection of the global phase diagrams for the reported binary mixtures in the vicinity of (a) N2 and (b) CO2 vapor pressure curves. Solid lines () represent calculated pure-compound vapor pressures, open circles (○) represent experimental gas−liquid critical points,91,92 dashed lines (- - -) represent calculated critical loci (PC-SAFT (black), PR (blue), and PR (red)), solid circles (●) represent experimental three-phase points,75 solid diamond symbols (◆) denote experimental azeotropic points,92 and dashed-dotted lines (− · −) represent the calculated three-phase, vapor−liquid−liquid lines in panel a or calculated azeotropic line in panel b; the open triangle (△) denotes the calculated UCEP. The largest experimental threephase point is the experimental UCEP in panel a. k12 values are taken from tables in the Supporting Information.

point, UCEP, is reasonably close to the experimental value, with better results again for PC-SAFT. CO2 + n-alkanes mixtures show Type I behavior for the studied n-alkanes. Despite the azeotropic behavior of CO2 + ethane mixture, EOS predictions showed reasonable predictions, as can be seen in Figure 10b. In contrast, the azeotropic line is more accurately reproduced by PC-SAFT, compared to PR/PT. Analysis of the global phase behavior leads to the conclusions as the studied EOS predict successfully the global phase behavior for the studied mixtures and, as a general observation, the larger the n-alkane chain, the higher the deviations. Moreover, the performance of cubic EOS is similar with slightly better results for PR, whereas PC-SAFT leads to the best results among the molecular-based EOS with a slight trend to overestimate critical loci.

5. CONCLUSIONS A collection of binary interaction parameters (BIPs) for 16 equations of state (EOS) is obtained for those binary mixtures from the correlation of reliable literature VLE data in wide pressure/temperature ranges. These BIPs will be used in the future manuscript for prediction of phase equilibria of multicomponent natural gas mixtures. Correlative ability of the studied EOS are quite similar for the studied systems, except RK and SAFT. Most of the remaining EOS may be used 9695



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Figure 11. Comparison of BIPs obtained in this work with literature data obtained from Danesh et al.9 for SRK, PR, and PT EOS, along homologous series for the reported EOS. m stands for the number of carbon atoms of the n-alkane. Filled symbols represent BIPs obtained in this work, and empty symbols represent data taken from the literature.

with acceptable accuracy for binary systems. Nevertheless, PR, ALS, and PC-SAFT lead to the lower deviation, and thus they are recommended. The detailed analysis of the correlative ability of the studied EOS shows that an increasing complexity

in the form of the studied cubic EOS, through modifications both on the attractive or repulsive terms and increasing number of parameters does not necessarily lead to an improvement of the predictive capability and simple cubic EOS such as PR might show better prediction performance for some cases. The BIPs reported in this work seem to improve EOS performance for the prediction of phase equilibria in multicomponent mixtures.

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ASSOCIATED CONTENT

S Supporting Information *

Literature references and temperature ranges used in the binary vapor−liquid equilibria (VLE) correlations (Table S1), and binary interaction parameters and absolute average deviations for the binary VLE correlations for all of the equations of state and one-fluid monoparametric mixing rule (Tables S2−S17), are reported as supporting information. This information is available free of charge via the Internet at http://pubs.acs.org/

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Figure 12. Comparison between experimental and PR EOS calculated phase envelope for a multicomponent natural-gas-like mixture. Experimental data obtained from Atilhan et al.96 for a mixture with compositions: x (methane) = 0.89975, x (ethane) = 0.02855, x (propane) = 0.01427, x (i-butane) = 0.00709, x (n-butane) = 0.00722, x (i-pentane) = 0.00450, x (n-pentane) = 0.00450, x (nitrogen) = 0.01713, x (carbon dioxide) = 0.01699.

AUTHOR INFORMATION

Corresponding Author

*E-mail addresses: [email protected] (M.A.), sapar@ubu. es (S.A.). Notes

The authors declare no competing financial interest. 9696



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LIST OF SYMBOLS

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REFERENCES

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a = attractive force parameter in cubic EOS, co-pressure A = Helmholtz energy b = repulsive force parameter in cubic EOS, co-volume c = third parameter in three-parametric cubic EOS d = fourth parameter in four-parametric cubic EOS kij = binary interaction parameter L, M, N = parameters of the TRK and TPR EOS m = parameter of the α(T,ω) function in cubic EOS NM = parameter of the MMM EOS P = pressure R = gas constant T = temperature Tr = reduced temperature v = molar volume xi = mole fraction of compound i

Greek Letters

α(T,ω) = temperature attractive dependent term in cubic EOS α(0), α(1) = parameters of the TRK and TPR EOS ε = parameter of the MMM EOS ω = acentric factor Superscripts

A = dispersive contribution to total Helmholtz energy in BACKONE EOS chain = chain contribution to total Helmholtz energy in SAFT and PC-SAFT EOS disp = dispersive contribution to total Helmholtz energy in SAFT and PC-SAFT EOS hs = hard−spheres contribution to total Helmholtz energy in SAFT and PC-SAFT EOS H = hardcore contribution to total Helmholtz energy in BACKONE EOS POL = polar contribution to total Helmholtz energy in BACKONE EOS seg = segment contribution to total Helmholtz energy in SAFT and PC-SAFT EOS Abbreviations

ALS = Adachi−Lu−Sugie equation of state BACKONE = equation of state by Muller et al. EOS = equation of state GD = Guo−Du equation of state KU = Kubic equation of state MMM = Mohsen−Modarress−Mansoori equation of state NRTL = nonrandom-two-liquid model PC−SAFT = perturbed chain statistical associating fluid theory PR = Peng−Robinson equation of state PRSV = Peng−Robinson−Stryek−Vera equation of state PT = Patel−Teja equation of state PTV = Patel−Teja−Valderrama equation of state PVT = pressure−volume−temperature relationship RK = Redlich−Kwong equation of state SAFT = statistical associating fluid theory SRK = Soave−Redlich−Kwong equation of state SW = Schmidt−Wenzel equation of state TRK = Twu−Redlich−Kwong equation of state TPR = Twu−Peng−Robinson equation of state VLE = vapor−liquid equilibria 9697



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Optimized Binary Interaction Parameters for VLE Calculations of

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