Modeling of Gas Solubility in Hydrocarbons Using the Perturbed

Article pubs.acs.org/IECR

Cite This: Ind. Eng. Chem. Res. 2019, 58, 12347−12360

Modeling of Gas Solubility in Hydrocarbons Using the PerturbedChain Statistical Associating Fluid Theory Equation of State Huashuai Wu,†,‡,§ Ke Zheng,†,‡,§ Gang Wang,*,‡ Yong Yang,†,‡ and Yongwang Li†,‡ †

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State Key Laboratory of Coal Conversion, Institute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan 030001, People’s Republic of China ‡ National Energy R&D Center for Coal to Liquid Fuels, Synfuels China Company, Limited, Huairou District, Beijing 101400, People’s Republic of China § University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China S Supporting Information *

ABSTRACT: Fitted functions that correlate binary interaction parameters (BIP) for the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) with the carbon number of n-alkanes were established for gas solubility in n-alkanes using solubility data of gases in a few selected nalkanes typically with 10−28 carbon atoms. Data screening was made to ensure accuracy and consistency of data used for parametrization and validation. The PC-SAFT EoS using the BIPs by the established fitted functions was evaluated against experimental data. It was found that the PC-SAFT EoS is quantitatively accurate for the entire homologous series of nalkanes, regardless of whether the data of n-alkanes were used in the parametrization. It was shown that the PC-SAFT EoS successfully reproduces the two types of the temperature dependence of gas solubility: (1) gas solubility decreases with an increasing temperature and then increases; (2) gas solubility always increases with an increasing temperature. In particular, the PC-SAFT EoS is capable of predicting an experimentally observed linear relationship between gas solubility in n-alkanes and the carbon number of n-alkanes. To put it into perspective, the PC-SAFT EoS was compared to a couple of popular benchmark group-contribution (GC) EoSs, namely, the predictive Soave−Redlich−Kwang (SRK) EoS and the SRK EoS with the modified Huron−Vidal second-order mixing rule. It was found that the two models generally deliver much worse results than the PCSAFT EoS despite their satisfactory performance for gas solubility in lighter n-alkanes. The PC-SAFT EoS and other cubic EoSs with improved accuracy in modeling of asymmetric mixtures compared to the above benchmark models were validated against experimental data of synthetic gas condensates. It was found that the PC-SAFT EoS is superior to the other models. The comparison was also made among different parametrization strategies, namely, GC-PC-SAFT, the one proposed in this study, and a four-parameter correlation for the simplified PC-SAFT EoS. It was shown that the PC-SAFT EoS parametrized in this study yields more accurate results, although the other two approaches also give satisfactory performance.

1. INTRODUCTION

biomedical technology, medicine and pharmacology, metallurgy, materials science, and so on.3 It is time consuming and costly to measure solubility of gases experimentally. Therefore, the development of estimation methods is essential for reliable estimates of gas solubility over a wide range of temperatures and pressures and for a large variety of solutes and solvents. Modeling of solubility of gases has a long history. Henry’s law was first formulated more than two centuries ago. It is simple to use but holds only at low pressures. Cubic equations of state (EoS), such as Soave− Redlich−Kwang (SRK) and Peng−Robinson (PR), have been

Solubility of gases is one of the most important concepts in science and engineering. Solubility data are basic to the fundamental understanding about a rather diverse area of pure and applied sciences. For instance, for many gas−liquid and gas−liquid−solid reactions, reliable estimates of solubility of gases are often essential for the reliable design, optimization, and control of chemical reactions, e.g., the Fischer−Tropsch synthesis.1,2 Study on gas solubility in aqueous solutions can be used to provide information on hydrophobic effects, which is crucial for life support systems.3 Solubility of CO2, N2, and H2S in high-molecular-weight solvents is very important for engineering processes, such as hydrofining of oil and coal, enhanced oil recovery, and removal of acid gases.4−6 Other areas where solubility data are frequently needed include © 2019 American Chemical Society

Received: Revised: Accepted: Published: 12347

March 12, 2019 June 11, 2019 June 17, 2019 June 17, 2019 DOI: 10.1021/acs.iecr.9b01383 Ind. Eng. Chem. Res. 2019, 58, 12347−12360

Article

Industrial & Engineering Chemistry Research

The perturbed-chain statistical associating fluid theory (PCSAFT) EoS has been the subject of many investigations due to its sound theoretical basis and strong predictive power.57−62 It finds applications in rather diverse areas, such as petroleum,63 biofuels,64 polymers,65 electrolytes,66,67 and ionic liquids.6,68,69 One key advantage that greatly benefits the parametrization of the PC-SAFT EoS is its well-behaved parameters, paving the way for the development of GC methods and BIP correlations. Such a phenomenon has been found in many studies.70−76 GC methods have been devised to parametrize the EoS, leading to the so-called GC-PC-SAFT and critical-point-based PCSAFT.77−88 Ghosh et al.71 evaluated the correlative and predictive capability of the PC-SAFT EoS on gas solubility in hydrocarbons. It was found that a single BIP, independent of both temperature and the carbon number of the solvent, is enough for accurate modeling of the gas solubility. Nonetheless, other studies72,73,89,90 show that the modeling accuracy is improved systematically via introduction of the temperature and carbon-number dependence of the BIP. GC-PC-SAFT has been extended to model gas solubility.74−76,82,91−94 However, the BIP used remains independent of temperature, although the carbon-number dependence is added. Ma et al.72,73 employed the simplified PC-SAFT (sPC-SAFT) EoS and came up with a four-parameter correlation, expressing the BIP as a function of temperature and the molecular weight of nalkanes. It was demonstrated that the results are satisfactory. Nonetheless, their discussions are confined to a few selected gases only, namely, CH4, C2H6, and CO2. In this study, our 3-step parametrization strategy proposed earlier95 for the PC-SAFT EoS was extended to the area of gas solubility. Data screening was made to ensure data accuracy and consistency. The basic criteria are whether Henry’s law holds for experimental data measured at low pressures and whether the Henry’s law constants acquired by fitting data from different sources but obtained at the same temperature and for the same solute and solvent are consistent. Gas solubility obtained at the same temperature and pressure was plotted as a function of the carbon number of n-alkanes. The resulting curves should be smooth in the case of consistent data. Although gas solubility has been studied extensively using the GC EoSs and the PC-SAFT EoS, very few studies have been devoted to comparing the two types of approaches. To put it into perspective, the PC-SAFT EoS and a couple of popular GC EoSs (namely, SRK/MHV2 and PSRK) were compared and evaluated against experimental data. Applications of the GC-SRK EoSs to gas solubility have been studied extensively.45,46,96 Their advantages and disadvantages are well known. Our intention here is not to reiterate what has been found about the GC-SRK EoSs but to advocate the practical use of the PC-SAFT EoS by a comprehensive evaluation and comparison. The PC-SAFT EoS and other cubic EoSs (namely, LCVM, UMR-PRU, and PPR78) with improved accuracy in modeling of asymmetric mixtures compared to the two GC EoSs were validated against experimental data of synthetic gas condensates to further test its predictive power for multicomponent mixtures. The comparison was also made among different parametrization strategies for PC-SAFT, namely, GC-PC-SAFT, the one proposed in this study, and the four-parameter correlation for the sPC-SAFT EoS. Calculations using the PC-SAFT or sPC-SAFT EoS were performed with our in-house FORTRAN code, while those using the PSRK or SRK/MVH2 model with the commercial

widely used in the modeling of solubility of gases at a large variety of conditions due to their ease of implementation and limited computing requirements.7 Typically, binary interaction parameters (BIPs) need to be properly adjusted to obtain reasonable estimates due to a significant difference in molecular size and shape between solutes and solvents. BIPs were fitted with respect to experimental data on a case-by-case basis in most studies.8−16 Nonetheless, this is not a practical option, since it is tedious to fit so many fluid-specific parameters, not to mention the lack of accurate experimental data for weakly characterized components. Many attempts have been made to correlate BIPs with the carbon number or molecular weight of a homologous series of solvents, e.g., nalkanes.5,9,17−28 Most BIP correlations apply to a specific gas only or suffer from limited extrapolative capabilities. For instance, Kordas et al.25 proposed a piecewise function to correlate a temperature-independent BIP with the acentric factor of n-alkanes. However, their study is limited to methane only, and the discontinuity arising from the piecewise function is problematic. Three EoSs (namely, PR, SRK, and the one proposed by Valderrama and Cisternas29) were applied to calculate the vapor−liquid equilibrium of different gases (H2S, CO2, and H2) and n-alkanes.22−24 BIP correlations were developed for different gases and different models. However, such obtained correlations are applicable to lighter n-alkanes only. One popular model in this category is the so-called predictive PR EoS (PPR78). The acronym comes from the fact that such a model was developed based on the version of the PR EoS30 published in 1978 and with the addition of a group contribution (GC) method for estimating required BIPs.31 The GC method correlates BIPs with temperature and properties (namely, critical temperatures, critical pressures, and acentric factors) of the pure components. The application of the PPR78 model was limited to mixtures containing n-alkanes initially and has been extended to cover more solvents (e.g., branched alkanes, aromatics, and naphthenes) and gases (e.g., CO2, H2, N2, and H2S).31−41 Another popular methodology to parametrize cubic EoSs is to use the Huron−Vidal-type mixing rule coupled with the universal quasi-chemical functional-group activity coefficients (UNIFAC) model, leading to a so-called GC EoS.42,43 Such models have been used extensively in practical applications, since they are fully predictive due to the availability of all BIPs via the UNIFAC model and combine the advantages of both cubic EoSs and activity coefficient models.42−50 On one hand, GC EoSs enjoy their EoS identity and therefore can handle applications involving noncondensable gases. On the other hand, GC EoSs are capable of dealing with cases involving highly polar fluids by taking advantage of activity coefficient models. A couple of GC EoSs, namely, the SRK EoS with the modified Huron−Vidal second-order mixing rule (MHV2) and the predictive SRK (PSRK) EoS, have been available in many commercial process simulators (e.g., Aspen plus) and served as benchmark models for many applications.45−50 Nonetheless, it was found that these two models yield poor results for asymmetric mixtures where different components differ significantly in molecular size.51−53 As a result, a few GC EoSs, namely, linear combination of the Vidal and Michelsen mixing rules (LCVM),54 universal mixing rule PR UNIFAC (UMR-PRU),55,56 and volume-translated PR (VT-PR),52 have been developed to remedy such a deficiency. It was observed that these new models yield better results for asymmetric mixtures.51,52,54−56 12348

DOI: 10.1021/acs.iecr.9b01383 Ind. Eng. Chem. Res. 2019, 58, 12347−12360

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Industrial & Engineering Chemistry Research process simulator, Aspen plus 8.4. The BIP values for the PSRK and SRK/MHV2 EoSs can be found elsewhere.47,97

2. PARAMETRIZATION OF PC-SAFT EOS 2.1. PC-SAFT Parameters. The three pure-component parameters, namely, (1) the segment diameter σ, (2) the depth of the potential ε/k, and (3) the number of segments per chain m, are required for the PC-SAFT EoS to describe nonpolar fluids. One more parameter (the quadrupole moment Q) is required for quadrupolar components, e.g., N2 and CO2. The quadrupolar term derived by Gross62 was used in this work to account for the quadrupolar contribution. Since measured quadrupole moments are used, no additional adjustable parameters are introduced. The pure-component parameters for gases involved are presented in Table 1, while those for nalkanes can be found elsewhere.95

m

σ [Å]

ε/k [K]

CH4 C2H6 C2H4 H2 CO N2 CO2

1 1.6069 1.5831 1.3060 1.3097 1.1504 1.5131

3.7039 3.5206 3.4138 2.6010 3.2507 3.3848 3.1869

150.03 191.42 177.82 23.42 92.15 91.40 163.33

Q [DA]

ref

1.43 4.40

57 57 57 92 57 62 62

In the case of mixtures, conventional Lorentz−Berthelot mixing rules as given in eqs 1 and 2 are used to determine the size and energy parameters for a pair of unlike segments σi + σj σij = (1) 2 εij = (1 − kij) εiεj

(2)

The BIP kij is used to correct the segment−segment interactions for unlike segments. The BIP is obtained as a function of the temperature T and the carbon number Cn of nparaffins kij = kij0 +

kijTT 1000

(4)

kijT = bi1 − bi2biC3n

(5)

where ai1, ai2, ai3, bi1, bi2, and bi3 are adjustable parameters. Our 3-step parametrization methodology is briefly discussed here for completeness. Take the BIPs of hydrogen-n-alkanes pairs for example: (1) estimate kij0 and kijT for a few selected nalkanes with different carbon atoms by fitting gas solubility data; (2) determine ai1, ai2, and ai3 by fitting the kij0 acquired from step 1 to eq 4; determine bi1, bi2, and bi3 by fitting the kijT acquired from step 1 to eq 5; (3) obtain a BIP value for any given n-alkane by the fitted function acquired from step 2. More details on our parametrization strategy can be found elsewhere.95 It should be stressed that parametrization by fitted functions derived from the above procedure is equivalent to that by GC methods. Data regressions were used to determine the six parameters (ai1, ai2, ai3, bi1, bi2, and bi3) in the BIP fitted functions, which can subsequently be used to provide BIP values for the PC-SAFT EoS. Likewise, data regressions were also used to determine BIPs between two different functional groups, and subsequently, GC methods (e.g., UNIFAC) with the estimated BIPs for functional groups can be used to parametrize cubic EoSs. Therefore, a meaningful comparison can be made between the PC-SAFT EoS parametrized by fitted functions and GC EoSs by the UNIFAC model. 2.2. Data Screening. A reliable, consistent data set is crucial for parametrization and validation of thermodynamic models. Data screening was performed in this study to ensure the quality of the data used. The basic criteria are whether Henry’s law holds for experimental data measured at low pressures and whether the Henry’s law constants acquired by fitting data from different sources but obtained at the same temperature and for the same gas−solvent pair are consistent. As expected, most experimental data collected in this study are accurate and consistent with each other. Figure 1 gives two examples where data from two or more sources can be characterized with a single Henry’s law constant. More examples can be found in Figures S1−S4. Some outliers were also detected. For instance, as shown in Figure 2, data measured at low pressures disobey Henry’s law, since the linear fits clearly give nonzero intercepts, implying much higher vapor pressures caused by the presence of more volatile impurities in n-hexadecane. Other examples can be found in

Table 1. PC-SAFT Pure-Component Parameters for Gases component

kij0 = ai1 − ai2aiC3n

(3)

Figure 1. Solubility of various gases in n-alkanes as a function of pressure at 373 K. Symbols are experimental data98−101 and lines linear fits. 12349


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holds between the logarithm of the gas solubility and the logarithm of the temperature over a temperature range within which ΔS remains unvaried. All of the estimations were made inside the region where both Henry’s law and the approximation that eq 6 can be treated as a linear relationship hold. More details of the estimations can be found in Figures S9−S31. Figure 3 displays an example of the variation of the measured (solid symbols) and estimated (empty symbols) solubilities of CH4 in nalkanes with increasing chain length of n-alkanes. The results for other gases are shown in Figures S32−S41. Clearly, a linear relationship between gas solubility in n-alkanes and the carbon number of n-alkanes holds for all of the studied cases, implying the universality of such a linear relationship for each gas on one hand and accuracy and consistency of the data used for parametrization and validation in this study on the other hand. Table 2 presents the data used in this study for CH4. A selected set of data was used for parametrization only and the remainder for validation. The data used for the other gases (C2H6, C2H4, H2, CO, N2, and CO2) can be found in Tables S1−S6.

Figure 2. Solubility of CO2 in n-hexadecane as a function of pressure at different temperatures (black for 303 K, red for 313 K, and blue for 323 K). Symbols are experimental data102 and lines linear fits.

Figures S5 and S6. Figures S7 and S8 display examples showing the inconsistency among data from different sources. To further test the consistency and accuracy of the data collected, gas solubility obtained at the same temperature and pressure was plotted as a function of the carbon number of nalkanes. It was found that a linear relationship approximately holds between the solubility of C2H6 and SO2 in n-alkanes and the carbon number of n-alkanes.103,104 It is not clear whether this is universally true for all other gases. In our opinion, the relationship between the two variables should be characterized by a smooth curve when data used are consistent and reliable. Most solubility data were measured at different temperatures and pressures. In order to obtain solubility data at the same temperature and pressure, Henry’s law was used to estimate solubilities at different pressures and eq 6 derived by Hilderbrand and co-workers105,106 solubilities at different temperatures ij ∂ ln x1 yz ΔS jj zz = R k ∂ ln T { P

3. RESULTS AND DISCUSSION According to our 3-step parametrization strategy, step 1 was to estimate kij between the studied gases and a few selected nalkanes typically with 10−28 carbon atoms by fitting experimental data. Figure 9 gives which specific set of kij was estimated for each gas. As mentioned earlier, Table 2 summarizes the data used for the parametrization. BIPs for lighter or heavier n-alkanes were intentionally excluded in parameter estimation here. The purpose was to test whether the fitted functions in eqs 4 and 5 can yield reliable extrapolation for lighter or heavier n-alkanes; step 2 was to use the kij obtained from step 1 to estimate the parameters (ai1, ai2, ai3, bi1, bi2, and bi3) in the fitted functions in eqs 4 and 5. The temperature-independent part and the temperaturedependent part of kij are shown in Figure 9. A greater value of kijT indicates a stronger temperature dependence of kij. Both kij0 and kijT are well behaved and easy to fit as a function of the carbon number of n-alkanes. The same phenomenon was observed when the PC-SAFT EoS was parametrized for water, hydrocarbons, alcohols, and many other oxygenates.95 Note

(6)

where x1 is the gas solubility in mole fraction, R the gas constant, and ΔS the entropy of solution. A linear relationship

Figure 3. Solubility of CH4 in n-alkanes as a function of the carbon number (Cn) of n-alkanes at different temperatures and different pressures (black for 10 bar, red for 30 bar, and blue for 50 bar). Solid circles are experimental data,101,107−110 empty circles estimated values using both Henry’s law and eq 6 based on the experimental data,101,107−119 and lines linear fits. 12350


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Industrial & Engineering Chemistry Research Table 2. Selected Solubility Data of CH4 in n-Alkanes n-alkanes

authors

used for parametrization Reamer et al. n-decane (C10) (1942) Srivastan et al. n-dodecane (C12) (1992) n-hexadecane (C16) Lin et al. (1980) Huang et al. n-eicosane (C20) (1988) Huang et al. n-tetracosane (C24) (1992) n-octacosane (C28) Huang et al. (1988) used for validation Reamer et al. n-propane (C3) (1950) Berry et al. n-pentane (C5) (1970) n-heptane (C7) Reamer et al. (1956) Stepanova et n-tridecane (C13) al. (1972) Nourozieh et n-tetradecane (C14) al. (2012) n-octadecane (C18) Kariznovi et al. (2012) n-dotriacontane Huang et al. (C32) (1992) n-hexatriacontane Huang et al. (C36) (1988) Darwish et al. n-tetratetracontane (1992) (C44)

temperature range [K]

pressure range [bar]

ref

310−510

1−394

111

323−373

13−104

112

462−623

20−252

116

373−573

10−50

110

373−573

10−50

108

373−573

10−50

101

277−360

6−99

120

277−411

1−165

121

277−511

13−241

122

323−423

9−441

114

294−447

20−95

115

323−447

20−95

118

373−573

10−50

108

373−573

10−50

107

373−423

10−50

119

The BIPs increase with an increasing carbon number before reaching asymptotic values, implying a reasonable extrapolation for heavier alkanes. Compared with the values of kij for H2,CO, and N2, those for CH4, C2H4, and C2H6 are much closer to zero. The value for CO2 lies in between. This is possibly because H2,CO, and N2 are all inorganic, CO2 exhibits properties of both organic compounds and inorganic ones, and all of the other gases are organic. The closer to zero the value of kij is, the closer to ideal mixtures the related pair is. The most nonideal is the pairs between n-alkanes and inorganic gases. CO2/n-alkanes pairs are in between. The least nonideal is the pairs between n-alkanes and organic gases. The value of kij for CH4 is much closer to zero than those for C2H4 and C2H6. This is attributed to the fact that the size difference between CH4 and heavier n-alkanes is bigger. As mentioned earlier, many BIP correlations have been published for cubic EoSs.5,21−28,31,33,123 It is interesting to compare such correlations to the ones proposed in this study. As shown in Figure 5, kij increases monotonically with an increasing carbon number in most cases. Such a carbonnumber dependence can be explained as follows. A BIP value is a measure for the nonideality mainly due to the difference in molecular size for asymmetric mixtures. The size difference increases with an increasing carbon number. Therefore, BIP values should increase monotonically with the carbon number as well. For lighter n-alkanes, the size difference almost disappears as does the nonideality. Therefore, BIP values should be much smaller for such cases. The carbon-number dependence is different for BIP correlations developed for cubic EoSs. One phenomenon many of such BIP correlations share in common is that there exists the maximum of BIP values located around n-hexadecane,24,25,31,33,123 which is in contradiction with the above explanation. The carbon-number dependence in the region of heavier hydrocarbons is also problematic for many BIP correlations developed for cubic EoSs. For instance, kij was found to decrease from positive values to negative ones.25 This means that the nonideality decreases first and then increases in the opposite direction. Another example is that there exists the minimum of BIP values around n-eicosane.123 The carbon-number dependence of the PC-SAFT BIP correlations is beneficial from a practical

that this is not applicable for most thermodynamic models but an extra benefit for the PC-SAFT EoS due to its sound theoretical basis; step 3 was to obtain kij for any given carbon number using the fitted functions. It is generally accepted that calculations using BIPs other than the fitted BIPs (symbols in Figure 4) are not correlations but predictions. The fitted parameters in eqs 4 and 5 are given in Table 3 for all of the gases studied. As shown in Figure 5, kij is presented as a function of the carbon number of n-alkanes for different gases.

Figure 4. Variation of BIPs (kij0 and kijT) for gases/n-alkanes with carbon number (Cn) of n-alkanes (black for CH4, red for C2H6, blue for C2H4, magenta for H2, olive for CO, orange for N2, and navy for CO2). Symbols are fitted BIPs and lines (solid for interpolation and dash for extrapolation) fitted functions. 12351


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Industrial & Engineering Chemistry Research Table 3. Summary of Parameters for Fitted Functions for BIPs gases

ai1

ai2

ai3

bi1

bi2

bi3

CH4 C2H6 C2H4 H2 CO N2 CO2

−0.05007 −0.00382 0.00217 0.03595 0.07011 0.08166 −0.00619

−0.08766 −0.00642 −0.07430 −0.51913 −0.15199 −0.30573 −0.22167

0.86766 0.92728 0.83169 0.86101 0.86224 0.76976 0.86089

0.31109 0.09832 0.08089 0.395 0.19611 0.26494 0.23249

0.23609 0.08028 0.13325 1.54308 0.86625 0.57519 0.69016

0.95599 0.94296 0.87148 0.88168 0.85957 0.87952 0.88102

Figure 5. Variation of BIP kij for gases/n-alkanes with carbon number (Cn) of n-alkanes at different temperatures (black for CH4, red for C2H6, blue for C2H4, magenta for H2, olive for CO, orange for N2, and navy for CO2).

Figure 6. Solubility of gases in n-eicosane as a function of pressure at different temperatures (black for 373 K, red for 473 K and blue for 573 K). Symbols are experimental data110,124 and lines (solid for temperature-dependent kij and dashed for temperature-independent kij) results by PCSAFT.

imental data. More results for the other gases (C2H6, C2H4, N2, H2, and CO2) can be found in Figure S42. The goodness of fit is improved for all the cases with temperature-dependent BIPs, although a reasonable fit is obtained with temperatureindependent ones. In particular, as shown in Figure 6a, model predictions with a temperature-independent kij are qualitatively incorrect: the solubility of CH4 at 573 K is experimentally observed to be lower than that at 373 K, whereas the model with a temperature-independent kij gives the opposite result. To put it into perspective, the PC-SAFT EoS parametrized in this study and the two popular benchmark models (namely, PSRK and SRK/MHV2) were compared and evaluated using

perspective, since the parametrization of a monotonic function is easier and requires less experimental data. It should be stressed that BIPs are often used to compensate or correct for the deficiencies and limitations of thermodynamic models. The superiority of our correlations is rooted in the sound theoretical basis of the PC-SAFT EoS. Data regression with a temperature-independent kij was performed to justify the temperature dependence. When kij is independent of temperature, kijT is set to zero but ai1, ai2, and ai3 are allowed to be adjusted freely in data regression. Figure 6 presents the calculated solubilities of gases (namely, CH4 and CO) in n-eicosane with a temperature-dependent kij and those with a temperature-independent one together with exper12352


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Figure 7. AARD% in bubble-point pressure for solubility of CH4 and C2H6 in n-alkanes.

Figure 8. Vapor−liquid equilibrium for binary mixtures at different temperatures. Comparison of PC-SAFT (solid lines), PSRK (dashed lines), and SRK/MHV2 (dotted lines) to experimental data120,122,125,126 (symbols).

intentionally excluded for parameter estimation as mentioned earlier, such a distinction is not reflected in the distribution of the deviations of the PC-SAFT EoS. The evenly distributed deviations over the entire spectrum of n-alkanes imply that the PC-SAFT EoS parametrized with the fitted functions can perform well not only within the region where the parameters were estimated directly by fitting experimental data but also outside the region where the BIPs were given with the use of the fitted functions. The comparison and evaluation were also made among different parametrization strategies for the PC-

experimental data. The deviations of the calculated solubilities of CH4 and C2H6 in n-alkanes from experimental data are shown in Figure 7 for the three models. The results for the other gases (namely, C2H4, CO2, CO, H2, and N2) can be found in Figure S43. It was found that the PC-SAFT EoS is significantly better than the other two models and yields quantitatively accurate results in all cases. The SRK/MHV2 model is the worst of all. Although much better than the SRK/ MHV2 model, the PSRK model also delivers rather poor results. Although the data for lighter or heavier n-alkanes were 12353


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Figure 9. Solubility of gases in different n-alkanes (black for n-eicosane and red for n-hexatriacontane) as a function of temperature at 50 bar. Comparison of PC-SAFT (solid lines), PSRK (dashed lines), and SRK/MHV2 (dotted lines) to experiments data107,110,124(symbols).

Figure 10. Solubility of CH4 in n-alkanes as a function of the carbon number (Cn) of n-alkanes at different temperatures and pressures. Comparison of PC-SAFT (solid lines), PSRK (dashed lines), and SRK/MHV2 (dotted lines) to experimental data101,107,108,110,129 (symbols).

are equally accurate for the solubility of CH4 in n-alkanes. The former is noticeably better than the latter in the case of the solubility of C2H6 in n-alkanes. Phase diagrams are shown in Figure 8 for binary mixtures of lighter n-alkanes and different gases. It was found that all three models yield accurate results except in the region close to the critical point for some cases. The same phenomenon was observed for other gases as shown in Figure S44. It is well

SAFT EoS, namely, GC-PC-SAFT, the one proposed in this study, and the four-parameter correlation for the sPC-SAFT EoS. As shown in Figure 7, the results by the three parametrization strategies are all acceptable. It was shown that GC-PC-SAFT gives the largest deviations. This is because the BIPs by GC-PC-SAFT are independent of temperatures. The PC-SAFT EoS parametrized in this study and the sPCSAFT EoS parametrized with the four-parameter correlation 12354


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Figure 11. Phase envelopes for synthetic gas condensates. Comparison of PC-SAFT (solid lines) and PPR78 (dashed lines) to experimental data130 (symbols).

curves for binary mixtures of different gases and lighter alkanes, particularly up to the critical point in many cases. Despite excellent correlations/predictions by the other two models on the dew-point branches of saturation curves as well, the same statement that applies to the PC-SAFT EoS may not be applicable to the PSRK and SRK/MHV2 models, since it is vague whether dew points were used to parametrize the GC method that is used to obtain the BIPs for the two models. As discussed by Poling et al.,128 gas solubility decreases with an increasing temperature and then increases for most gases. However, there are a few exceptions that gas solubility always increases with an increasing temperature. Figure 9 presents measured and calculated solubilities at different temperatures. As temperature changes, the solubility of CH4 changes little with a shallow minimum and that of CO2 changes much more appreciably with a shallow minimum as well. In contrast to the cases for CH4 and CO2, the solubility of H2 increases with an increasing temperature. It was found that the PC-SAFT EoS is quantitatively accurate in all cases and captures all of the minima. Though predicting the qualitative trends correctly, the PSRK model gives large deviations and fails to capture the minima for the solubilities of CH 4 and CO 2 in nhexatriacontane. The SRK/MHV2 model is the worst of all without any captured minimum and predicts the incorrect trend that the solubility of H2 decreases with an increasing temperature. The results for C2H6 and C2H4 are similar to those for CO2 and are shown in Figure S45. The results for

Figure 12. AARD% in bubble-point pressure of model predictions from experimental measurements130 for synthetic gas condensates (SGC). Compositions of the SGCs are given in Figure 11.

known that the PC-SAFT EoS is less reliable in the region close to the critical point.88,127 Special treatment is needed to remedy this defect.86−88,127 This topic is out of the scope of this study and thus is not further discussed. It should be noted that only bubble points were used in parametrizing the fitted functions in this study, and kij is available by extrapolation for lighter alkanes. It is remarkable that the PC-SAFT EoS can accurately reproduce the dew-point branches of saturation 12355


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Figure 13. Phase envelopes for synthetic gas condensates. Comparison of PC-SAFT (solid lines) and UMR-PRU (dashed lines) to experiments data131,132 (symbols).

deviations of model predictions from the experimental data. The deviations were reported for UMR-PRU and LCVM elsewhere.51,133 Clearly, the PC-SAFT EoS is better than the two EoSs. Novak et al.56 compared model predictions by the UMR-PRU model and the PC-SAFT EoS (with the use of a set of BIPs different from the one proposed in this study) to experimental data of synthetic gas condensates with high molar fractions of methane. It was concluded that the UMR-PRU model is superior and the PC-SAFT EoS consistently overpredicts the bubble-point pressures with large deviations. The same calculations were carried out using the PC-SAFT EoS but with the use of the BIPs proposed in this study. Figure 13 presents model predictions by the PC-SAFT EoS parametrized in this study and the UMR-PRU model. The results by the UMR-PRU model are from Figure 2 in another paper.56 It was shown that the PC-SAFT EoS still overpredicts the bubble-point pressures but with smaller deviations. Furthermore, the deviations of the two models are about the same.

CO and N2 are similar to those for H2 and are shown in Figure S46. Figure 10 compares the calculated solubility of CH4 in nalkanes by the three models to experimental data. The results for H2 can be found in Figure S47. It was found that the PCSAFT EoS is quantitatively accurate and yields reasonable extrapolation for lighter or heavier n-alkanes. The other two models generally deviate significantly from experimental measurements. Both the PC-SAFT EoS and the PSRK model predict a linear relationship between the gas solubility and the carbon number of n-alkanes, whereas the SRK/MHV2 model fails to do so. Interestingly, as shown in Figures 7, 8, and 10, the PSRK and SRK/MHV2 models perform much better for lighter n-alkanes than for heavier ones. The PC-SAFT EoS was validated against experimental data of synthetic gas condensates to further test its predictive power for multicomponent mixtures. Similar calculations were also performed using other cubic EoSs (namely, LCVM, UMRPRU, and PPR78) with improved modeling capabilities for asymmetric mixtures elsewhere.31,51,133 Figure 11 compares the predictions by the PC-SAFT EoS and the PPR78 model to the experimental data. The results by the PPR78 model are from Figure 14 in another paper.31 It was demonstrated that both models are sufficiently accurate, though the PPR78 model predicts a larger two-phase region at high temperatures. Bear in mind that the λ parameters required by the PPR78 model were fitted to the experimental data used in the validations. Therefore, it is fair to conclude that the PC-SAFT EoS performs better in these test cases. Figure 12 summarizes the

4. CONCLUSIONS Following its success in parametrizing the BIPs for water, hydrocarbon, alcohols, and many other oxygenates for the PCSAFT EoS, our 3-step parametrization strategy was extended to the area of gas solubility. The BIPs obtained directly by data fitting are well behaved, paving the way for easy parametrization for different gases and the entire homologous series of n-alkanes. It was demonstrated that the fitted 12356



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ACKNOWLEDGMENTS The authors acknowledge financial support from the National Key Research and Development Program of China (No. 2017YFB0602402).

functions can provide reliable BIPs outside the region where experimental data were used for parametrization. Data screening is inherently an integrated part of our 3-step parametrization strategy and was performed to ensure the accuracy and consistency of the data used. Our 3-step parametrization strategy is expected to work well for other types of fluids and will be extended to more areas. In addition, the BIP correlations proposed in this study and those developed for cubic EoSs were compared. The PC-SAFT correlations are monotonically increasing functions of the carbon number of n-alkanes and thus easy to parametrize. Such a phenomenon is attributed to the fact that the difference in molecular size increases with increasing carbon number, leading to a rise in nonideality and consequently an increasing BIP value, whereas there exist the maximum and/or minimum of BIP correlations developed for cubic EoSs, which is in contradiction with the monotonically increasing nonideality. Furthermore, the parametrization of such BIP correlations requires more experimental data, which is unfavorable from a practical perspective. The PC-SAFT EoS was compared to the two popular benchmark models (namely, PSRK and SRK/MHV2) to put it into perspective. It was found that the PC-SAFT EoS yields much better results in almost all of the cases, whereas the two models give significant deviations despite satisfactory results for lighter n-alkanes in a limited number of cases. The PCSAFT EoS can accurately predict a linear relationship between the gas solubility and the carbon number of n-alkanes and two types of the temperature dependence of gas solubility: (1) gas solubility decreases and then increases with an increasing temperature; (2) gas solubility always increases with an increasing temperature. The PC-SAFT EoS and other cubic EoSs (namely, LCVM, UMR-PRU, and PPR78) with improved accuracy in modeling of asymmetric mixtures compared to the above benchmark models were validated against experimental data of synthetic gas condensates. It was found that the PC-SAFT EoS delivers more satisfactory results than the other models. From a practical perspective, a strong predictive power and easy parametrization makes the PCSAFT EoS an ideal substitute for its classical predecessors.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b01383.

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Article

Detailed results about data screening; vapor−liquid equilibrium about gases/n-alkanes; AARD% in pressure for solubility of gases (C2H4, H2, CO, N2, and CO2) in n-alkanes; variation of solubility of C2H6, C2H4, CO, and H2 in n-alkanes with temperature; variation of solubility of H2 in n-alkanes with the carbon number of n-alkanes (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86 01060684022. Fax: +86 010-60684023. ORCID

Gang Wang: 0000-0003-1222-2893 Notes

The authors declare no competing financial interest. 12357


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