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Modeling Water Containing Systems with the Simplified PC-SAFT and CPA Equations of State Xiaodong Liang,† Ioannis Tsivintzelis,†,‡ and Georgios M. Kontogeorgis*,† †

Center for Energy Resources Engineering (CERE), Department of Chemical and Biochemical Engineering, Technical University of Denmark, 2800 Kongens Lyngby, Denmark ‡ Department of Chemical Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece S Supporting Information *

ABSTRACT: Numerous studies have been presented for modeling of water containing systems with the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EOS), and more than 20 water parameter sets have been published with emphasis on different applications. In this work, eight of these sets and new estimated parameters with different association schemes are systematically compared on describing properties of pure water, the liquid−liquid equilibria (LLE) of water with hydrocarbons, and the vapor−liquid (VLE) and/or vapor−liquid−liquid equilibria (VLLE) of water with 1-alcohols. An interactive procedure is further proposed for including the LLE of water with hydrocarbons into the pure fluid parameter estimation. The results show that it is possible for PC-SAFT to give an accurate description of the LLE of water and hydrocarbons while retaining satisfactory accuracy for both vapor pressure and saturated liquid density of water. For the aforementioned aqueous systems, the PC-SAFT correlations using the newly developed parameters are compared with the corresponding correlations of the cubic plus association EOS. The two models show comparable results for phase equilibria, and both of them fail to describe second-order derivative properties of water, i.e., residual isochoric heat capacity and speed of sound. The ability of the models to predict the monomer (free site) fractions of saturated pure water is investigated and discussed from various aspects. The results suggest that more experimental or theoretical studies are needed.

1. INTRODUCTION

vapor pressure and saturated liquid density during parameter estimation. In the work of extending PC-SAFT to associating systems, Gross and Sadowski8 published a 2B parameter set for water along with other associating fluids, i.e., alcohols, amines and one acid. To have a better description of experimental data in a narrow temperature range, Cameretti et al.17 refitted the pure component parameters using also the 2B scheme. Later, Cameretti et al.18 proposed to use a temperature dependent segment diameter to obtain an excellent description of the density of water, including the maximum of density of saturated water versus temperature. This later water parameter set has been recently applied for biological systems.19−21 To find an appropriate association scheme for water, Kleiner22 fitted the pure component parameters using the 3B and 4C schemes as well, which were tested for the binary systems of water with different hydrocarbons along with the 2B parameters from Gross and Sadowski.8 It was found that the mutual solubility of water and hydrocarbons could be satisfactorily described only with the 4C scheme. To investigate the performance of PC-SAFT for describing spectroscopy data, seven 4C parameter sets with gradually fixed segment number (m = 2.00−3.50) were proposed by von Solms et al.,23 and they are compared with the 2B parameters of Gross and Sadowski.8 The 4C scheme was found to be more

Modeling of water containing systems with equations of state (EOS) is vital for both research and industrial applications, and extensive studies have been conducted in the past few decades.1 Water is, in many respects, a unique molecule and it is a challenge for any EOS to simultaneously model the physical properties and phase equilibria of water containing systems with satisfactory accuracy.1 Hydrogen-bonding and the associated tetrahedral structure are considered to be the dominant factors for the unusual and complex behavior of water containing systems.2 The association models, which explicitly account for hydrogen bonding interactions, show advantages over the classical ones, especially on the predictive aspect.1 The statistical associating fluid theory (SAFT)3−5 and cubic plus association (CPA)6 EOS are two of most successful and widely used models. In the past three decades, numerous SAFT variants have been proposed, among which the perturbed-chain SAFT (PC-SAFT)7,8 has gained widespread acceptance with great successes in the fields of petroleum, chemicals, biochemicals, electrolytes, polymers, and so on.1,9 Water has been modeled as a two-site (2B), three-site (3B) or four-site (4C) molecule within the SAFT framework. Within the SAFT framework, the water containing systems have been studied by many researchers,1,10−15 and especially, for PCSAFT, more than 20 sets of pure component parameters have been published with emphasis on different applications. This is because, as clearly demonstrated by Clark et al.,16 the five pure component parameters have a degeneracy when only fitted to © XXXX American Chemical Society

Received: May 16, 2014 Revised: August 17, 2014 Accepted: August 21, 2014

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dx.doi.org/10.1021/ie501993y | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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appropriate by using the spectroscopy data as a guide in finding suitable model parameters. Using physically justified values for the association energy and the dispersion energy, Grenner et al.24 proposed a 4C parameter set with a fixed segment number (m = 1.5). They showed that in this way good results are obtained for the phase equilibria of water containing systems.25,26 To comment on parameter estimation of water, Grenner et al.27 published a new 4C parameter set by fitting model predictions to vapor pressure, liquid density and enthalpy of vaporization data. They pointed out that the use of mixture data, especially for mixtures of associating and inert compounds, is possibly the ultimate test for obtaining optimum parameters. Kontogeorgis et al.28 published three parameter sets for 2B, 3B and 4C directly using monomer fraction in the parameter estimation, and it was found that the 4C scheme is the best choice for water. Meanwhile, it is still an open question whether the phase behavior calculations could be improved by including monomer fraction data in the parameter estimation.28,29 Aparicio-Martı ́nez et al.30 adjusted the parameters to vapor pressure and saturated liquid data with the consideration of hydrogen-bonding energy for the association schemes 2B, 3B and 4C. They found that the 3B scheme seems to be the most appropriate one from the structural point of view. They also commented that the experimental spectroscopic data seems to be helpful for selecting the most adequate association scheme. Then the 3B parameters were further rescaled to match the critical points with the association parameters retained, and with this new parameter set, promising results were shown for modeling aqueous mixtures with CO2, N2 and n-alkanes. Diamantonis and Economou31 published a 4C parameter set that shows overall satisfactory results for the physical properties of pure water, and has found applications in carbon capture and sequestration (CCS) related systems. The CPA equation of state1,6 is a valuable EOS in modeling aqueous systems. It can predict satisfactorily multicomponent, multiphase equilibria for mixtures containing water, hydrocarbons and alcohols or glycols.1,6,32−34 More specifically, CPA has been previously shown to perform very well in correlating with one adjustable parameter (per binary) LLE for water− alkanes35 and with two adjustable parameters (as cross association is accounted for) LLE and VLE for water−aromatic mixtures.36 A characteristic application of the model that reveals its predictive capabilities is the LLE of aqueous multicomponent systems with glycols and hydrocarbons.34 Since CPA is a well established model for water mixtures, it is used in this work for comparing the results obtained with the PC-SAFT model. The purposes of this work are (1) to investigate the performance of the published parameters of PC-SAFT on the properties of pure water and phase equilibria of binary systems of water with hydrocarbons and with 1-alcohols; (2) to compare for PC-SAFT the two association schemes 2B and 4C on the properties of water and phase equilibria of water with nhexane; (3) to propose an interactive optimization procedure, which is suitable for association models, to estimate the pure component parameters by taking LLE of water with hydrocarbons into account; (4) to investigate the performance of PCSAFT with the new developed parameters, and to compare the results with CPA; (5) to investigate the free site fractions of saturated pure water from different aspects.

2. MODELS 2.1. PC-SAFT EOS. The PC-SAFT EOS was developed by Gross and Sadowski7 by extending the perturbation theory of Barker and Henderson37 to a hard-chain reference. The reduced residual Helmholtz free energy for mixtures containing associating fluids in PC-SAFT can be formulated as a r = (a hs + achain) + adisp + aassoc hs

(1)

chain

where a and a are the contributions from hard sphere segment−segment interaction and chain formation, of which the summation is the reference to build the dispersion force adisp. The term aassoc represents the contributions of association forces of sites, which can be expressed as aassoc =

∑ xi[∑ (ln X A i

i

− X A i /2) + Mi /2] (2)

Ai

where Mi is the association site number of molecule i, and XAi is the fraction of molecules i not bonded at site A, given by X A i = [1 +

∑ ∑ ρj X B ΔA B ]−1 j

j

i j

Bj

(3)

In this work, the simplified PC-SAFT version proposed by von Solms et al.38 will be used. It is not a new EOS, rather a simplified version in terms of mixing rules of the original PCSAFT EOS, which aims to simplify and reduce the computational time of the PC-SAFT EOS. The original and the simplified PC-SAFT have the same pure component parameters, while a simple conversion is needed for the association volume parameter due to a slightly different expression for the association strength employed in the simplified PC-SAFT: ⎤ ⎡ ⎛ ε A iBj ⎞ π ΔA iBj = Navσij3g hs κ A iBj⎢exp⎜ ⎟ − 1⎥ 6 ⎦ ⎣ ⎝ kT ⎠

(4)

In other words, the association volume κAiBj from the original PC-SAFT should be divided π/6 when used into the simplified PC-SAFT. The following combining rules are used for crossassociating mixtures: ε A iBj = κ A iBj =

1 A iBi (ε + ε AjBj) 2 κ A iBiκ AjBj

(5) (6)

More details can be found in the original literature7,8,38 or the book of Kontogeorgis and Folas.1 2.2. CPA EOS. The CPA EOS, proposed by Kontogeorgis et al.,6 is a combination of the SRK (or other cubic) EOS, widely used in the petroleum industry (e.g., for mixtures with gases and hydrocarbons), and of the association term of the SAFT type models. The CPA model reduces to SRK in the absence of hydrogen bonding compounds (water, alcohols, acids, etc.), thus achieving a balance between accuracy and simplicity and gaining acceptance in the oil, gas and chemical industries. The CPA EOS can be expressed for mixtures in terms of pressure P as P=

∂ln g ⎞ a(T ) RT 1 RT ⎛ − − ⎜1 + ρ ⎟ ∑ xi Vm − b Vm(Vm + b) 2 Vm ⎝ ∂ρ ⎠ i

∑ (1 − X A ) i

Ai

B

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Table 1. Water Pure Component Parameters with the Simplified PC-SAFT EOS

a

sets

m

σ (Å)

ε/k (K)

εHB/k (K)

κHB

scheme

T range (K)

ref.

W2B_1 W2B_2 W3B_1 W3B_C W4C_1 W4C_2 W4C_3 W4C_4 TWa

1.0656 1.3112 1.7960 2.3753 1.5 2.1945 3.0 1.5725 2.0

3.0007 2.7613 2.4697 2.5609 2.6273 2.2290 2.0135 2.6270 2.3449

366.51 372.37 327.62 275.81 180.30 141.66 182.92 291.13 171.67

2501.00 2123.10 1558.40 1558.40 1804.22 1804.17 1259.00 1334.20 1704.06

0.06659 0.09356 0.1304 0.1304 0.1800 0.3894 0.8188 0.1420 0.3048

2B 2B 3B 3B 4C 4C 4C 4C 4C

273−647 273−634 273−634 273−634 324−583 275−640 275−640 273−634 280−620

8 30 30 30 24 31 23 30

TW denotes This Work, i.e., the simplified PC-SAFT with the new proposed water parameters.

where ρ is the molar density (ρ = 1/Vm). More details of CPA can be found in the original article6 and in a recent monograph of Kontogeorgis and Folas.1 2.3. Second-Order Derivative Properties. The secondorder derivative properties considered in this work are residual isochoric and isobaric heat capacities, and speed of sound. They can be calculated by the following equations (eqs 8−10). The ideal gas heat capacity is also needed for speed of sound calculation, and it is taken from the DIPPR database.39 ⎛ ∂ 2a r ⎞ C Vr = −T ⎜ 2 ⎟ ⎝ ∂T ⎠V , n

(8)

⎛ ∂ 2a r ⎞ ⎛ ∂P ⎞ ⎛ ∂V ⎞ C Pr = −T ⎜ 2 ⎟ + T ⎜ ⎟ ⎜ ⎟ − R ⎝ ∂T ⎠V , n⎝ ∂T ⎠ P , n ⎝ ∂T ⎠V , n

(9)

u=

−

V 2 CP ⎛⎜ ∂P ⎞⎟ M w CV ⎝ ∂V ⎠T , n

3. RESULTS AND DISCUSSION 3.1. Comparisons for Literature Parameters. Eight of the PC-SAFT parameter sets for pure water reviewed above will be systematically compared in this work. The parameters are listed in Table 1, and the association volume has been converted to those used in the simplified PC-SAFT EOS.24,38 It can be readily seen that the five pure component parameters cover wide ranges, e.g., the segment number from 1 to 3, the dispersion energy from 140 to 372 K, and the association energy from 1259 to 2501 K. These parameter sets are selected because, besides the wide ranges, they are reported from different groups, and are optimized from different criteria, e.g., best pure properties or monomer fractions. Also, we can reproduce the published deviations for vapor pressure and liquid density. Because there are eight different parameter sets, we name them in a common way for easy reference in the following discussions. The parameter sets are named by starting with the letter “W” (water), then followed with the association scheme and ending with a number or letter “C” (critical). For example, the parameter set from Gross and Sadowski8 is defined as W2B_1 (the first 2B scheme parameter set), and the parameter set from Aparicio-Martıń ez et al.30 with nonassociating parameters matched to the critical point is named as W3B_C. The names for other sets can be found in Table 1. First, the physical properties of pure water are calculated using these eight parameter sets with the simplified PC-SAFT

(10)

2.4. Deviations. The percentage average absolute deviation will be used to evaluate the quantitative performance in this work, defined as %AAD(Ω) =

1 N

N

∑ i=1

Ωicalc − 1 × 100% Ωiexp

(11)

where Ω is vapor pressure, liquid molar volume, speed of sound, residual isochoric and isobaric heat capacities, or composition in LLE. The following equation is used for temperature: %ΔT =

1 N

Table 2. %AADs for Pure Water Properties Using the Parameters of Table 1a %AAD of different properties against NIST data in 280−620 K40

N

∑ |Ticalc − Texp i | i=1

(12)

And for vapor composition, it is %Δy =

1 N

N

| × 100% ∑ |y icalc − y exp i i=1

(13)

The percentage average relative deviation is also used in certain cases, and it gives the information to show if the deviations are positive or negative, and if the modeling results match well the experimental data qualitatively. It is defined as ⎛ Ωcalc ⎞ 1 %ARD(Ω) = ∑ ⎜⎜ iexp − 1⎟⎟ × 100% N i=1 ⎝ Ωi ⎠

sets

vap. pres.

density

res. CV

res. CP

speed of sound

dP/ dV

sum

W2B_1 W2B_2 W3B_1 W3B_C W4C_1 W4C_2 W4C_3 W4C_4 TW CPA

2.30 0.82 0.71 3.92 1.85 2.03 0.30 0.70 1.46 0.75

6.22 4.28 4.02 61.1 3.50 0.86 1.41 4.85 2.14 1.16

22.8 19.8 29.1 18.4 20.1 23.8 34.5 18.9 21.8 15.1

25.9 14.5 14.2 26.4 29.7 20.1 9.33 13.8 20.6 11.0

41.0 55.3 66.9 55.7 19.1 7.20 45.6 63.3 21.1 9.05

94.4 137 156 29.0 61.1 16.9 89.5 176 49.7 18.4

193 232 271 195 135 70.9 181 278 117 55.5

a

Note: (1) The bold and italic values are the smallest %AAD. (2) The italic values are slightly worse than the best ones, but they are quite satisfactory. (3) The underline values are the largest %AAD. (4) If the result of CPA is best, it is also marked.

N

(14) C



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EOS. The %AADs of the predictions against from the NIST data,40 in the temperature range of 280−620K, are listed in Table 2. Almost all parameter sets give quite reasonable and similar deviations for vapor pressure, whereas they show different deviations for the saturated liquid density. The set W3B_C, as expected, presents the worst deviations for these two properties, because the parameters are forced to match the critical properties. Most of the sets fail to represent the secondorder derivative properties within 10%. It is shown in Figure 1a that none of the parameter sets satisfactorily describe the residual isochoric heat capacity from both the quantitative and qualitative points of view. This property is directly related to the derivatives of Helmholtz free energy with respect to temperature as shown in eq 8. Table 2 shows that the parameter set W4C_2 presents the smallest deviation for the speed of sound. However, as shown in Figure 1b, all sets fail to capture the curvature of speed of sound against temperature, especially the maximum around 350 K. Both results of residual isochoric heat capacity and speed of sound indicate that the temperature dependences of the model must be improved for water. The parameter sets with 4C association scheme tend to provide an overall better description of these properties from a quantitative point of view, as seen in Table 2. As discussed by von Solms et al.,23 the monomer fraction of pure associating fluids can be calculated by the following equation.

where X1 is the monomer fraction, XA is the free site fraction, S is the total site number. The calculated percentage monomer fractions using the eight parameter sets are plotted in Figure 2a. It can be seen that the predictions from the sets W4C_3 and W4C_4 are closest to the experimental data at the low and high temperature regions, respectively. The parameters with both 2B and 3B association schemes overpredict the monomer fractions. In the original article41 and a later publication,42 as discussed and verified by von Solms et al.,23 Luck assumed four sites on water to calculate the monomer fractions. So it might be unfair to compare the monomer fractions predicted from 2B or 3B schemes to the “experimental 4C data”. It is, however, possible to obtain the experimental free site fraction from monomer fraction by applying eq 15, whereas the free site fraction can be directly calculated from the association models using eq 3. This indicates that it is more straightforward to compare the free site fractions instead of monomer fractions if different association schemes are to be compared at the same conditions, so the free site fractions will be used hereafter in the discussions. As shown in Figure 2b, the two 2B parameter sets underpredict the free site fractions.

Figure 1. Experimental and calculated properties with PC-SAFT using different model parameters (a) residual isochoric heat capacity of saturated water and (b) speed of sound in saturated water. The experimental data are taken from NIST.40

Figure 2. Experimental and calculated with PC-SAFT (a) percentage monomer fraction and (b) free site fraction of saturated water. Black, blue and red lines are the results from 2B, 3B and 4C schemes, respectively. The original data are taken from Luck.41,42

X1 =

∏ XA A=1

D

(15)



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interaction parameter (kij) is fitted to the solubilities in both phases. The fitted kij values are sorted from smallest to largest, and plotted against the parameter sets in Figure 4. It can be concluded that (1) the behavior for the three binary systems is quite similar for all the parameter sets as shown by the %AAD in Table S2 (Supporting Information), and also indicated by the kij values in Figure 4. (2) The correlations of the solubilities of hydrocarbons in the water rich phase show a weak dependence on the parameters within the same association scheme, i.e., different parameters with the same association scheme have quite similar results; the minimum in the solubilities of hydrocarbons in water are not captured for any set. (3) The parameter sets with the 2B or 3B association schemes overpredict the solubility of water in the hydrocarbon rich phase. (4) The two parameter sets W4C_1 and W4C_2 are the only ones able to simultaneously describe the solubilities in both phases, and W4C_1 gives the best results (at the slight cost of the density prediction shown in Table 2). (5) The parameter set W4C_3, which has the best representation of vapor pressure and quite accurate description of liquid density for pure water, shows difficulties in simultaneously capturing the solubility in both phases of the binary mixtures. This is mainly because it significantly underpredicts the solubility of hydrocarbon in the water rich phase. (6) The parameter set W3B_C overpredicts the solubility of water in the hydrocarbon rich phase most. As seen from Table S2 (Supporting Information) and Figure 3, the two solubility lines move in the same direction, and the sign of kij value is determined by the solubility of hydrocarbon in the water rich phase for the parameters discussed above. Figure 3 also shows that the low solubility lines, i.e., the solubility of hydrocarbons in the water rich phase, have quite similar slopes for 2B and 3B association schemes, whereas they are significantly different from those of the 4C association scheme. This leads to large differences on the deviations of the solubility of hydrocarbons in the water rich phase for the 2B and 3B schemes, as listed in Table S2 (Supporting Information). On the basis of this fact, it can also be anticipated that quite different results might be obtained when the data in different temperature ranges are used to fit the kij values. The prediction and correlation of VLE or VLLE of water and 1-alcohols are reported in the Support Information, Tables S3 and S4. The VLE results are based on bubble point calculations, while the LLE results are based on isothermal−isobaric flash. Typical correlation results are presented in Figure 5a,b for the

The investigations on properties of pure water presented above show that the parameter sets with 4C scheme present better performance, but none of them seem to be clearly superior to the others. To have a more complete and clearer picture of their performance, it is also crucial to investigate the phase equilibria of water, e.g., with hydrocarbons and with alcohols. The binary systems of water with n-hexane, n-octane, cyclohexane, methanol, ethanol, 1-propanol, 1-butanol and 1pentanol are chosen in this study, because of the wide variations of molecular interaction strength and the variety of phase equilibrium types. In these systems, the VLE, LLE and VLLE types of phase equilibria are observed, and azeotrope points also appear in some cases. In order to conduct systematic comparisons, the pure component parameters of the other compounds are taken from the same authors, i.e., Gross and Sadowski,7,8 which are listed in the Supporting Information, Table S1, for easy reference. The prediction and correlation of LLE of binary systems of water with n-hexane, n-octane or cyclo-hexane43,44 are shown in the Support Information, Table S2, in which both %ARD and %AAD are reported for the mutual solubility of water and hydrocarbons. The %ARD is helpful to distinguish a positive or negative deviation, and give an intuitive idea about how good or bad the results are. Typical prediction and correlation results of the binary system of water with n-hexane are presented in Figure 3a,b, respectively. A temperature independent binary

Figure 3. Experimental and calculated with PC-SAFT mutual solubilities of water and n-hexane. (a) Model predictions, and (b) correlations with kij shown in the parentheses. The data are taken from Tsonopoulos et al.43

Figure 4. Binary interaction parameters kij of water−hydrocarbons for the investigated PC-SAFT parameter sets. E



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critical point and poor description of vapor pressure and liquid density, shows comparably good prediction results. In general, as shown in Table S3 (Supporting Information), the phase equilibria of water and 1-alcohols are easier to tune than the LLE of water and hydrocarbons, using either large or small kij values. The most obvious example is the parameter set W4C_3, as listed in Tables S3 and S4 (Supporting Information). This is also demonstrated in Figure 5a, in which relatively large kij values are used to correlate the VLE of water and methanol for the parameter sets that show poor predictions for this system. It is interesting to note that the parameter set W2B_1 presents small positive kij values, close to zero, for water with 1-propanol and with 1-pentanol, whereas all of other sets give negative ones. As reported in Tables S3 and S4 (Supporting Information), the parameters with 4C association schemes do perform better than those with 2B and 3B on correlating VLE of water with 1-alcohols if taking both pressure or temperature and vapor composition into account. As seen from Table S4 (Supporting Information), the two sets showing the best description for LLE of water with hydrocarbons, i.e., W4C_1 and W4C_2, have difficulties in matching the water rich phase in the LLE of water with 1butanol and with 1-pentanol, whereas W4C_3 gives the best results in balancing three phases with relatively large kij values. Even though the parameters with 2B and 3B show acceptable deviations for the 1-alcohols rich phases, they have difficulties in describing the phase equilibria from the qualitative point of view, as shown in Figure 5b. 3.2. 2B versus 4C. As shown in Table 1 and discussed above, the model parameters from different sources cover wide ranges, thus it is of interest to compare the association schemes in some systematic ways from both qualitative and quantitative points of view. We have decided to investigate the water parameters with fixed values of association energy. The assumed ranges 1000−2500 K and 1000−2000 K are, respectively, chosen for 2B and 4C for the association energy. The other four parameters are fitted to vapor pressure and saturated liquid density of water in the temperature range 280− 620 K, with 10 K as the interval, based on the data from NIST40 using the following objective function:

Figure 5. Experimental data and PC-SAFT correlations (kij shown in the parentheses) for the phase behavior of water with (a) methanol and with (b) 1-butanol. The experimental data are taken from Butler et al.,45 Griswold et al.,46 Boublik et al.47 and DECHEMA data series.48

binary systems of water with methanol45,46 and with 1butanol,47,48 respectively. In Tables S3 and S4 (Supporting Information), quite large deviations from the predictions are seen for some cases, for which incorrect phase behavior is predicted. These large deviations are still reported simply because the same calculation procedures are used for both predictions and correlations, which are good examples to show the deficiencies of the corresponding parameter sets and to verify the significances of correlations, i.e., using the binary interaction parameter kij. As reported in Table S3 (Supporting Information), the parameter sets W4C_1 and W3B_C give the best predictions for VLE of binary systems of water with methanol45,46 and with ethanol,49−51 respectively. The parameter sets W2B_1 and W4C_1 give best predictions for VLE of water with 1propanol.52 It can be seen from Table S4 (Supporting Information) that the parameter sets W2B_1 and W3B_C show comparable better predictions than other sets for the VLLE of water with 1-butanol,47,48 whereas W2B_1 results in the best predictions for the system of water with 1pentanol.48,53 It is surprising that the parameter sets with the best description of vapor pressure of water, i.e., W4C_3, show worst predictions for all of these systems. It is worth noting that the predictions of parameter sets W3B_1 and W4C_4, both from Aparicio-Martı ́nez et al.,30 are not satisfactory, whereas it is interesting to see that the set W3B_C, with rescaling to the

obj(m , σ , ε , κ HB) ⎛ Ωexp − Ωcalc(m, σ , ε , ε HB , κ HB) ⎞2 i ⎟⎟ = ∑ ⎜⎜ i Ωiexp ⎝ ⎠ i

(16)

where Ω is vapor pressure or saturated liquid density. The %AADs for vapor pressure and liquid density are plotted against association energy in Figure 6a. It can be seen that 2B and 4C have quite similar performances for vapor pressure from the qualitative point of view, but 2B gives smaller deviations in the range of reasonable values for the association energy (close to the experimental value around 1800 K). The parameters with 4C association scheme present smaller deviations for saturated liquid densities in the whole range. The %AADs for residual isochoric and isobaric heat capacities are presented against association energy in Figure 6b. It is revealed that 4C sets show smaller deviations for residual isochoric heat capacity, whereas 2B sets seem to give a better description of the residual isobaric heat capacity, again in the region close to the experimental value of the association energy (around 1800 K). The %AADs for speed of sound and dP/dV are presented in Figure 6c, which clearly shows that 4C sets present smaller deviations for F



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can be readily seen that the relationships of this property on association energy are quite similar for these two schemes. The calculated and experimental speed of sound in saturated water are plotted in the Support Information, Figure S2a,b. Figure S2 shows that the slope of the calculated speed of sound curve can be slightly changed for both association schemes by changing the parameters, but none of the sets can capture the curvature of speed of sound against temperature. This fact suggests that it is not feasible to put speed of sound directly in parameter estimation as the curvature change occurs in a wide temperature range. The free site fractions that are predicted using these five parameter sets are presented in Figure 7a,b for both association schemes. It can be seen that these two association schemes perform quite similarly for this property as well. The same trends as seen here for vapor pressure, speed of sound and free site fractions against association energy are also observed for other properties, e.g. liquid density, residual heat capacities. This observation reveals that it is hard to determine which association scheme is superior to the other one, in terms of describing the properties of pure water. The correlated LLE for water with n-hexane is shown in Figure 8 for these five parameter sets, for both 2B and 4C. It confirms that it is relatively easy to tune the solubility of hydrocarbons in the water rich phase for both association schemes, while with the 2B scheme some difficulties in describing the solubility of water in the hydrocarbon rich phase are observed without strongly deteriorating the

Figure 6. %AADs for vapor pressure (Pres), liquid density (LiqD), residual isochoric (Res. CV) and isobaric (Res. CP) heat capacities, speed of sound (SoS) and the derivative dP/dV (dP/dV) calculated with PC-SAFT using the 2B and 4C schemes.

both properties. As discussed in some previous works,54,55 speed of sound is dominated by the derivative property dP/dV when density is described well. To further study the relationship of water properties on association energy, five parameter sets are chosen for each association scheme. These selected parameter sets cover wide association energy ranges and have enough differences, e.g., larger than 200 K, to distinguish each other. The ratios of the calculated and experimental vapor pressure of water are presented in the Support Information, Figure S1a,b for these two association schemes. Though the best representation of vapor pressure locates at different association energy regions, it

Figure 7. Predicted with PC-SAFT and experimental free site fractions (a) 2B and (b) 4C. The label ε_ass = 2000 denotes the parameters with fixed association energy equal to 2000 K. The experimental data are taken from Luck.41,42 G



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where EHB is the hydrogen bonding energy and εHB is the association energy in perturbation theory based models. The experimental hydrogen bonding energy is reported as 3.7 kcal/mol from IR measurements,41,42 whereas Luck reported 3.4 ± 0.1 kcal/mol from the two-state theory.41 If the hydrogen bonding energy range 3.3−3.7 kcal/mol is taken, the corresponding association energy range will be 1660−1860 K by applying eq 17. Therefore, the parameters with association energy in the range 1660−1860 K will be investigated in this work. The five pure component parameters are obtained with the following objective function: ⎛ Ωexp − Ωcalc(m , σ , ε , ε HB , κ HB) ⎞2 i ⎟⎟ obj(σ , x1 , x 2) = ∑ ⎜⎜ i exp Ω ⎝ ⎠ i i (18)

where Ω is vapor pressure or saturated liquid density. The estimation procedure is as follows: with a given fixed association energy and another fixed parameter, the other three parameters are fitted to vapor pressure and liquid density. The segment diameter (σ) is adjustable in all cases, while the three parameters segment number (m), dispersion energy (ε) and association volume (κHB) are fixed sequentially, which means that the fitting parameters x1 and x2 could be the combinations {ε, κHB} if m is fixed, {m, κHB} if ε is fixed, or {m,ε } if κHB is fixed. The association energy is taken gradually from 1660 to 1860 K with an interval 20 K, whereas the ranges for the other three parameters are, respectively, m = [1.5, 3.5], ε = [120, 190] and κHB = [0.1, 0.6]. A quite similar procedure was adopted in the work of Clark et al.,16 in which the association energy and the dispersion energy were fixed in wide ranges. This approach makes the optimization procedure be of more or less global character in a rather manual way. The parameters from these three different combinations are presented in Figure 9. It can be seen that the parameters are consistent with each other. This indicates that a unique solution can be obtained for the three parameters to match the two

Figure 8. Mutual solubilities of water and n-hexane. Calculations with PC-SAFT using the (a) 2B and (b) 4C schemes. The label ε_ass = 2000 denotes the parameters with fixed association energy equal to 2000 K. The data are taken from Tsonopoulos et al.43

description of the other phase. The binary systems of water with normal hydrocarbons are perfect candidate systems to study the associating interactions of water, as the normal hydrocarbons are considered to be inert compounds. The solubility of water in the hydrocarbon rich phase is a few orders of magnitude higher than the solubility of hydrocarbon in the water rich phase, mainly due to the self-associating interactions of water. If we consider the phase equilibria of associating and inert mixtures to be the ultimate test for obtaining optimum parameters, there is no doubt that the 4C scheme is superior to the 2B scheme. 3.3. New Water Pure Component Parameters with PCSAFT. The two parameter sets W4C_1 and W4C_2 have almost the same association energy, but the other four parameters are quite different. They show comparable performance for the phase behaviors of binary systems investigated in this work. Inspired by this fact, following the analysis of association scheme above, we propose a procedure to take the LLE of water and hydrocarbons into account for the estimation of water pure component parameters with the PC-SAFT EOS. This procedure may also be suitable for other SAFT variants. According to the works of Gupta et al.56 and Pfohl et al.,57 the association energy can be approximately connected to the hydrogen bonding energy: EHB = NAε HB = Rε HB/k

Figure 9. Comparison of water parameters obtained using the procedure developed for PC-SAFT with three combinations for fixing some parameters. The parameter being fixed, besides the association energy, is shown in the parentheses, which is also denoted by the colors. Black, Red and Blue are used to mark the dispersion energy (ε/ k), segment number (m) and association volume (κ) being fixed, respectively. The association energy is fixed gradually from 1660 to 1860 K with an interval 20 K.

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properties, i.e., vapor pressure and saturated liquid density, when the other two model parameters are fixed. The scenario that both the association energy and association volume are fixed is preferable according to our experience. So the parameters are refitted with this scenario in the range of association energy from 1660 to 1860 K with an interval 10 K, and association volume from 0.1 to 0.6 with an interval 0.01. The final parameters are chosen for each given association energy manually based on the deviations of the solubilities of water in the hydrocarbon rich phase. The main criteria are (1) if it is possible to keep the deviations of the solubility of water in all three systems less than 10%, the parameter set is chosen with the smallest sum of the deviations of water with n-octane and with cyclohexane. (2) Otherwise, the parameter set with smaller sum of the deviations in either the binary systems of water with n-hexane and with cyclohexane, or the binary systems of water with n-octane and with cyclohexane, but the later one is given a higher priority, is chosen. This is because nhexane and cyclohexane both have six carbon numbers, and quite similar results are obtained for the systems of water with n-hexane and with n-octane, whereas the correlated “experimental” data of the system with n-octane is more reliable according to Mac̨ zyński et al.58 The %AADs for vapor pressure and saturated liquid density of pure water, and the %AAD for the solubilities of water in the n-hexane, n-octane and cyclohexane rich phases are presented in Figure 10. With this parameter estimation strategy, the % AADs for vapor pressure and saturated liquid density increase linearly with the association energy, whereas the changes of the %AAD for vapor pressure are smoother. It is also readily seen that the solubility of water in all three systems can be correlated with quite good accuracy, i.e., less than 15%, using either small or large kij values, in the whole investigated association energy range. It is noticed that the parameters with association energy below 1740 K can present the %AAD for water solubility in the hydrocarbon rich phases less than 10% and the %AADs for vapor pressure and saturated liquid density less than 2% and 3%, respectively. As presented in Figure 11, the kij values are relatively small for the parameters with association energy in the range of 1660−1740 K, and they increase linearly with the association energy. Moreover, very good linear correlations can

Figure 11. Binary interaction parameters of water−hydrocarbons using the water parameters obtained by the procedure developed for PC-SAFT.

be obtained between the other four parameters and association energy in this range, as shown in Figure 12. All these results indicate that it is hard to determine a unique parameter set if only based on the information available here. Even if the prediction of the mutual solubility of water and hydrocarbons is used as an extra constraint, it is still a difficult decision to make, since mixtures with different hydrocarbons (n-hexane, n-octane or cyclohexane) are described better with different parameter sets. As shown in Figure 11, the sets that present better predictions for mixtures of water with n-octane or n-hexane have much higher association energies (around 1700 K) than the sets that present better predictions for water with cyclohexane (less than 1660 K). As shown in Figure 11, the parameter sets with association energy around 1700 K present best predictions for the LLE of water with n-hexane and with n-octane, whereas the corresponding segment number is around 2. So without loss of generality, we have decided to assume the segment number m to be equal to 2, instead of assuming association energy to be equal to 1700 K. The association energy can be reversely calculated by the correlation of m, and then the other parameters can be calculated sequentially using the correlations shown in Figure 12. The pure component parameters (m = 2, σ = 2.3449 Å, ε = 171.67K, εHB = 1704.06 K, κHB = 0.3048) will

Figure 10. %AADs for the solubility of water in hydrocarbon rich phases, vapor pressure and liquid density against the association energy for parameters obtained using the procedure developed in this work for PC-SAFT.

Figure 12. Linear correlations for the PC-SAFT parameter trends against the association energy in the range of 1660−1740 K. I



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be used hereafter in this work. Again, here the association volume κHB is based on the simplified PC-SAFT EOS, and it should be multiplied by π/6 (thus κ = 0.1596) if it is used with the original PC-SAFT EOS. It needs to be pointed out that it would not be surprising that equally good LLE correlation results could be obtained using the parameter sets with association energy 1690, 1700 or 1710 K. The calculated deviations of the properties of water from this new water parameter set are also given in Table 2. 3.4. Comparison with CPA. The deviations of CPA EOS predictions for the properties of water are also reported in Table 2. The water parameters are from Kontogeorgis et al.59 With the new water parameters, the PC-SAFT EOS gives 1.45% and 2.12% deviations for vapor pressure and saturated liquid density, respectively, against from the NIST data,40 in the temperature range of 280−620 K, which are larger than those of CPA, i.e., 0.73% and 1.16%, respectively. They are satisfactory, however, compared to those from the PC-SAFT EOS with the literature available parameters, as shown in Table 2. The calculated and experimental residual isochoric and isobaric heat capacities are compared in the Support Information, Figure S3a,b. Even though CPA gives much smaller %AAD values, we cannot conclude that CPA performs better than PC-SAFT for these two properties. It can be seen that PC-SAFT captures the trends slightly better than CPA from the qualitative point of view. The calculated and experimental speed of sound in liquid water at saturated, isobaric and isothermal conditions are presented in the Support Information, Figure S4a,b,c, respectively. It is shown, on one hand, that both models have apparent difficulties in describing the temperature dependence of speed of sound, even though CPA gives much smaller quantitative deviations. On the other hand, PC-SAFT describes the pressure dependence of speed of sound at constant temperature quite well from a qualitative point of view, which can be demonstrated using a simple translation strategy, as shown in Figure S4c (Supporting Information). The whole calculated line could successfully match the experimental data, if a multiplying factor was used, which equals to the ratio of the experimental and the calculated speed of sound at atmospheric pressure, i.e., the starting point of the line. The prediction and correlation LLE results for water− hydrocarbon systems from the simplified PC-SAFT with the new parameters and CPA are also reported in Table S2 (Supporting Information). The results obtained with the new parameters proposed in this work are denoted as TW. Typical correlation results are shown in Figure 13a,b for the systems of water with n-octane and with cyclohexane, respectively. The two models apparently show quite satisfactory deviations for both phases for these systems. They give similar results for the systems of water with n-hexane and with n-octane, whereas PCSAFT seems to have a better description of mutual solubility of water and cyclohexane. The two models also show slightly different slopes of the solubility of hydrocarbons in the water rich phase. Compared to the results in Figure 3b, this indicates that the association term plays a more important role than the physical (nonassociation) terms in describing the solubility of hydrocarbons in the water rich phase. The prediction and correlation VLE and VLLE results for water with 1-alcohol systems are also presented in Tables S3 and S4 (Supporting Information), and typical correlation results are plotted in Figure 14a,b for the systems of water

Figure 13. Mutual solubilities of water with (a) n-octane and with (b) cyclohexane (kij shown in the parentheses). Experimental data are from Tsonopoulos et al.43,44

with ethanol and with 1-pentanol, respectively. In terms of prediction results, on one hand, CPA shows much better accuracy than the simplified PC-SAFT with the new parameters for all systems, and with other parameters for most of the systems as well. On the other hand, in terms of temperature or pressure in VLE correlations, PC-SAFT shows better results for the systems of water with ethanol, 1-propanol or 1-butanol, whereas CPA shows better performance for the systems of water with methanol and 1-pentanol. PC-SAFT with the new parameters shows, however, smaller deviations for vapor composition for all systems. CPA results in better LLE correlations of water with 1-butanol on both phases, while the two models describe best one of the two sides of the binodal for the LLE of water with 1-pentanol. 3.5. Impacts of 1-Alcohols Model Parameters. The modeling of 1-alcohols containing systems presented above has been conducted using the parameters from Gross and Sadowski.8 There are also some other parameters available for the 1-alcohols in the literature, which have shown good performance for some applications. To investigate if it is possible to have better description of water−alcohol systems, the same calculations are conducted for the 1-alcohols model parameters from other sources with the new proposed water parameters in this work. The best correlation is selected for each system, as reported in Tables S3 and S4 (Supporting Information) using the name TW_1, which denotes water parameters from this work (TW) and an extra one (1) alcohol parameter set from another source. The parameters of methanol and ethanol are from J



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Figure 15. Free site fractions of saturated water with PC-SAFT (using the new proposed water parameters) and CPA. The experimental data (o) are taken from Luck.41,42

water. The same behavior is also observed in the binary system of water with n-octane. The results presented above suggest that either the experimental free site fraction could be much smaller than what are currently being used, or the current association framework prevents them from a simultaneous satisfactory description of the free site fractions and LLE of water− hydrocarbon systems. It is also interesting to note that the trends of free site fractions are slightly different from below and above 450 K, as shown in Figure 16, based on the experimental data of Luck.41 If this is not within the experimental uncertainty, it could indicate that, to some extent, the water is not stabilized in one structure in wide temperature ranges. More systematic experimental investigations are much needed, based on all these discussions, especially due to the fact the LLE of water and hydrocarbons have been measured and critically evaluated by several groups.43,44,58 Tsivintzelis et al.29 also suggested that the data for the monomer fractions of methanol and ethanol from the same study of Luck41 also have to be validated from other groups. The free site fractions calculated from the three parameter sets with association energies of 1660, 1760 and 1860 K and the set W4C_1 are presented in Figure 17. It can be seen that these

Figure 14. Phase behavior of water with (a) ethanol at 343.15 K and with (b) 1-pentanol at 1 atm (kij shown in the parentheses). Experimental data from Pemberton et al.,51 Beregovykh et al.,53 and DECHEMA data series.48 The new proposed water parameters are used for PC-SAFT.

Liang et al.,55,60 for which speed of sound were used in the parameter estimation. The parameters of 1-propanol and 1pentanol are from Grenner et al.,61 but the “optimized” set is used for 1-propanol, whereas the “generalized” set is used for 1pentanol. The parameter set of 1-butanol is from Kontogeorgis et al.,62 where they successfully applied the simplified PC-SAFT into the complex polymer systems. It can be seen that the results with TW_1 are slightly better than those with TW, which is using the 1-alcohols parameters from Gross and Sadowski,8 in terms of correlations, whereas these 1-alcohol parameters are obtained in different ways. These results deliver an important message that systematic investigations are needed to be conducted on alcohols as well, because they are modeled using five pure component parameters with the SAFT models. 3.6. Final Comments on Free Site (Monomer) Fraction. The calculated and experimental monomer fractions are presented in Figure 15 for CPA and PC-SAFT with the new parameters. Similar results are obtained, as those show good performance for LLE of water with nonaromatic hydrocarbons, that the models predict smaller free site fraction. Comparing Figures 7 and 8, it can be seen that all five 2B parameter sets under- and overpredict the monomer fractions and the solubility of water in the n-hexane rich phase, respectively. However, the watershed happens around association energy 1700 K for the 4C parameter sets, from where the parameter sets underpredict both the free site fraction and the solubility of

Figure 16. Different trends of free site fraction of pure saturated water below and above 450 K. The experimental data are taken from Luck.41,42 K



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water with hydrocarbons, and is also more effective in obtaining better correlations of the phase equilibrium for aqueous 1alcohols mixtures. The results also reveal that the phase behavior of binary aqueous systems with 1-alcohols is easier to be described (using one adjustable parameter) than the phase behavior of water−hydrocarbon mixtures. Moreover, the solubility of hydrocarbons in the water rich phase is easier to be correlated than the solubility of water in the hydrocarbon rich phase. It is also necessary to point out that the 4C parameter sets, which present a better description of LLE for water with hydrocarbons, give larger deviations on the water rich phase for the LLE results of water with 1-butanol and with 1-pentanol than other parameter sets, whereas the 2B parameters have difficulties in describing the 1-alcohol rich phases from the qualitative point of view. This suggests that systematic investigation needs to be conducted on alcohols as well, since the alcohols, as associating fluids, are also described using five pure component parameters. The binary systems of water with hydrocarbons, without accounting for cross association (solvation), present a good way to investigate the effect of the self-association of water. This is because the hydrocarbons are normally considered to be inert compounds, and the solubility of water in the hydrocarbon rich phase is a few magnitudes higher than the solubility of hydrocarbon in the water rich phase due to the self-association interactions of water. An interactive optimization procedure is proposed to take LLE of water with hydrocarbons into account when estimating the water pure component parameters with the simplified PC-SAFT EOS. It is found that numerous parameter sets could give comparably good results in wide parameter ranges. A new parameter set is obtained with the segment number being fixed to 2, which coincidentally presents kij values close to 0.0 for the systems of water with n-hexane and with n-octane. The new parameters give 1.45% and 2.12% deviations for vapor pressure and saturated liquid density, respectively, in the temperature range of 280−620 K. The new parameter set shows accurate description for the mutual solubility of water and hydrocarbons, which is superior to the other parameter sets available in literature on this aspect. It shows quite satisfactory VLE correlations for the binary systems of water with 1-alcohols. Furthermore, it describes the 1alcohols rich phases with quite high accuracy, but it gives large deviations on the water rich phase, for the LLE of water with 1alchols, especially with 1-butanol. The CPA EOS shows, on one hand, overall best predictions for the properties of pure water, and the phase equlilibria of the systems investigated in this work in terms of deviations. Moreover CPA can give quite satisfactory description of the LLE of water with hydrocarbons and with 1-alcohols. On the other hand, CPA also has difficulties in describing the secondorder derivative properties, i.e., residual heat capacities and speed of sound. The significant deficiency of perturbation theory based models, i.e., CPA and PC-SAFT here, on residual isochoric heat capacity of and speed of sound in pure water indicates that the temperature dependency is not described well within the current frameworks. This suggests that it is not recommended to directly put these two properties in the parameter estimation for water. Finally, more systematically experimental and theoretical works are highly needed for measuring and explaining the free site fractions or monomer fractions of water, and their relationships with the hydrogen-bonding structure of liquid water.

Figure 17. Free site fraction of water with PC-SAFT using different parameter sets. The experimental data are from Luck.41,42 The parameter sets 1660 K, 1760 K and 1860 K are obtained in this work with the consideration of the LLE of water with hydrocarbons, and the parameter set W4C_1 is taken from Grenner et al.24

four parameter sets perform quite similarly from both the quantitative and qualitative points of view. This reveals that the predicted values for the free site fractions or monomer fractions are insensitive to the pure component parameters, when they are estimated under the same constraints, e.g., equally good description of mutual solubility of water and hydrocarbons in this work. Therefore, it would not be surprising that free site or monomer fractions could not provide much help to find a unique parameter set for associating fluids, especially when the experimental uncertainties are not reported.

4. CONCLUSIONS In this work, the performance of eight parameter sets from literature is investigated on properties of pure water, LLE of water with hydrocarbons, and VLE or VLLE of water with 1alcohols for the simplified PC-SAFT EOS. To have a more complete and clearer picture of which association scheme is a better choice for water, the pure component parameters are obtained for the 2B and 4C association schemes by fitting to vapor pressure and saturated liquid density with fixed association energies covering wide ranges. Then they are systematically studied for the properties of pure water and the LLE of water with n-hexane on both qualitative and quantitative aspects. The results show that it is hard to determine which scheme (2B or 4C) performs better if we compare them based only on the properties of pure water. This is because 2B tends to have smaller deviations for vapor pressure and residual isobaric heat capacity in the “experimentally” reasonable association energy ranges, whereas 4C shows smaller deviations for liquid density, residual isochoric heat capacity and speed of sound. The most important finding is that the two association schemes perform quite similarly from the qualitative point of view for all the properties investigated in this work. For neither scheme do we get parameters able to yield acceptable deviations for the residual isochoric heat capacity, nor could they capture the maximum of speed of sound in water against temperature. It is shown, however, that the association scheme 2B presents significant difficulties in describing the mutual solubility of water and hydrocarbons. Meanwhile, the association scheme 4C presents definitely better performance on phase equilibria of L



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ASSOCIATED CONTENT

S Supporting Information *

Ratio of correlated by PC-SAFT and experimental vapor pressure values (assuming the 2B and 4C schemes) against temperature (Figure S1); Predicted with PC-SAFT (using the 2B and 4C schemes) and experimental speed of sound data (Figure S2); Modeling results of PC-SAFT with the new proposed water parameters and CPA for the (a) residual isochoric heat capacity and (b) residual isobaric heat capacity (Figure S3); Experimental and calculated speed of sound in pure water with PC-SAFT using the new proposed water parameters and CPA (Figure S4); The pure component parameters of other compounds with the simplified PC-SAFT EOS (Table S1); %AAD (%ARD) for the mutual solubility of water and hydrocarbons with PC-SAFT and CPA (Table S2); %AAD for the VLE of water with methanol, ethanol or 1propanol (Table S3) and %AAD for the VLE and LLE of water with 1-butanol or 1-pentanol (Table S4). This material is available free of charge via the Internet at http://pubs.acs.org.

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xi = molar fraction ΔAiBj = Association strength of association site Ai and Bj εAiBj = mixing association energy of association site Ai and Bj κAiBj = mixing association volume of association site Ai and Bj ρ = molar density σij = mixing segment diameter of segment i and j

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*G. M. Kontogeorgis. Tel.: 045-45252859. Fax: 045-45882258. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge Prof. Nicolas von Solms and Prof. Michael L. Michelsen for many useful discussions. The Ph.D. project of Xiaodong Liang is funded by the Danish National Advanced Technology Foundation (DNATF) and the Department of Chemical and Biochemical Engineering, Technical University of Denmark, to whom the authors express their gratitude.

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NOMENCLATURE %AAD = percentage average absolute deviation %ARD = percentage average relative deviation ar = molar residual Helmholtz free energy assoc = association term of reduced residual Helmholtz free energy CrV = residual isochoric heat capacity CrP = residual isobaric heat capacity CPA = cubic plus association disp = dispersion term of reduced residual Helmholtz free energy EOS = equation(s) of state ghs = radial distribution function of hard sphere fluid hc = chain term of reduced residual Helmholtz free energy hs = hard sphere (segment) term of reduced residual Helmholtz free energy Mw = molecular weight Nav = Avogadro’s number P = pressure PC-SAFT = perturbed-chain statistical associating fluid theory R = gas constant SAFT = statistical associating fluid theory T = temperature u = speed of sound Vm = molar volume M



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