Article pubs.acs.org/IECR
An Improved Group Contribution Method for the Prediction of Second Virial Coefficients Giovanni Di Nicola,*,† Matteo Falone,† Mariano Pierantozzi,† and Roman Stryjek‡ †
DIISM, Università Politecnica delle Marche, Via Brecce Bianche 60131, Ancona, Italy Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, 01-224, Poland
‡
S Supporting Information *
ABSTRACT: Techniques most commonly used for the estimation of the second virial coefficients are based on the corresponding states principle. They are generally semiempirical correlating methods, and their validity is usually limited to nonpolar gases or to small polar molecules. In this paper, an estimation technique for the second virial coefficients based on a modified group contribution method is presented. This method can be applied to a wide range of organic compounds. In fact, the method depends on the knowledge of reduced temperature and acentric factor, and is based only on summing the products of the group contributions and their respective number of occurrences. Each group contribution is derived by analysis of the second virial coefficient data available in the literature.
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Tsonopoulos1 modified the Pitzer−Curl’s equation2 to improve its predictive capability. In particular, for nonpolar gases, he introduced the following terms:
INTRODUCTION The deviation of real gas PVT behavior from the ideal gas equation of state shows that the knowledge of the PVT relationship for real gases is needed. Many modifications of the perfect gas equation of state have been proposed to represent the PVT behavior of real gases, but the most satisfactory form regarding organic compounds at low and moderate pressures is the virial equation of state. The following equation expresses the deviation from the perfect gas equation as an infinite power series in V, the molar volume: PV B C C =1+ + 2 + 3 + ... RT V V V
fTS(0) (Tr) = fP − C(0) (Tr) − 0.000607Tr−8
(1) f TS (Tr) = 0.0637 + 0.0331Tr −2 − 0.423Tr −3 − 0.008Tr −8
(3)
He also introduced a corrective function to estimate the second virial coefficients of polar gases: fTS(2) (Tr) = a TTr −6 − bTTr −8
(4)
where Tr is the reduced temperature, and aT and bT are tabulated parameters related to the reduced dipole moment, μr, of the compound studied. The final form of the Tsonopoulos’ equation is
(1)
where B is the second virial coefficient, C is the third virial coefficient, and D is the fourth virial coefficient. Even when truncated at the second coefficient, the virial equation of state gives a remarkable estimate of the PVT relationship of real gases at low and moderate pressures (generally less than 1 MPa). Because of this, estimation of second virial coefficients is an important task in thermodynamics. In addition, virial coefficients are an important factor because they form a link between the microscopic and macroscopic points of view, experimental results, and the knowledge of molecular interactions. In fact, the second virial coefficient represents the deviation from perfection due to interactions between pairs of molecules, the third virial coefficient reflects the effects of interactions of molecular triplets, and so on. Many theoretical or semiempirical models have been proposed for second virial coefficients based on the theoretical interpretations of intermolecular interactions between molecules in the gaseous states. They are reasonably successful for simple molecules. To overcome the limitations of theoretical and semitheoretical models as applied to more complex molecules, many empirical models were proposed in the literature and they will be briefly described below. © 2014 American Chemical Society
(2)
BPc (0) = fB = f TS (Tr) + ωfTS(1) (Tr ) + fTS(2) (Tr) RT
(5)
where Pc is the critical pressure (Pa) and ω is the acentric factor. Vetere3 proposed a new version of the Pitzer−Curl’s correlation.2 To this equation he added a new factor, ωV, and an additional term, f(2)(Tr), in which
ωV =
Tb1.72 − 263 M
(6)
fV (2) (Tr) = 0.1042 − 0.2717Tr −1 − 0.2388Tr −2 − 0.0716Tr −3 − 0.0001502Tr −8 Received: Revised: Accepted: Published: 13804
(7)
June 10, 2014 August 18, 2014 August 18, 2014 August 18, 2014 dx.doi.org/10.1021/ie502334h | Ind. Eng. Chem. Res. 2014, 53, 13804−13809
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Figure 1. Histogram, data distribution, and statistical summary of the second virial coefficient experimental data.
⎛ RT ⎞ (b) f BL (Tr) = ⎜ C ⎟( −E′)Tr−m ⎝ PC ⎠
Tb and M in eq 6 are, respectively, the normal boiling point temperature and molar mass. He obtained a correlation that describes the temperature dependence of B for polar and nonpolar gases: BPc = fB = f P(0) (T ) + ωf P(1) (T ) + ω V f V(2) (Tr) −C r −C r RT
D′,E′, and m are adjustable parameters for polar compounds. Weber6 modified the Tsonopoulos’ correlation1 to better describe the temperature dependence of the second virial coefficients. He removed the last term in the Tsonopoulos’ eq 2, and defined f (1) TS (Tr) as follows:
(8)
O’Connell and Prausnitz4 adopted both the first and the second terms, f(0)(Tr) and f(1)(Tr), of the Pitzer−Curl’s correlation for nonpolar gases. For polar gases, they introduced two additional functions, fμ(μr,Tr) and fa(Tr), based on the extended theory of corresponding states, in order to consider the polarity of compounds and their self-association trend. Their equation is
(1) f TS (Tr) = 0.0637 + 0.0331Tr−2 − 0.423Tr−3
a T = −9·10−7μr2
3
(ln μr ) − 0.2525373(ln μr )
fa (Tr) = exp[6.6(0.7 − Tr)]
(9) (10)
The O’Connell−Prausnitz’s complete equation reads as follows: fB = f P(0) (T ) + ωHf P(1) (T ) + fμ (μr , Tr) + ηfa (Tr) −C r −C r (11)
where ωH is the acentric factor of the polar component’s homomorph and η is a constant of association for the compound. Black5 suggested a van der Waals-type equation of state: v=
RT ξa +b− P RT
ΔBi = ai +
(12)
where ξ expresses the effects of temperature and pressure on the molar cohesive energy. Starting from this equation of state, Black proposed a correlation for B: B=
RTC (a) (b) + f BL (Tr) + f BL (Tr) 8PC
bi c d e + i3 + i7 + i9 Tr Tr Tr Tr
(18)
For most groups, only the first four terms were required. The second virial coefficients for any organic compound can be calculated from these group contributions and critical temperatures, for reduced temperatures from 0.5 to 5, by summing up the products of the group contribution (primary and secondary) and their respective number of occurrences, n:
(13)
B=
where:
∑ niΔBi + ∑ (ni − 1)2 ΔBi pri
⎛ 27 ⎞⎛ RT ⎞ (a) f BL (Tr) = ⎜ ⎟⎜ C ⎟(0.396Tr−1 + 1.181Tr−2 ⎝ 64 ⎠⎝ PC ⎠ − 0.864Tr−3 + D′Tr−4)
(17)
In light of new data appearing in the open literature, especially for polar fluids such as haloalkanes, detailed analysis on second virial coefficients predictions by the Tsonopoulos and Weber equations were performed.8,9 Important modifications to the polar term were recently proposed,10 with a particular attention to associated fluids, such as alcohols, amines, water, and quantum fluids.11 A different approach to finding a correlation capable of estimating the second virial coefficients is based on the group contribution method. The basic principle of this method is that every compound can be divided into simpler subgroups and that these groups contribute to the property examined. McCann and Danner12 proposed an equation capable of predicting the second virial coefficients based on group additivity. Group contributions were derived by analyzing available second virial coefficient data, and are represented by the equation
+ 1/Tr[5.76977 − 6.181427(ln μr ) + 2.28327(ln μr )2 − 0.2649074(ln μr )3 ]
(16)
He also found a new method to calculate aT in the 7 Tsonopoulos’ function f (2) TS (Tr). Weber obtained good results introducing the aT function:
fμ (μr , Tr) = −5.023722 + 5.65807(ln μr ) − 2.133816 2
(15)
sec
(19)
The secondary group was defined only for the following groups: C−C2H2, C−C2F2, and Cb−F. In any case, the ΔBi term, (cm3/mol), was calculated with eq 18 both for the primary and the secondary group.
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Figure 2. Scatter plot of the complete data set of second virial coefficients vs reduced temperature.
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DEVELOPING THE NEW MODEL All estimation techniques generally need a large set of experimental data. In fact, through experimental data analysis, it is possible to find a function that describes the correlation between the second virial coefficients and the physical/ molecular properties. Therefore, a data bank was created by collecting experimental data on the second virial coefficients available in the literature.13,14 It includes 3441 experimental data of second virial coefficients and their respective experimental temperatures. Data were collected for 157 compounds belonging to 19 families. In Figure 1, a statistical survey of the data distribution of second virial coefficients is reported. From Figure 1, information on the data set distribution and on the presence of the outliers inside the data set are reported. Each datum lying outside the defined bounds can be considered as outlier. In our case, it is evident that the considered data are only out of the lower fence. In particular, 55 outliers were detected. All the experimental data by chemical family are shown in Figure 2, where the scatter plot of the second virial coefficients versus the reduced temperature is reported. In Figure 2, a common behavior of the second virial coefficients for all compounds present in the data bank is clearly evident. In particular, at low reduced temperatures, all values of B are widely negative, growing rapidly for reduced temperatures close to unity. Then, the second virial coefficients values slowly increase up to Boyle’s temperature. Information on all data in the database is presented in Table 1, grouped by families. For each group, the number of experimental data, the range of second virial coefficients, and the respective reduced temperature range are summarized. This paper presents an estimation technique based on a modified group contribution method for the second virial coefficients developed by McCann and Danner,12 following the Benson group method.15 In fact, the proposed method starts from the Benson group contribution method based on summing the products of the groups and their respective number of occurrences as in eq 19, replacing eq 18 with a new equation containing the reduced temperature and the acentric factor, ω, as follows: ⎛ ⎛ b c ⎞ e ⎞ ΔBi = ⎜ai + i2 + i4 ⎟ω + ⎜di + i8 ⎟ Tr ⎠ Tr Tr ⎠ ⎝ ⎝
Table 1. Summary of Experimental Data for Each Family name
no. of points
no. of compounds
maximum molecular mass
acetates
66
4
102.13
alcohols
179
10
102.17
25
3
72.10
alkanes
1121
16
114.22
alkenes
415
13
112.21
alkynes
62
4
amines
100
10
101.19
aromatics
500
13
186.055
cycloalkanes
100
6
200.03
cycloalkenes
5
1
68.12
dialkenes
40
4
68.12
epoxides
25
4
88.10
ethers
79
9
130.23
formates
56
4
102.13
haloalkanes
442
24
338.04
haloalkenes
22
2
118
9
100.16
mercaptans
28
9
116.22
sulfides
58
12
112.19
aldehydes
ketones
54.090
64.034
Tr range
B range (cm3/mol)
0.61 to 0.85 0.55 to 1.22 0.56 to 1.02 0.55 to 2.04 0.48 to 1.67 0.59 to 1.17 0.55 to 1.28 0.48 to 1.21 0.56 to 1.60 0.66 to 0.74 0.56 to 0.90 0.52 to 0.78 0.58 to 0.93 0.60 to 0.81 0.54 to 1.31 0.89 to 1.40 0.58 to 0.93 0.56 to 0.67 0.53 to 0.67
−2200 to −500 −3500 to −755 −2210 to −260 −2710 to −17.9 −2179 to −42.9 −1100 to −133 −2000 to −118 −2760 to −247 −2050 to −82 −820 to −630 −1513 to −260.9 −1430 to −375 −2650 to −306 −1738 to −440 −2320 to −106 −299 to −85.8 −2860 to −400 −1960 to −839 −2410 to −660
According to eq 20, the coefficients ai, bi, ci, di, and ei were calculated for each group by a nonlinear regression fitting procedure. Since the families of alkanes and aromatics were found to be the most numerous in terms of data, their groups were regressed separately. The obtained coefficients for alkanes
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Figure 3. Scatter plot of absolute deviations vs reduced temperature for all the families excluding alkanes and aromatics.
Figure 4. Scatter plot of absolute deviations vs reduced temperature for alkanes and aromatics.
Figure 5. Scatter plot of relative deviations vs reduced temperature for all the families excluding alkanes and aromatics.
and aromatics were then kept as fixed values during the regression of the complete set of data. As compared to the McCann and Danner approach, the present method, besides the new reduced temperature dependence and the acentric factor introduction as a parameter, considered a larger number
of fluids. In addition, four new groups were added, namely cyclopentene ring corrections, C−(CO)(C)3, Cb−(NI)(Cb)(H), and Cb−(NI)(Cb)(C). To optimize the coefficients, the Levenberg−Marquardt curve-fitting method was adopted.16 This method is a combination of two minimization methods: 13807
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Figure 6. Scatter plot of relative deviations vs reduced temperature for alkanes and aromatics.
the gradient descent method and the Gauss−Newton method. The Levenberg−Marquardt method acts more like a gradientdescent method when the parameters are far from their optimal value, and more like the Gauss−Newton method when the parameters are close to their optimal value and the solution typically converges rapidly to the local minimum. This way, it was possible to identify the parameters of the proposed equation which guarantee the lowest deviation of the predicted virial coefficients. The final equation proposed, even if very simple, produced only minor deviations, as presented in Figures 3 and 5, in which absolute and relative deviations are reported as a function of reduced temperature, respectively, for all the families excluding alkanes and aromatics. Deviations for alkanes and aromatics are presented in Figures 4 and 6. From Figures 3 and 4, it is evident that deviations decrease with increasing temperature, while from Figures 5 and 6 it is clear that most deviations are within 5%. To compare the results obtained with the equations discussed above, the second virial coefficients were recalculated with eqs 5, 8, 11, 16, 18, and 20. Because of the four groups added, the AAD% was not calculated for eight fluids for eq 18. The results are reported in Table 2 in terms of AAD%. In Table 2, improvements of the proposed equation on the calculation of the second virial coefficients are clearly evident, with a lower AAD% for almost all of the chemical families studied.
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Table 2. Summary of deviations for eqs 5, 8, 11, 16, 18 and 20 family
eq 5 AAD%
eq 8 AAD%
eq 11 AAD%
eq 16 AAD%
eq 18 AAD%
eq 20 AAD%
acetates alcohols aldehydes alkanes alkenes alkynes ammines aromatics cycloalkanes cycloalkenes dialkenes epoxides ethers formates haloalkanes haloalkenes ketones mercaptans sulfides avg
14.05 12.85 10.95 3.14 4.37 370.60 10.92 12.34 5.55 3.95 8.23 5.33 12.50 10.33 4.57 6.87 15.88 3.72 12.07 13.51
6.18 92.61 68.12 3.48 5.97 27.40 37.55 83.71 20.97 29.68 21.31 129.41 9.08 12.79 6.72 6.69 38.63 40.48 83.44 26.38
51.90 49.02 19.73 3.30 13.38 54.50 46.56 11.57 8.39 5.28 21.21 32.59 49.97 43.20 30.84 46.68 21.35 54.35 42.63 18.67
18.16 14.94 15.02 4.06 4.70 38.36 10.67 13.67 3.63 7.48 9.22 6.64 12.81 12.27 4.34 7.13 15.60 9.64 16.29 8.26
10.80 15.03 18.97 5.99 4.77 13.85 14.40 16.47 22.64 30.65 9.22 39.98 16.70 19.85 11.79 8.57 6.89 8.38 22.62 10.71
2.92 6.61 10.14 2.93 1.79 1.47 2.32 3.87 1.34 0.18 2.12 8.85 8.42 2.87 2.83 3.16 3.71 2.07 5.02 3.28
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ASSOCIATED CONTENT
* Supporting Information S
Number of occurrences and coefficients ai, bi, ci, di, and ei for each group. This material is available free of charge via the Internet at http://pubs.acs.org/
CONCLUSIONS
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An estimation technique based on a modified group contribution method was presented to calculate the second virial coefficients. The method can be applied to a wide range of compounds. In fact, the method depends on the knowledge of reduced temperature and acentric factor, and is based only on summing the products of the group contribution and their respective number of occurrences. The method was compared with the method existing in the literature, giving the lowest deviations for almost all of the families studied and a total AAD of 3.28%.
AUTHOR INFORMATION
Corresponding Author
*Tel.: +390712204277. Fax: +390712204770. E-mail: g.
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Tsonopoulos, C. An empirical correlation of second virial coefficients. AIChE J. 1974, 20, 263.
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(2) Pitzer, K. S.; Curl, R. F. The volumetric and thermodynamic properties of fluids. III. Empirical equation for the second virial coefficient. J. Am. Chem. Soc. 1957, 79, 2369. (3) Vetere, A. An improved method to predict the second virial coefficients of pure compounds. Fluid Phase Equilib. 1999, 164, 49. (4) O’Conell, J. P.; Prausnitz, J. M. Empirical correlation of second virial coefficients for vapour-liquid equilibrium calculations. IEC Proc. Des. Dev. 1967, 6, 245. (5) Black, C. Vapor phase imperfections in vapor−liquid equilibria. Ind. Eng. Chem. 1958, 50, 391. (6) Weber, L. A. Estimating the virial coefficients of small polar molecules. Int. J. Thermophys. 1994, 15, 461. (7) Tsonopoulos, C. Second Virial Coefficients of Polar Haloalkanes. AIChE J. 1975, 21, 827. (8) Dymond, J. H. Second virial coefficients and liquid transport properties at saturated vapour pressure of haloalkanes. Fluid Phase Equilib. 2000, 174, 13. (9) Tsonopoulos, C. Second virial coefficients of polar haloalkanes− 2002. Fluid Phase Equilib. 2003, 211, 35. (10) Meng, L.; Duan, Y. Y.; Li, L. Correlations for second and third virial coefficients of pure fluids. Fluid Phase Equilib. 2004, 226, 109. (11) Meng, L.; Duan, Y. Y. An extended correlation for second virial coefficients of associated and quantum fluids. Fluid Phase Equilib. 2007, 258, 29. (12) McCann, D. W.; Danner, R. P. Prediction of second virial coefficients of organic compounds by a group contribution method. IEC Proc. Des. Dev. 1984, 23, 529. (13) Dymond, J. H.; Marsh, K. N.; Wilhoit, R. C.; Wong, K. C. The Virial Coefficients of Pures Gases and Mixtures. Landolt-Börnstein; Springer: New York, 2002. (14) DIPPR-801, Project 801, Evaluated Process Design Data, Public Release Documentation, Design institute for Physical Properties (DIPPR); American Institute of Chemical Engineers, AIChE: New York, 2006. (15) Benson, S. W.; Buss, J. H. Additivity rules for the estimation of molecular properties. Thermodynamic properties. J. Chem. Phys. 1958, 29, 546. (16) Marquardt, D. W. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 1963, 11, 431.
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