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Fluid Phase Equilibria 260 (2007) 354–358
Short communication
Binary interaction parameter k ijij for calculating the second cross-virial coefficients of mixtures Long Meng a , Yuan-Yuan Duan a,∗, Xiao-Dong Wang b a
b
Key Laboratory Laboratory for Thermal Science and Power Engineering of MOE, Tsinghua University, Beijing 100084, PR China Department of Thermal Engineering, School of Mechanical Engineering, University of Science and Technology echnology Beijing, Beijing 100083, PR China
Received 1 June 2007; received in revised form 15 July 2007; accepted 16 July 2007 Available Available online 20 July 2007
Abstract
The binary binary interacti interaction on paramete parameters, rs, k ijij , of 119 mixtur mixtures es were were deter determin mined ed by fitting fitting thesecond thesecond crosscross-vir virialcoef ialcoeffici ficient entss of mixtur mixtures es with with correl correlati ations ons for pure compounds [L. Meng, Y.Y. Duan, L. Li, Fluid Phase Equilib. 226 (2004) 109–120; L. Meng, Y.Y. Duan, Fluid Phase Equilib., 258 (2007) 29–33] and classical mixing rules. The mixtures included nonpolar/polar (associated), polar/polar, quantum/nonpolar (quantum) binaries. Very simple correlations for k ijij of H 2 O/ n-alkane, CO/nonpolar and quantum/nonpolar (quantum) binaries were successfully developed. developed. The results show that the present correlations can accurately predict the second cross-virial coefficients. © 2007 Elsevier B.V. All rights reserved. Mixtures; k ijij ; Correlation Keywords: Second cross-virial coefficients; Mixtures;
1. Introducti Introduction on
Meng Meng et al. al. [1] present resented ed an empiri empirical cal correl correlati ation on for the secsecond virial virial coeffi coefficie cients nts of both both nonpol nonpolar ar and polar polar fluids fluids based based on the corresponding-states principle and then extended the correlation to associated and quantum fluids [2] fluids [2].. Meng and Duan [3] Duan [3] also reported the values of k k ijij for nonpolar/nonpolar mixtures. The correlation for pure compounds has the form [1] form [1]:: BP c RT c
= f (0) (T r ) + ωf (1) (T r ) + f (2) (T r )
(1)
where B is the second virial coefficient, T r (=T / T Tc ) the reduced temperature, Pc and T c the critical pressure and temperature, 8.3144 4471 71 J mol mol−1 K−1 the universal gas constant, and ω is R = 8.31 the acentric factor. factor. f (0) is the spherical term, f (1) is the nonpolar term and f (2) is the polar, associated or quantum term [2] term [2].. For binary mixtures, the classical mixing rule is Bm =
xi xj Bij
i
∗
j
Corresponding Corresponding author. Tel.: +86 10 6279 6318; fax: +86 10 6277 0209. E-mail address: yyduan@m
[email protected] ail.tsinghua.edu.cn u.cn (Y.-Y (Y.-Y.. Duan).
0378-3812/$ 0378-3812/$ – see front matter © 2007 Elsevier B.V. B.V. All rights reserved. doi:10.1016/j.fluid.2007.07.044 doi:10.1016/j.fluid.2007.07.044
(2)
where x i is the mole fraction of component i , B m is termed the mixture virial coefficient and Bij is the second cross-virial coefficient ficient.. When When Eq. (1) is used used to calcul calculate ate Bij , Pc , T c and ω should be replaced by P cij , T cij and ω ij defined by T cij = (T ci T cj )1/2 (1 − kij ) P cij =
ωij =
4T cij (P ci vci /T ci + P cj vcj /T cj ) 1/3 (vci
ωi + ωj
2
1/3 3 + vcj )
(3) (4)
(5)
where v ci and v cj are the critical volumes of components i and j. The characteristic parameter k ijij expresses the deviation from the geomet geometric ric mean mean for T cij . The The valu values es of k ijij obtain obtained ed using using difdifferent ferent method methodss can differ differ.. Inappr Inappropr opriat iatee use of these these result resultss may then lead to misleadingobserva misleadingobservations tions concerningthe concerningthe inherent inherent law for k ijij . Therefore, only second cross-virial coefficient data were used to evaluat evaluatee the binary binary interactio interaction n parameters parameters k ijij in this this work work to improv improvee the repres represent entati ation on of the second second virial virial coeffi coefficie cients nts for binary mixtures. Many new, high-quality experimental data for the second cross virial coefficient has been published since the 1980s. Most of them were recently collected by Dymond et al. [4].. This data with new correlations for the second virial coef[4] ficients of pure fluids fluids [1,2] [1,2] can be used to update the binary
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L. Meng et al. / Fluid Phase Equilibria 260 (2007) 354–358
Table 1 Optimum values of k ij and deviations for binary mixtures Compound i
Compound j
No. of points
k ij
RMSD (cm3 mol−1 ) Present work a
Nonpolar +polar Argon Argon Argon Benzene Benzene Benzene Benzene Benzene Benzene Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Cyclohexane Nitrous oxide Nitrous oxide Nitrogen Nitrogen Nitrogen Nitrogen Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Carbon monoxide Butane Methane Ethane Propane Hexane
Chloromethane Fluoromethane 1,1-Difluoroethane Propanone Diethyl ether Chloromethane Trichloromethane Dichloromethane 3-Pentanone Propanone Dichloromethane Diethyl ether Chloromethane Trichloromethane 3-Pentanone Naphthalene Phenanthrene Propanone Trichloromethane Dichloromethane Nitric oxide Argon Carbon dioxide Krypton Nitrogen Methane Ethene Propane Butane 2-Methylpropane Pentane Benzene Octane Acetonitrile Propanone Chloromethane 1,1,1,2-Tetrafluoroethane Diethyl ether
9 4 6 14 10 11 8 11 8 6 11 10 11 8 8 13 11 8 5 11 9 22 18 5 13 11 7 11 5 5 8 11 7 4 7 7 8 4
0.266 0.153 0.145 0.024 −0.019 −0.054 −0.067 −0.060 0.009 0.143 0.084 0.030 0.135 0.024 0.033 0.158 0.199 0.142 0.192 −0.019 0.035 −0.006 −0.045 0.011 0.002 0.0002 0.014 0.020 0.032 −0.002 0.074 0.049 0.100 0.313 0.120 0.030 0.125 0.005
18.2 1.8 3.6 77.5 7.2 10.5 4.4 5.2 2.2 10.6 11.9 13.0 1.2 6.5 10.6 14.7 17.2 16.8 28.3 4.6 8.0 2.1 3.8 1.2 1.3 1.1 0.4 0.9 2.2 4.5 4.0 3.6 10.0 12.3 32.9 1.9 2.0 10.3
Nonpolar+ associated Water Water Water Water Water Water Water Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Methanol Ethanol Ethanol Ethanol Ethanol Ethanol
Argon Oxygen Nitrogen Carbon dioxide Ethene Benzene Cyclohexane Argon Nitrogen Carbon dioxide Benzene Cyclohexane Hexane Methane Ethene Ethane Argon Cyclohexane Nitrogen Benzene Carbon dioxide
16 45 29 36 8 10 12 5 15 11 29 16 11 10 5 7 4 15 5 17 4
0.288 0.405 0.276 0.167 0.233 0.276 0.468 0.063 0.043 −0.087 0.178 0.294 0.313 0.060 0.098 0.110 0.159 0.224 0.079 0.133 0.076
4.8 2.5 4.4 12.8 4.5 6.5 14.4 3.9 5.5 99.3T 6.5 9.8 14.9 27.8 9.3 9.3 3.8 14.1 22.3 10.8 23.0
Present work b
2.1 11.5 5.1 1.4 1.2 0.4 2.7 4.2 11.8 6.2 3.6 10.0
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L. Meng et al. / Fluid Phase Equilibria 260 (2007) 354–358
Table 1 (Continued ) Compound i
Compound j
No. of points
k ij
RMSD (cm3 mol−1 ) Present work a
Ammonia Ammonia Ammonia 1-Propanol Nitrogen
Present work b
Argon Nitrogen Methane Heptane 1-Butanol
10 6 6 7 4
0.153 0.113 0.154 0.198 0.059
Polar + polar Chloromethane Trichloromethane Trichloromethane Trichloromethane Chlorodifluoromethane Chlorodifluoromethane Chlorodifluoromethane Chlorodifluoromethane Difluoromethane Pentafluoroethane Pentafluoroethane Pentafluoroethane
Propanone Propanone Dimethyl ether Diethyl ether Dichlorodifluoromethane 1,2-Dichloro-1,1,2,2-tetrafluoroethane 1-Chloro-1,1-difluoroethane 1,1-Difluoroethane 1,1,1,2-Tetrafluoroethane Difluoromethane 1,1,1,2-Tetrafluoroethane 1,1,1,2-Tetrafluoroethane
6 26 6 11 13 21 8 8 3 11 4 5
−0.004 −0.090 −0.157 −0.131
0.070 0.055 −0.007 −0.040 0.028 0.029 0.017 0.011
7.0 127.4 36.7 82.8 17.2 13.0 4.4 13.8 0.7 1.3 2.8 3.3
Quantum+ nonpolar Helium Helium Helium Helium Helium Helium Helium Helium Helium Helium Helium Neon Neon Neon Neon Neon Neon Neon Neon Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen Hydrogen
Argon Krypton Nitrogen Oxygen Xenon Tetrachloromethane Tetrafluoromethane Methane Butane Sulfur hetrafluoride Carbon dioxide Argon Oxygen Xenon Methane Krypton Nitrogen Carbon dioxide Sulfur hetrafluoride Carbon dioxide Propane Argon Octane Benzene Krypton Nitrogen Xenon Tetrachloromethane Tetrafluoromethane Methane Ethane
21 15 56 19 9 5 19 18 12 16 35 30 10 28 8 48 12 6 6 17 10 36 11 9 15 42 19 6 9 16 14
0.283 0.425 0.281 0.248 0.570 0.700 0.180 0.483 0.792 0.212 0.169 0.142 0.095 0.353 0.298 0.220 0.167 0.252 0.175 −0.151 0.094 −0.112 0.058 −0.009 −0.060 0.037 0.038 0.153 −0.087 −0.038 0.020
1.7 0.9 2.5 17.5 3.1 9.3 2.9 1.7 4.0 1.7 10.1 1.5 2.8 3.8 1.5 2.9 1.2 2.8 1.5 8.7 6.3 1.9 6.1 1.4 2.1 3.4 3.2 13.2 4.6 1.9 3.3
2.2 2.0 2.7 29.4 3.1 9.5 3.3 1.8 4.0 2.4 11.5 2.9 22.2 4.2 2.6 2.9 2.6 2.8 3.4 9.7 6.7 2.6 20.2 20.7 2.6 25.3 3.4 18.5 5.9 2.0 4.1
Quantum+ quantum Helium Helium Helium Hydrogen
Neon Hydrogen Deuterium Neon
18 28 5 8
−0.100
2.9 4.2 0.3 0.2
4.6 4.2 3.1 0.7
a b
Calculated with optimum values of k ij . Calculated using the corresponding correlations.
0.328 0.210 0.173
2.1 0.9 2.2 48.3 T 2.1
L. Meng et al. / Fluid Phase Equilibria 260 (2007) 354–358
357
interaction parameters, k ij and extend their range. Before analyzing the data, the experimental data were carefully selected, mostly following Dymond’s recommendations [4]. 2. Nonpolar/polar (associated) binaries
When fitting the second cross-virial coefficient for k ij using Eqs. (1)–(5), the polar or associated f (2) term should be set equal to zero as recommended by Tsonopoulos [5]. The optimum values of k ij for 38 nonpolar/polar binaries and 34 nonpolar/associated binaries were determined by minimizing the deviations between the calculated and experimental second cross-virial coefficients, with the results listed in Table 1. All the property parameters used for the correlation in this study, namely, P c , T c , v c and ω were taken from the DIPPR database [6]. For binaries containing CO, which has a small dipole moment, the simple correlation can be used: kij = −0.0086 + 0.27ω
(6)
where ω is the acentric factor of the nonpolar component. Fig. 1 presents the optimum k ij ’s for CO/nonpolar binaries together with the straight line calculated with Eq. (6), which shows that Eq. (6) givesa goodfit of the optimum valuesof k ij for most binaries except CO/CO2 and CO/C4 H10 (2-methylpropane) binaries. The predicted and experimental Bij values for the CO/Ar binary, for which we have the most extensive information, are shown in Fig. 2 with the RMSDs listed in Table 1. For binaries containing H2 O, the k ij are difficult to correlate, however, there is an excellent linear relationship between k ij and the carbon number, n c , of H2 O/ n-alkane binaries as shown in Fig. 3. kij = 0.31 + 0.0264nc
(7)
This correlation is compared in Table 2 with that of Tsonopoulos and Dymond [7] whose correlation for H2 O/ n-alkane binaries used the critical volume vc which shows that the two correlations are roughly equivalent. Note that experimental
Fig. 1. Optimum values of k ij for CO/nonpolar binaries: ( ) optimum values; (—) Eq. (6).
Fig.2. Secondcross-virialcoefficientsfor CO/Ar binary: () experimentaldata [4]; (—) calculated with Eq. (6).
data with large uncertainties may lead to large uncertainties in k ij . 3. Polar/polar binaries
For polar/polar binaries, the parameter a in f (2) term can be used with the following mixing rule: aij = 0.5(ai + aj )
(8)
with the values of a given by Meng et al. [1]. The optimum k ij for 12 polar/polar binaries are listed in Table 1. 4. Quantum/nonpolar (quantum) binaries
For binary mixtures containing a quantum component, the quantum term, f (2) , in Eq. (1) must be included as suggested by Meng and Duan [2]. The quantum contribution to B ij was then calculated by assuming that Λ∗ = 0.5(Λ∗i + Λ∗j )
(9)
Fig. 3. Optimum values of k ij for H2 O/ n-alkane binaries: () optimum values; (—) Eq. (7).
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L. Meng et al. / Fluid Phase Equilibria 260 (2007) 354–358
Table 2 Optimum values of k ij and deviations for binary mixtures containing H 2 O Mixture
H2 O/CH4 H2 O/C2 H6 H2 O/C3 H8 H2 O/C4 H10 H2 O/C5 H12 H2 O/C6 H14 H2 O/C7 H16 H2 O/C8 H18
RMSD (cm 3 mol−1 )
No. of points
k ij
0.321 0.353 0.409 0.430 0.445 0.470 0.489 0.512
29 19 12 8 5 16 19 12
Present work
Tsonopoulos and Dymond [7]
6.5 6.7 15.4 10.5 8.3 18.6 47.3 24.5
6.2 10.1 14.4 7.9 4.5 17.0 38.0 33.4
by regression of only second cross-virial coefficient data. Simple correlations were developed for CO/nonpolar, H2 O/ n-alkane and quantum/nonpolar (quantum) binary mixtures. The predicted k ij accurately represent the experimental data for k ij . List of symbols second virial coefficient B (0) (1) f , f , f (2) dimensionless functions of T r binary interaction parameter k ij pressure P universal gas constant, R = 8.314471 J mol −1 K−1 R
RMSD root mean square deviation, temperature T specific volume v
Fig. 4. Optimum values of k ij for binaries containing quantum component: ( ) optimum values; (—) Eq. (10).
where Λ is the reduced de Broglie wavelength taken from Hirschfelder et al. [8]. The optimum k ij values are also listed in Table 1. Hiza andDuncan [9] developed a correlation using the ionization potential (IP) parameter I . However, since the Hiza–Duncan correlation was developed using rather different, limited k ij data sets, their correlation cannot be compared with the k ij values of this work using values of I taken from the CRC handbook [10]. However, there is a strong relationship between k ij and ( I i / I j ) as shown in Fig. 4, which can be correlated by kij = −0.88 + 0.714
I i
I j
(I i > I j )
(10)
as shown in Fig. 4. The RMSDs in Table 1 f o r most of the binary mixtures containing a quantum componentare very small, which indicatesthat Eq. (10) gives goodpredictionsof k ij for this group. 5. Conclusions
The optimum k ij values for 119 binary mixtures are presented using the correlations for pure compounds given by Meng and co-workers [1,2]. Optimum k ij ’s were determined
n i=1 (Bexp
− Bcal )2 /n
Greek letters reduced de Broglie wavelength Λ* acentric factor ω Subscripts
c r
critical property reduced property
Acknowledgement
This work was supported by the National Natural Science Foundation of China (No. 50636020). References [1] L. Meng, Y.Y. Duan, L. Li, Fluid Phase Equilib. 226 (2004) 109–120. [2] L. Meng, Y.Y. Duan, Fluid Phase Equilib. 258 (2007) 29–33. [3] L. Meng, Y.-Y. Duan, Fluid Phase Equilib. 238 (2005) 229–238. [4] J.H. Dymond, K.N. Marsh, R.C. Wilhoit, Virial coefficients of pure gases and mixtures, Subvolume B, Virial coefficients of mixtures, LandoltB¨ornstein Series IV/21B, 2002. [5] C. Tsonopoulos, AIChE J. 20 (1974) 263–272. [6] Design Institute for Physical Properties: DIPPR 801 Database (2005 release), Brigham Young University, 2005. [7] C. Tsonopoulos, J.H. Dymond, Fluid Phase Equilib. 133 (1997) 11–34. [8] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, New York, 1954. [9] M.J. Hiza, A.G. Duncan, AIChE J. 16 (1970) 733–738. [10] D.R. Lide, CRC Handbookof Physics andChemistry, 85th ed., CRCPress, New York, 2005.
