2110
Ind. Eng. Chem. Res. 1999, 38, 2110-2118
A Cubic Equation of State with Group Contributions for the Calculation of Vapor-Liquid Equilibria of Mixtures of Hydrofluorocarbons and Lubricant Oils Nicola Elvassore and Alberto Bertucco* Istituto di Impianti Chimici, Universita` di Padova, via Marzolo 9, I-35131 Padova, Italy
A ° sa Wahlstro1 m Department of Heat and Power Technology, Chalmers University of Technology, S-41296 Gothenburg, Sweden
A method for calculating the vapor-liquid equilibria of mixtures between hydrofluorocarbons and lubricant oils is presented. A cubic equation of state is used (Sako, T.; et al. J. Appl. Polym. Sci. 1989, 38, 1839), containing three parameters: the attractive one, a, the volume parameter, b, and the number of external degrees of freedom per molecule, c. To allow calculation of the parameters of the high molecular weight components, whose critical constants and vapor pressure are unknown, a group-contribution approach is developed for a, b, and c. The extension to mixtures is achieved by applying Huron-Vidal mixing rules. A modified Uniquac model is used to evaluate infinite-pressure activity coefficients. With the proposed method, the density of pure heavy components (such as n-hexadecane and pentaerythritol esters) is predicted as a function of temperature. Vapor-liquid equilibria calculations are presented for binary mixtures between several hydrofluorocarbons and pentaerythritol esters or hexadecane; a comparison with the results obtained by available models is also outlined. Introduction The replacement of chlorofluorocarbons as working fluids in refrigeration cycles is a matter of current research for a number of scientific and technological issues. In most cases, new refrigerants are mixtures of hydrofluorocarbons (HFC) or hydrocarbons. With this respect, one problem to address is the evaluation of thermophysical properties of such mixtures. On the other hand, the presence of a lubricant oil in the refrigeration cycle may reduce dramatically the efficiency of the plant, because it can modify the refrigerant composition, that is, both the equilibrium pressure and the volumetric properties. The lubricant oil considered here is made up of many high molecular weight compounds, but pentaerythritol esters (POEs) represent 95% of this mixture. For these reasons, there is an urgent need for a model able to calculate thermodynamic properties of mixtures of new refrigerants and lubricant oils (such as POEs), that is, of mixtures containing at the same time small molecules as well as long-chain or high molecular weight compounds. Because the operating pressure of refrigeration cycles may be as high as 40 bar, this model must be an equation of state (EOS). The problem of developing a model suitable for a refrigerant-oil system has been addressed in the open literature only recently. Among them, excess Gibbs energy (GE) model types have been proposed2,3 for vapor-liquid equilibria (VLE) calculation of systems containing POEs. Instead, simple EOSs suitable for this purpose are still mixing; a modification of the Redlich* To whom correspondence should be addressed. E-mail:
[email protected]. Phone: [+39] (049) 8275457. Fax: [+39] (049) 8275461.
Kwong-Soave (RKS) EOS has been proposed by Yokozeki,4 in which specifically tailored mixing rules and oil parameter determination have been suggested for correlating refrigerant-oil systems. On the other hand, a number of models, which are relatively complex, have been proposed to represent systems containing long-chain or high molecular weight compounds; among them, we cite equations especially devised to treat polymer solutions, such as the latticefluid theory (LF; Panayiotou and Vera,5,6 Lee and Danner7), the statistical associating fluid theory (SAFT; Chapman et al.8,9), the perturbed hard-chain theory (PHCT; Beret and Prausnitz10), and the perturbed hardsphere-chain theory (PHSC; Song et al.,11 Hino and Prausnitz12). We note that classical cubic EOSs cannot be applied in this case. In fact, according to the usual approach, the knowledge of properties such as the critical constants and the vapor pressures of all components involved in the mixture is essential to evaluate the EOS parameters. If some of these data are unknown, as happens in our case of interest, the pure-component parameters have to be evaluated in a different way. This idea was already applied, for example, to treat polymeric solutions with the van der Waals EOS (Kontogeorgis et al.13); however, because cubic EOSs were invented to represent fluids of spherical particles, such an approach leads to parameter values lacking any physical meaning (Bertucco and Mio14). In one case only was a cubic EOS derived taking into account some aspects of the molecular theory of chainlike molecules (namely, the PHCT theory), under simplified assumptions. This is the EOS proposed by Sako et al.,1 here referred to as SP EOS, which was conceived on a segment basis, rather than on a spherical molecule
10.1021/ie980570w CCC: $18.00 © 1999 American Chemical Society Published on Web 04/15/1999
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2111
basis, and is therefore structurally suitable to be applied also to long-chain molecules. We have modified the SP EOS by developing a groupcontribution approach for the evaluation of purecomponent parameters, so that neither critical constants nor vapor pressures are needed for high molecular weight compounds to perform calculations with this EOS. Accordingly, the values of all three EOS parameters can be predicted with only the knowledge of the compound’s chemical structure. Although our approach is referred to the groups involved in mixtures of refrigerants and POEs, it can be easily applied, in general, with other groups and compounds. The extension to mixtures has been carried out with Huron-Vidal mixing rules at infinite pressure (Huron and Vidal15), to improve the EOS flexibility in treating solutions containing polar components. The infinitepressure activity coefficient has been evaluated by a modified Uniquac model. It will be shown that, with the presently proposed model, it is possible to perform accurate calculations of VLE for mixtures of refrigerants and POEs by a cubic EOS. The Model The SP EOS is a semiempirical cubic equation that belongs to the so-called perturbed hard-sphere EOSs, because it is based on the fundamental assumption that the microstructure of a fluid is determined primarily by the repulsive forces between molecules. The effect of attractive forces is taken into account by a mean-field assumption. Sako et al.1 showed that by inserting in the partition function Q the rotational and vibrational contributions of Beret and Prausnitz10 (derived following the Prigogine idea16), the van der Waals free-volume expression, and the RKS potential field, a cubic EOS can be obtained. For details we refer to Sako et al.1 The final view of the EOS is
RT(v - b + bc) a P) v(v - b) v(v + b)
(1)
(BA - 1 - B - (c - 1))Z A B ( + (c - 1)) ) 0 (2) B 2
A ) aP/R2T2
(3)
B ) bP/RT
(4)
Note that parameter A in eq 2 appears only as the ratio A/B, which is independent of the pressure. For pure components, parameters A/B, B, and c can be evaluated by
R(T) 3 A ) f(D0) B Tr D0 B)
(5)
D0 Pr 3 Tr
(6)
where f(D0) and D0, which are a functions of c, are reported elsewhere;1 Tr and Pr are the reduced temperature and pressure, respectively. The dimensionless function R(T), which becomes unity at T ) Tc, is used to reproduce exactly the vapor pressure value at T. Alternately, the parameters of the dimensional SP EOS, i.e., eq 1, can be obtained according to
a ) acR(T) ac )
(7)
R2Tc2 f(D0) Pc
(8)
D0 RTc 3 Pc
(9)
b)
The fugacity coefficient of a pure fluid in terms of dimensionless parameters is given by
ln(φ) ) -ln(Z - B) -
where P, T, and v are the pressure, temperature, and molar volume and R is the universal gas constant. With symbols a, b, and c, the attractive parameter, the van der Waals covolume, and the external degrees of freedom are indicated, respectively. Thanks to its theoretical basis, this EOS is applicable not only to small, spherical molecules but also to large, structurally complex ones. Also the choice of the expression of the RKS potential field comes up from theoretical consideration. Using the Peng-Robinson expression for the attractive part, the EOS is fourth order in volume; on the other hand, the simplest case of the van der Waals attractive term leads to a cubic EOS, but the critical compressibility factor, ZC ) PCvC/RTC, becomes a function of parameter c. In the case of eq 1 only, regardless of the value of c, the SP EOS gives a constant critical compressibility factor ZC ) 1/3. It is useful to rewrite eq 1 in terms of Z and other dimensionless parameters:
Z3 - Z2 + B
where
Z+B A ln + B Z
(
)
(Z -Z B) (10)
(Z - 1) - (c - 1) ln
Note that for spherical molecules (i.e., c ) 1) eqs 1-10 reduce to the original RKS EOS. Sako et al.1 applied their EOS to polymer solutions by writing the pure-component parameters on a segment basis. The van der Waals volume and specific potential energy parameter of the segment were used to obtain the values of segment parameters, so that the need of critical constant and vapor pressure values was overcome. In this way, Sako et al.1 could use successfully their EOS to describe VLE of polymer solutions at high pressure. In our work, a group-contribution method is developed to obtain the pure parameter values by using chemical structure data only, as will be outlined in the section Parameter Determination.
Extension to Mixture To evaluate mixture parameters for eq 2, the HuronVidal procedure at infinite pressure was applied (Huron
2112 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
and Vidal15) to the SP EOS. Accordingly, the following mixing rules were derived:
A B
)
[
Ai
∑i zi B
i
-
ln(γi∞) ln 2
-
]
1
(ci ln Bi - ln Bi - ci ln ci) ln 2 1 [c ln c - (c - 1) ln B] (11) ln 2 B)
∑i ziBi
(12)
c)
∑i zici
(13)
where i is the component index, γ∞ is the activity coefficient at the infinite-pressure reference state, and z is the mole fraction in the phase considered. The expression of the fugacity coefficient of component i in the mixture is
(Z - 1) ∂nTB ∂(nTA/B) × B ∂ni ∂ni ∂(nTc - nT) Z+B Z-B ln ln (14) Z ∂ni Z
ln(φi) ) -ln(Z - B) +
(
)
(
)
where, considering eqs 11-13, we have
(
ci ∂(nTA/B) Ai ln(γi∞) 1 + ) ci ln - (ci - 1) × ∂ni Bi ln 2 ln 2 c Bi Bi + (c - 1) - (ci - 1) (15) ln B B
)
∂nTB ) Bi ∂ni
(16)
∂nTc ) ci ∂ni
(17)
We have expressed the activity coefficients at infinite pressure by the Uniquac model. However, with respect to its original version (Abrams and Prausnitz17), only the residual part was considered, so that we have for binary mixtures
gE,∞ ) RT[-q1z1 ln(θ1 + θ2τ21) - q2z1 ln(θ2 + θ1τ12)] (18) where gE,∞ is the molar excess Gibbs energy of the mixture and θi and τij are the area fraction of component i and the interaction parameter between components i and j, respectively. Another modification with respect to the original Uniquac is that, here, parameters θi have been calculated from the covolume bi of component i, according to
θi )
∑j zjbj
(19) 2/3
compound
Tc [K]
Pc [bar]
F293 [g/cm3]
Psat293 [mbar]
pentane hexane heptanea octanea nonanea decanea undeacnea dodecanea tridecanea tetradecanea pentadecanea hexadecanea 2-methylbutane isohexane 3-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylhexane 3-methylhexane 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 3-ethylpentane 2,2,3-trimethylbutane 2-methylheptane 3-methylheptane 4-methylheptane 2,2-dimethylhexane 2,3-dimethylhexane 2,4-dimethylhexane 2,5-dimethylhexane 3,3-dimethylhexane 3,4-dimethylhexane 3-ethylhexane 2,2,3-trimethylpentane 2,2,4-trimethylpentane 2,3,3-trimethylpentane 2,3,4-trimethylpentane 2-methyl-3-ethylpentane 3-methyl-3-ethylpentane 2-methyloctane 2,2-dimethylheptane 2,2,3-trimethylhexane 3,3-diethylpentane 2,2,3,3-tetramethylpentane 2,2,3,4-tetramethylpentane 2,2,4,4-tetramethylpentane 2,3,3,4-tetramethylpentanea 2,4,4-trimethylhexanea n-propyl acetatea ethyl propionatea n-butyl acetatea isobutyl acetate n-propyl propionatea n-heptyl acetatea n-octyl acetatea n-nonyl acetatea
469.7 507.3 540.3 568.76 594.6 617.7 639.0 658.2 675.0 693.0 708.0 723.0 460.39 497.45 504.40 488.73 499.93 530.31 535.19 520.44 537.29 519.73 536.34 540.52 531.11 559.57 563.60 561.67 549.80 563.42 553.45 549.99 561.95 568.78 565.42 563.43 543.89 573.49 566.34 567.09 576.6 587.0 576.8 588.0 610.0 607.7 592.7 574.7 607.7 581.0 549.73 546.0 579.15 564.0 568.6 637.0 652.0 661.0
33.69 30.12 27.36 24.87 22.90 21.20 19.50 18.20 16.80 15.70 14.80 14.00 33.81 30.10 31.22 30.80 31.27 27.34 28.14 27.73 29.08 27.37 29.46 28.91 29.54 24.84 25.46 25.42 25.29 26.28 25.56 24.87 26.54 26.92 26.08 27.30 25.68 28.20 26.94 27.00 28.1 23.1 23.5 24.9 26.7 27.4 26.0 24.9 27.16 24.60 33.60 33.62 31.10 30.2 30.60 23.30 21.50 19.90
0.626 0.659 0.684 0.703 0.718 0.730 0.740 0.748 0.757 0.763 0.769 0.773 0.620 0.653 0.664 0.649 0.662 0.679 0.687 0.674 0.695 0.673 0.693 0.698 0.690 0.698 0.706 0.705 0.695 0.712 0.700 0.694 0.710 0.719 0.714 0.716 0.692 0.726 0.719 0.719 0.727 0.713 0.711 0.729 0.752 0.757 0.739 0.719 0.755 0.724 0.888 0.889 0.881 0.875 0.882 0.871 0.869 0.866
562.619 161.944 46.792 14.008 4.235 1.279 0.386 0.122 0.038 0.011 0.004 0.001 760.951 229.404 203.280 349.495 253.971 68.614 64.102 111.197 71.991 103.796 87.186 60.299 108.731 20.613 19.562 20.591 34.782 23.706 30.952 30.934 29.244 21.840 20.174 32.915 51.347 27.736 27.228 24.196 23.431 6.286 10.832 11.192 7.281 9.388 12.709 20.425 8.735 14.379 34.930 45.944 10.417 17.905 13.278 0.382 0.114 0.047
a Compounds used for regression of the temperature dependency constants.
The temperature dependence of the model is embedded in the coefficients τij and was derived from Gmehling et al.:18
(
τij ) exp -
2/3
zibi
Table 1. Critical Temperature, Critical Pressure, Density, and Vapor Pressure at 293 K (Daubert and Danner19 and Reid et al.20) of the Compounds Used for the Regression of the Group-Contribution Parameters
)
∆uij RT
∆uij ) aijT - bijT2
(20) (21)
where aij and bij are binary adjustable parameters. They
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2113 Table 2. Pure Compound Parameters Calculated from Experimental Data (calc) and with Group Contributions (GC)a ac/Nc pentane hexane heptane octane nonane decane undecane dodecane tridecane tetradecane pentadecane hexadecane 2-methylbutane isohexane 3-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylhexane 3-methylhexane 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 3-ethylpentane 2,2,3-trimethylbutane 2-methylheptane 3-methylheptane 4-methylheptane 2,2-dimethylhexane 2,3-dimethylhexane 2,4-dimethylhexane 2,5-dimethylhexane 3,3-dimethylhexane 3,4-dimethylhexane 3-ethylhexane 2,2,3-trimethylpentane 2,2,4-trimethylpentane 2,3,3-trimethylpentane 2,3,4-trimethylpentane 2-methyl-3-ethylpentane 3-methyl-3-ethylpentane 2-methyloctane 2,2-dimethylheptane 2,2,3-trimethylhexane 3,3-diethylpentane 2,2,3,3-tetramethylpentane 2,2,3,4-tetramethylpentane 2,2,4,4-tetramethylpentane 2,3,3,4-tetramethylpentane 2,4,4-trimethylhexane n-propyl acetate ethyl propionate n-butyl acetate isobutyl acetate n-propyl propionate n-heptyl acetate n-octyl acetate n-nonyl acetate obj. function a
(RNc)293
b
c
calc
GC
calc
GC
calc
GC
calc
GC
4.182 4.603 4.965 5.366 5.706 6.031 6.490 6.816 7.325 7.782 8.106 8.434 3.935 4.371 4.287 4.014 4.183 4.724 4.635 4.391 4.466 4.467 4.351 4.589 4.220 5.141 5.054 5.017 4.766 4.816 4.779 4.898 4.704 4.765 4.926 4.545 4.517 4.542 4.720 4.710 4.631 5.417 5.031 4.882 4.861 4.639 4.682 4.580 4.697 4.819 6.082 6.006 5.893 5.797 5.794 6.423 6.600 6.811
4.234 4.582 4.930 5.278 5.626 5.974 6.322 6.670 7.018 7.366 7.714 8.062 4.005 4.353 4.353 4.045 4.124 4.701 4.701 4.393 4.472 4.472 4.393 4.701 4.164 5.049 5.049 5.049 4.741 4.820 4.820 4.820 4.741 4.820 5.049 4.512 4.512 4.512 4.591 4.820 4.741 5.397 5.168 4.860 5.089 4.552 4.631 4.552 4.631 4.860 5.652 5.652 6.000 5.771 6.000 7.044 7.392 7.740 1.050
6.399 8.012 9.764 11.504 13.375 15.336 17.102 19.097 20.740 22.551 24.609 26.766 6.384 7.974 8.098 7.963 8.006 9.736 9.840 9.694 9.863 9.690 9.811 9.860 9.780 11.481 11.573 11.590 11.483 11.647 11.543 11.445 11.589 11.731 11.648 11.634 11.435 11.705 11.543 11.703 11.737 13.471 13.530 13.482 13.692 13.755 13.550 13.418 13.758 13.361 6.862 6.729 8.920 8.662 8.861 14.781 17.029 18.689
6.303 8.096 9.889 11.682 13.475 15.268 17.061 18.854 20.647 22.440 24.233 26.026 6.272 8.065 8.065 8.047 8.034 9.858 9.858 9.840 9.827 9.827 9.840 9.858 9.809 11.651 11.651 11.651 11.633 11.620 11.620 11.620 11.633 11.620 11.651 11.602 11.602 11.602 11.589 11.620 11.633 13.444 13.413 13.395 13.426 13.377 13.364 13.377 13.364 13.395 6.888 6.888 8.681 8.650 8.681 14.060 15.853 17.646 0.460
83.67 98.48 113.67 128.15 143.2 158.37 172.23 187.28 199.86 213.92 228.53 243.81 84.88 99.41 99.44 101.29 99.52 114.73 114.56 116.30 114.30 115.92 115.07 113.33 115.79 129.23 128.79 129.07 130.75 128.96 130.21 130.23 129.71 128.62 128.27 129.68 132.17 128.87 128.00 128.63 128.53 145.24 146.80 144.53 142.29 142.34 144.08 147.05 142.47 144.91 87.39 86.46 105.76 104.53 104.77 151.45 166.11 174.30
84.12 98.87 113.62 128.37 143.12 157.87 172.62 187.36 202.11 216.86 231.61 246.36 84.60 99.35 99.35 100.06 99.82 114.10 114.10 114.81 114.57 114.57 114.81 114.10 115.28 128.85 128.85 128.85 129.56 129.32 129.32 129.32 129.56 129.32 128.85 130.03 130.03 130.03 129.80 129.32 129.56 143.59 144.07 144.78 144.31 145.49 145.26 145.49 145.26 144.78 89.11 89.11 103.86 104.33 103.86 148.10 162.85 177.60 1.34
1.709 1.839 1.922 2.070 2.163 2.251 2.466 2.567 2.858 3.062 3.180 3.280 1.534 1.698 1.591 1.433 1.540 1.781 1.692 1.561 1.568 1.628 1.474 1.662 1.396 1.940 1.865 1.844 1.699 1.698 1.700 1.803 1.612 1.642 1.779 1.499 1.529 1.463 1.641 1.614 1.513 1.960 1.726 1.617 1.538 1.410 1.474 1.440 1.443 1.605 2.360 2.382 1.995 2.077 2.035 2.137 2.204 2.477
1.695 1.812 1.929 2.046 2.163 2.280 2.397 2.514 2.631 2.748 2.865 2.982 1.531 1.648 1.648 1.420 1.484 1.765 1.765 1.537 1.601 1.601 1.537 1.765 1.373 1.882 1.882 1.882 1.654 1.718 1.718 1.718 1.654 1.718 1.882 1.490 1.490 1.490 1.554 1.718 1.654 1.999 1.835 1.607 1.771 1.379 1.443 1.379 1.443 1.607 2.084 2.084 2.201 2.037 2.201 2.552 2.669 2.786 1.34
The objective function for the regression of each pure compound parameter is defined by eq 28.
can be either symmetrical or asymmetrical. We will show that for the systems under investigation symmetrical parameters are enough to describe the behavior of the mixture over the whole composition range. Note also that the extension of eq 18 to multicomponent mixtures is straightforward and that only binary interaction coefficients are needed for multicomponent calculations. Parameter Determination Volatile Components. For the volatile, i.e., low molecular weight, fluids, critical temperature and pressure as well as vapor pressures are known. These data can be used both in eqs 5 and 6 and in eqs 7-9 for
evaluating the pure-component parameters, provided parameter c is known. The value of c was calculated in order to reproduce a reference liquid density at standard conditions (T ) 293 K and P ) 1 atm). This procedure requires the simultaneous calculation of both R and c values at the reference conditions. In Table 1 the database is summarized: 12 n-alkanes, 38 branched alkanes, and 8 esters were considered. Critical temperature and pressure as well as density and vapor pressure data at T ) 293 K were used. The parameter values calculated are reported in Table 2. The estimation of c according to this method gave good results for all compounds included in Table 1. However, the same procedure led to unexpected results
2114 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 Table 3. Critical Temperature, Critical Pressure, and Density at the Normal Boiling Point for the HFCs Considered (Data from McLinden22 and Elvassore23), Together with the Values of Parameters c and d HFC
critical temp [K]
critical pressure [bar]
boiling temp [K]
density [g/mL]
parameter c
parameter d [mL/g]
R134a R143a R152a
374.21 345.88 386.44
40.56 37.70 45.2
247 225.8 247.0
1.375 1.176 1.012
1.86 1.69 1.73
-2.69 -6.59 -6.43
linear relationship between them and the number of carbon atoms in each compound, Nc, i.e., to derive a group-contribution method. The result was that parameters b and c are directly proportional to Nc, as can be seen from Figure 2a,b. For parameter a this does not happen, because a is not linear with the carbon number. Then, it was convenient to split a into two factors, according to
a ) (ac/Nc)(RNc)
Figure 1. Correlation of external degrees of freedom c to the acentric factor ω for n-alkanes (methane to n-decane) for the SP EOS.
when applied to HFCs: unreasonably high c values, out of any physical interpretation, are obtained in this case, as can be seen when comparing them with the corresponding alkanes at the same number of carbon atoms. This is clearly due to a deficiency of the SP EOS in representing liquid densities of HFCs. To overcome this problem, we followed a suggestion by Beret and Prausnitz.10 For normal fluids, they pointed out that the c parameter of the PHCT EOS can be correlated well with the acentric factor ω. We applied this idea to the SP EOS and regressed the function c(ω) on our database of n-alkanes. As can be seen in Figure 1, a good linear correlation was obtained, which can be expressed by
c ) 2.655ω + 1
(22)
By using eq 22, physically consistent, yet approximate, c values for HFCs were obtained. Unfortunately, such values led to unsatisfactory liquid density calculations. Therefore, following the Peneloux idea (Peneloux et al.21), we introduced a volume shift d in the EOS
vshift ) v + d
(23)
so that the shifted volume vshift appears in eq 1 instead of v. The new value of b was then used in the SP EOS to allow correct volumetric calculations of HFCs; for all other compounds considered, d was set to 0. We recall that the introduction of a volume shift parameter does not affect VLE calculations (Peneloux et al.21). In Table 3 the data used and the results obtained for HFCs are summarized. High Molecular Weight Components. Because neither critical data nor vapor pressures are known for these compounds, their parameters cannot be determined in the same way as for the volatile fluids. Looking for an alternate and possibly predictive way to calculate them, an alkane series from n-pentane to hexadecane was considered. The parameter values obtained (Table 2) were examined thoroughly to find a
(24)
As is shown in Figure 2c,d, both ac/Nc and RNc have a linear dependence on the carbon atom number, so that it is possible to establish group contributions on both of them and to calculate parameter a from eq 24. In summary, according to the group-contribution method presently proposed, for high molecular weight compounds the pure-component parameters of the SP EOS are evaluated by
() ac
n
acompound )
n
∑ vk N k)1 ∑ vk(RNc)k k)1
(25)
c k n
bcompound )
∑ vkbk
(26)
k)1 n
ccompound )
∑ vkck
(27)
k)1
where the specific parameter contribution for each group k has index k and vk is the number of groups k in the compound considered. Because lubricant oils of our interest, besides CH3and CH2- groups, contain also CH-, C-, and ester groups (COO-), it was necessary to obtain the related group contributions from data of branched alkanes and esters, which are listed in Table 1. In summary, ac/Nc, RNc, b, and c were calculated by using density and vapor pressure data at T ) 293 K, as well as critical pressure and temperature, as reported in Table 1. The group-contribution concept was then applied for one parameter at a time (a, b, c), while the others were kept at their calculated value. For each parameter the contributions of all groups k were regressed simultaneously by minimizing the objective function:
obf )
x
NP
∑ i)1
(
)
Fcalc - Fexp Fexp NP
2
× 100
(28)
where F is the reference density and NP is the number of points (i.e., of compounds). In Table 2 the values of pure-component parameters, calculated by group contributions (GC), are compared with those obtained directly by experimental data. In Table 2 also the value of the objective function for the regression of each pure
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2115
Figure 2. Linearity of parameters (a) b (cm3/mol), (b) c, (c) ac/Nc × 10-6 (cm6‚bar/mol2), and (d) RNc versus the number of carbons for an alkane series from pentane to hexadecane. Table 4. Group-Contribution Table for the Groups Considered
Table 5. Splitting of Groups in the Heavy Molecular Weight Compounds Considered compound
group
ac/Nc [10-6 cm6‚bar/mol2]
(RNc)293
b [cm3/mol]
c
CH3-CH2>CH>C< -COO
1.595 0.348 -1.128 -2.683 1.766
0.462 1.793 3.093 4.406 2.378
19.937 14.749 10.038 5.561 19.735
0.672 0.117 -0.602 -1.385 0.506
compound parameter is reported. The values of group contributions are given in Table 4. As already outlined, ac, b, and c are all temperatureindependent while R is dependent of temperature. Therefore, also parameter RNc is a function of temperature. This linear dependency is plotted in Figure 3a for n-alkanes. In the temperature range considered, the slope of each compound plotted versus the value of RNc gives a straight line (Figure 3b). This temperature dependency was then expressed by the following equation:
(RNc)T ) (RNc)293 - (Sb + I)(293 - T)
(29)
where S and I are constants while (RNc)293 and b are estimated with group contributions at the reference conditions. Nineteen compounds with density and vapor pressure data at temperatures from 293 to 363 K were used to fit the constants to the values S ) -0.108 mmol/ cm3‚K and I ) 8.76 K-1.
hexadecane pentaerythritoltetrapentanoate pentaerythritoltetranonanoate pentaerythritoltetraethylhexanoate pentaerythritoltetraethylbutanoate
CH3- CH2- CH- C- COO2 4 4 8 8
14 16 32 12 20
4 4
1 1 1 1
4 4 4 4
The method presently developed was checked on liquid density data for hexadecane and four different POEs at temperatures from 298 to 363 K. Table 5 shows the splitting of groups of these components. In Table 6 relative deviations between calculated and experimental values are given at each temperature, together with the root-mean-square deviation (RMSD). Figure 4 shows a comparison between experimental and predicted densities. Note that the group-contribution approach has not been applied to HFCs. VLE Calculation Results To show the ability of our model to perform binary VLE calculations, we have compared results obtained for the system R134a-hexadecane, in terms of RMSD on boiling pressure, with different EOS models (the SP EOS and RKS EOS)20 and some activity coefficient models (Flory-Huggins,25 Wilson,25 NRTL,25 Uniquac,25 and Heil).26 Calculations for all models, except the SP
2116 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 Table 6. Density (G [g/cm3]) and Relative Deviation between Calculated and Experimental Density at Various Temperatures [K] for Hexadecane (Daubert and Danner19) and for Pentaerythritol Esters (Wahlstro1 m and Vamling24) hexadecane
pentaerythritol tetrapentanoate
pentaerythritol tetranonanoate
pentaerythritol tetraethylhexanoate
T [K]
Fexp
(∆F/F) %
Fexp
(∆F/F) %
Fexp
(∆F/F) %
298 303 308 314 318 323 328 333 338 343 348 353 359 363 368 RMS (∆F/F) %
0.770 0.776 0.763 0.759 0.757 0.753 0.750 0.746 0.743 0.740 0.736 0.733 0.729 0.726 0.722
-0.9 -0.89 -0.87 -0.85 -0.83 -0.82 -0.81 -0.81 -0.80 -0.79 -0.78 -0.75 -0.75 -0.74 -0.73 0.81
1.015 1.011 1.007 1.002 0.999 0.995 0.991 0.988 0.983 0.980 0.976 0.973 0.968 0.964 0.960
0.41 0.48 0.54 0.66 0.70 0.77 0.84 0.85 0.98 1.00 1.03 1.04 1.12 1.16 1.23 0.89
0.951 0.948 0.944
0.21 0.14 0.10
0.936 0.933 0.929
0.17 0.22 0.27
0.977 0.974 0.971 0.967
1.31 1.38 1.45 1.52
0.922 0.919 0.917
0.37 0.36 0.32
0.960 0.957 0.954
1.60 1.64 1.64
0.909 0.906 0.903
0.39 0.44 0.46 0.31
0.946 0.943 0.940
1.70 1.77 1.75 1.58
Fexp
(∆F/F) %
pentaerythritol tetraethylbutanoate Fexp
(∆F/F) %
0.960 0.958 0.955 0.950 0.947 0.943 0.940 0.936 0.933 0.930 0.927 0.924 0.919 0.917 0.914
0.45 0.48 0.53 0.70 0.74 0.83 0.86 0.95 0.99 1.01 1.06 1.08 1.10 1.10 1.15 0.90
Figure 4. Comparison between experimental (symbols) and predicted (s) density at various temperatures for pentaerythritol ester and n-C16 (data from Wahlstro¨m and Vamling24). Table 7. Comparison of Different Thermodynamic Models for the R134A-Hexadecane Mixture: Number of Fitting Parameters and Values of Average (∆P/P) % between Calculated and Experimental Pressure model
no. of param
average (∆P/P) %
Flory-Hugginsa Wilsona Heila NRTLa Uniquaca RKS EOS SP EOS + GC
1 2 2 3 2 1 2
5.2 13.0 10.2 9.7 3.8 5.2 2.5
a
Figure 3. (a) Dependence of parameter RNc on temperature (heptane to hexadecane), (b) slope of the temperature dependency as a function of RNc,293 (heptane to hexadecane).
EOS, were done in a previous work, for the same data, by Wahlstro¨m and Vamling.27 It is clear from Table 7 that the SP EOS is able to describe the system much better than other models. A more difficult task is to represent VLE of refrigerants and POEs. Data of this kind, where the structure of the heavy compound is known, are scarce in the open literature; to check our model, we have used some new
Activity coefficient models.
data between HFCs and POEs (Wahlstro¨m and Vamling24). The statistical deviations of the regression for the systems considered are summarized in Table 8 together with the values of interaction parameters. In Figure 5, VLE calculations are reported for mixtures containing R134a and POE (MW 550) over a wide temperature range: the SP EOS with group contributions is able to describe correctly the temperature dependence of this system. In this case, Takaishi and Oguchi28 made measurements of R134a mixed with a synthetic lubricant, consisting of POE (MW 600) with a structure similar to the one of Wahlstro¨m and Vamling. Using the regressed interaction parameters for the previous R134a-POE mixture, the data from Takaishi
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2117 Table 8. Binary VLE Database and Summary of Fitting Results for the Systems Considered system
ref
T [K]
P [bar]
aij [(bar‚cm3)/(mol‚K)]
bij [(bar‚cm3)/(mol‚K2)]
R152a-POE R143a-POE R134a-POE R134a-POEb
24 24 21 23
303-363 303-343 303-363 303
0.5-9.0 0.8-16.9 0.5-10.6 2.5-7.6
0.156 × 10-1 0.272 × 10-1 0.204 × 10-1 0.102 × 10-1
0.675 × 10-4 0.986 × 10-4 0.101 × 10-3 0.501 × 10-4
a
(∆P/P) % average biasa 2.59 2.84 2.87 3.45
0.37 0.40 0.46 2.29
The bias is the arithmetic mean of residuals. b Predictive calculation performed for this system.
Figure 5. System R134a-POE. Comparison between bubble point calculated (s) and experimental data (symbols) at four temperatures. Data from Wahlstro¨m and Vamling.24
Figure 6. System R134a-POE. Comparison between bubble point calculated (s) and experimental data of two different data sets in a whole composition range at 303 K. Data from (b) Wahlstro¨m and Vamling24 and (O) Takaishi and Oguchi.28
Oguchi,28
and which cover the other part of the composition range, can be predicted: see Figure 6 for details. Also for the systems R152a-POE (Figure 7) and R143a-POE (Figure 8), a satisfactory correlation of experimental data is achieved. Conclusions An extension of the cubic equation of state by Sako et al.1 was proposed for calculating vapor-liquid equilibria of mixtures between HFCs and lubricant oils such as POEs. Critical constants and acentric factors were used for the determination of the EOS-specific parameters of HFCs; a group-contribution method was developed for the parameters of the high molecular weight compounds. The extension to mixtures was done by applying Huron-Vidal mixing rules at infinite pressure, and the
Figure 7. System R152a-POE. Comparison between bubble point calculated (s) and experimental data (symbols) at four temperatures. Data from Wahlstro¨m and Vamling.24
Figure 8. System R143a-POE. Comparison between bubble point calculated (s) and experimental data (symbols) at three temperatures. Experimental data from Wahlstro¨m and Vamling.21
infinite-pressure activity coefficients were calculated by a modified Uniquac model. For pure POEs the proposed model was able to predict the liquid density with good accuracy on the basis of the molecular structure only. The model was tested on binary mixtures between HFCs and POEs. In all of the cases considered, a satisfactory correlation of vapor-liquid equilibrium data at different temperatures was obtained; when applied to data on the whole composition range, the model is able to predict the behavior at high oil concentration. Acknowledgment This work was partially supported by the Italian Ministero dell’Universita` e della Ricerca Scientifica e
2118 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
Tecnologica (40%). The financial support from the Swedish National Board for Industrial and Technical Development (NUTEK) is gratefully acknowledged. Literature Cited (1) Sako, T.; Wu, A. H.; Prausnitz, J. M. A Cubic Equation of State for High-Pressure Phase Equilibria of Mixture Containing Polymers and Volatile Fluids. J. Appl. Polym. Sci. 1989, 38, 1839. (2) Thomas, R. H. P.; Pham, H. T. Solubility and Miscibility of Environmentally Safer Refrigerant/Lubricant Mixtures. ASHRAE Trans. 1992, 98, 783. (3) Martz, W. L.; Burton, C. M.; Jacobi, A. M. Local Composition Modelling of the Thermodynamic Properties of Refrigerant and Oil Mixtures. Int. J. Refrig. 1996, 19, 25. (4) Yokozeki, M. A. Solubility and Viscosity of RefrigerantOil Mixtures. In Proceedings of the International Compressor Engineering Conference, Soedel, W., Ed.; Purdue University, West Lafayette, IN, July 19-22, 1994; Vol. 1, p 335. (5) Panayiotou, C.; Vera, J. H. An Improved Lattice-fluid Equation of State for Pure Component Polymeric Fluids. Polym. Eng. Sci. 1982, 22, 345. (6) Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of r-Mer Fluids and Their Mixtures. Polym. J. 1982, 14, 681. (7) Lee, B. C.; Danner, R. P. Prediction of Polymer-Solvent Phase Equilibria by a Modified Group-Contribution EOS. AIChE J. 1996, 42, 837. (8) Chapman, W.; Gubbins, K.; Jakson, G.; Radoz, M. SAFT: Equation of State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. (9) Chapman, W.; Gubbins, K.; Jakson, G.; Radoz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (10) Beret, S.; Prausnitz, J. M. Perturbed Hard-Chain Theory: An Equation of State for Fluids Containing Small or Large Molecules. AIChE J. 1975, 21, 1123. (11) Song, Y.; Lambert, S. M.; Prausnitz, J. M. Chem. Eng. Sci. 1994, 47, 2765. (12) Hino, T.; Prausnitz, J. M. A Perturbed Hard-Sphere-Chain Equation of State for Normal Fluids and Polymers Using the Square Well Potential of Variable Width. Fluid Phase Equilib. 1997, 138, 105. (13) Kontogeorgis, G.; Harismiadis, V. I.; Fredenslund, A.; Tassios, D. Application of the van der Waals Equation of State to Polymer. I. Correlation. II. Prediction. Fluid Phase Equilib. 1994, 96, 65. (14) Bertucco, A.; Mio, C. Prediction of Vapor-Liquid Equilibrium for Polymer Solutions by a Group-Contribution RedlichKwong-Soave Equation of State. Fluid Phase Equilib. 1996, 117, 18.
(15) Huron, M. J.; Vidal, J. New Mixing Rules in Simple Equations of State for Representing Vapour-Liquid Equilibria of Strongly Non-Ideal Mixtures. Fluid Phase Equilib. 1979, 3, 255. (16) Prigogine, I. The Molecular Theory of Solution; NorthHolland: Amsterdam, The Netherlands, 1957. (17) Abrams, D. S.; Prausnitz, J. P. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible System. AIChE J. 1975, 21, 116. (18) Gmehling, J.; Li, J.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. (19) Daubert, T. E.; Danner, R. P. Physical and thermodynamic Properties of pure Compounds: Data Compilation; Taylor & Francis: New York, 1993. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids; 4th ed.; McGraw-Hill Book Co.: New York, 1987. (21) Peneloux, A.; Rauzy, E.; Freze R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7. (22) McLinden, M. O. Thermodynamic Properties of CFC Alternatives: A Survey of the Available Data. Int. J. Refrig. 1990, 13, 149. (23) Elvassore, N. A Cubic EOS for the Prediction of VLE of Refrigerant Mixtures (in Italian). Chemical Engineering Thesis, Istituto di Impianti Chimici, Universita` di Padova, Padova, Italy, 1995. (24) Wahlstro¨m, A° .; Vamling, L. The Solubility of HFC32, HFC125, HFC134a, HFC143a and HFC152a in a Pentaerythritol Tetrapentanoate Ester. To be submitted to Int. J. Chem. Eng. Data 1998. (25) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall Inc.: Englewood Cliffs, NJ, 1986. (26) Martz, W. L.; Burton, C. M.; Jacobi, A. M. Local Composition Modelling of the Thermodynamic Properties of Refrigerant and Oil Mixtures. Int. J. Refrig. 1996, 19, 25. (27) Wahlstro¨m, A° .; Vamling, L. Prediction of Solubility for HFC Working Fluids in Model Substances for Compressor Oils. Can. J. Chem. Eng. 1997, 75, 551. (28) Takaishi, Y.; Oguchi, K. Solubility of the Solutions of HFC134a and Polyolester Based Oil. Proceedings of the IIR Conference, May 12-14, 1993, Gent, Belgium; Comm B1/2, 141.
Received for review September 4, 1998 Revised manuscript received February 10, 1999 Accepted February 11, 1999 IE980570W
