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Group-contribution equation of state for correlating and predicting thermodynamic properties of weakly polar and non-associating mixtures
Fluid Phase Equilibria, 68 (1991) 47-102 Elsevier Science Publishers B.V., Amsterdam
47
Group-contribution equation of state for correlating and predicting thermodynamic properties of weakly polar and non-associating mixtures. Binary and multicomponent systems W. Abdoul, E. Rauzy and A. PCneloux Laboratoire de Chimie Physique, Faculte’ des Sciences de Luminy, 13288 Marseille G5de.x 9 (France) (Received November 14, 1990; accepted in final form June 12, 1991)
ABSTRACT Abdoul, W., Rauzy, E. and PCneloux, A., 1991. Group-contribution equation of state for correlating and predicting thermodynamic properties of weakly polar and non-associating mixtures. Binary and multicomponent systems. Fluid Phase Equilibria, 68: 47-102. The applicability of the “Guggenheim quasi-lattice excess function equation of state” formalism is tested through a zeroth approximation group-contribution model and a Peng-Robinson-type equation of state. The temperature-dependent correlations of the group-contribution model are derived from low-pressure excess enthalpy (iYE) and vaporliquid equilibrium (VLE) binary data and are then extended to the prediction of high-pressure vapor-liquid equilibria of mixtures of weakly polar and non-associating compounds (i.e. hydrocarbons, CO,, H,S, N,). The numbers of binary systems, data sets and determinations used in the development and testing of the method are 169, 805 and 10634 for low-pressure VLE, 282, 789 and 11308 for HE; and 163, 1329 and 12272 for high-pressure VLE, respectively. The average relative deviation with experiment in bubble-point pressure is 1% and the average absolute deviation in vapor-phase composition is 0.005 for low-pressure VLE data. Heat effects are well reproduced with average deviations of 12 J for 7009 determinations of binaries with HE less than 300 J and 5.5% for 4299 determinations of binaries with higher HE. Owing to a term accounting for chain-length effects, the accuracy of the prediction of high-pressure VLE is not deteriorated by the group-contribution approach (average deviations of 3% in bubble-point pressure, 0.0103 in vapor-phase composition). Thus, in addition to its predictive power, the method is found to be an improvement over the classical Peng-Robinson procedure with constant binary interaction parameters, especially for mixtures of molecules of different sizes.
INTRODUCTION
The most common way to describe high-pressure fluid phase equilibria is from equations of state. The works of Vidal (19781, Huron and Vidal 0378-3812/91/$03.50
0 1991 Elsevier Science Publishers B.V. All rights reserved
48
(1979), Mollerup (1981) and Whiting and Prausnitz (1982) have shown the connection between equation of state (EOS) mixing rules and excess functions. As examples, several pairs may be mentioned: Redlich-KwongSoave EOS + NRTL (Huron and Vidal, 1979), Hole Theory EOS + NRTL (Takishima and Saito, 1985), Ishikawa-Chung-Lu, Redlich-Kwong-Soave, Peng-Robinson, Martin EOS + ASOG (Tochigi and Kojima, 1984; Tochigi et al., 19851, Simonet-Behar-Rauzy EOS + UNIQUAC (Lermite and Vidal, 1986), van der Waals EOS + UNIFAC (Gupte et al., 1986) and Perturbed Soft Chain Theory EOS + a group-contribution model (Jin et al., 1986). In a preliminary paper (Peneloux et al., 1989) we have shown such connections between the zeroth approximation of Guggenheim’s quasilattice model and various equations of state. Here, we apply the principles we reported previously to a specific excess function model and equation of state, namely a zeroth approximation group-contribution model coupled with a translated Peng-Robinson-like (PR) equation of state (PCneloux et al., 1982; Rauzy, 1982). Many temperature-dependent correlations included in the method stem empirically from correlating low-pressure vapor-liquid equilibrium and excess enthalpy data and are then extended to high-pressure vapor-liquid equilibria. The range of applicability of this method is restricted to mixtures of weakly polar and non-associating compounds such as hydrocarbons, hydrogen sulfide, carbon dioxide and nitrogen.
DESCRIPTION OF THE MODEL
As was reported previously (Peneloux et al., 1989) the model used is characterized by the two following conditions. (1) The pure-component Helmholtz energies are calculated from equations of state. (2) The excess functions are defined at constant packing fraction, the latter described by v’/u, u” being the close-packed volume. We assume that it is possible to define a covolume bi proportional to u”, which enables us to measure the packing fraction by 77= b/u. We choose the same packing fraction for pure components as for the mixture. Those assumptions lead us to write the molar Helmholtz energy of a mixture as follows: A(T, q, n) =A*(T,
7, n) - nRT ln(1 - 77)- 2 (niai/bi)*l(v)
+ ml:,
i=l (1)
The various working equations which are used in conjunction (1) are reported in a previous paper (PCneloux et al., 1989).
with eqn.
49
Pure-component equations of state
For the components’ equation of state we have chosen a translated Peng-Robinson cubic equation of the form P=RT/(u-b)-a(T)/(u2+ubu+wb2)
(2)
Harmens and Knapp (1980), Schmidt and Wenzel (1980) and Pate1 and Teja (1981) proposed correlations for the parameters u, w, a and b for components obeying the law of corresponding states. We use this equation according to a technique which was originally applied to the RedlichKwong-Soave equation (RKS), the general principles of which were given in previous papers (PCneloux et al., 1982; Rauzy, 1982; Rauzy and PCneloux, 1986). In particular, it was observed that a translation in linear dependence on component mole fraction along the volume axis leaves the results T, P, x of vapor-liquid equilibrium calculations unchanged. Thus it is possible to express eqn. (2) in terms of a translated volume 5, the pseudo-volume: P = RT/( fi - 6) - a(T)/fi(
i; + 76)
We call this the basic form of eqn. (21, which is defined by a parameter y which is characteristic of the equation of state under consideration (e.g. y=OforvdW,y=lforRKS,y = 4.82843 for PR). In this work y = 4.82843 is retained, i.e. we use the translated basic form of the Peng-Robinson (1976) EOS. It is well known that a good correlation of the pure-component vapor pressures is a requirement for accurate prediction of mixture vapor-liquid equilibria. The success obtained with the Soave (1972) equation of state is a good example. For low reduced temperatures, however, Soave’s function a(T) leads to relative deviations with respect to experimental vapor pressures well above 1%. These shortcomings may be overcome in two ways. (1) a(T) is adjusted at each temperature to reproduce pure-component vapor pressure well (Goral et al., 1981). The value obtained is then used in mixing rules for the description of mixtures. A major drawback with this procedure is that it does not allow excess enthalpy calculations. (2) A temperature-dependent correlation is designed, the parameters of which are fitted to provide good agreement with experimental pure-component vapor pressure data (Vidal, 1983; Gibbons and Laughton, 1984). Since we are concerned with low-pressure vapor-liquid equilibrium and excess enthalpy calculations, we are led to adopt the latter procedure and to define two classes of compounds. (1) For components which are likely to be encountered at very low pressures in our calculations, we resort to the Carrier-Rogalski-PCneloux correlation for a(T) (Carrier et al., 1988; Appendix A(a)>. The properties
50
of those components are defined by their critical temperature and pressure, their normal boiling temperature and a fourth parameter m fitted to the experimental pure-component vapor-pressure data. (2) For the remaining compounds, a(T) is still calculated by the procedure defined by Rauzy (1982) which makes use of a Soave-like expression (see Appendix A(b)). A list of the compounds considered in this study and the parameters defining their properties are given in Table 1. Residual excess Helmholtz energy A:,, For AFe, we choose a function in which the composition fraction variables are separated %7) = ln(l + Yrl)/Y to be a sum of two terms:
4% = E(T, -++7) E(T, X) is postulated E = (l/2)
and packing (4
i i x~&x~~~,!Z~~(T)/~+ (l/2) i i xixjEchij(T) ix1 j=l i-1 j=l
(5)
The first term on the right-hand side of eqn. (5), a Van Laar-like excess function, is related to the classical van der Waals mixing rules linear in b and quadratic in a: b = i xibi i=l
U= ~ i=l
~
XiXj(
aiaj)1’2(
1-
kij) (6)
j=l
TABLE l(a) Parameters of the compounds of class 2, defined by T,, PC, w Compound
MW
T, (K)
PC(bar)
o
Tb (K)
126.26 190.55 282.40 305.43 304.21 373.20 364.85 369.82 408.14 417.90 419.60 425.37 425.16 647.37
33.97 46.01 50.39 48.75 73.79 89.40 46.01 42.68 36.55 40.02 40.23 43.27 37.96 221.20
0.0400 0.0111 0.0850 0.0970 0.2250 0.1090 0.1440 0.1536 0.1846 0.1900 0.1870 0.1950 0.2008 0.3440
77.40 111.64 169.44 184.57 194.70 212.80 225.45 231.08 261.42 266.25 266.89 268.74 272.65 373.15
(g mol-‘I
Nitrogen Methane Ethylene Ethane Carbon dioxide Hydrogen sulfide Propylene Propane Isobutane Isobutene 1-Butene 1,3-Butadiene Butane Water
28.01 16.04 28.05 30.07 44.01 34.08 42.08 44.10 58.12 56.11 56.11 54.09 58.12 18.02
51 TABLE l(b) Parameters of the compounds of class 1, defined by Compound
Neopentane 3-Methyl-1-butene Isopentane I-Pentene 2-Methyl-1-butene Isoprene Pentane 2-Methyl-2-butene trans-13Pentadiene Cyclopentane 2,ZDimethylbutane 4-Methyl-1-pentene 2,3_Dimethylbutane 1,SHexadiene 2-Methylpentane 2-Methyl-1-pentene 3-Methylpentane 1-Hexene truns-3-Hexene trans-2-Hexene Hexane cis-2-Hexene Methylcyclopentane 2,ZDimethylpentane Benzene 2,4-Dimethylpentane Cyclohexane 2,2,3_Trimethylbutane Cyclohexene 2,3_Dimethylpentane 3-Methylhexane 1-Heptene Heptane Isooctane Methylcyclohexane 2,SDimethylhexane 2,CDimethylhexane Toluene 2,3,4_Trimethylpentane 4-Methylheptane 3,4_Dimethylhexane Cycloheptane truw1,4,-Dimethylcyclohexane 1-Octene
T,,PC,Tbrm
MW (g mol-‘I
T,
72.15 70.14 72.15 70.14 70.14 68.12 72.15 70.14 68.12 70.14 86.18 84.16 86.18 82.15 86.18 84.16 86.18 84.16 84.16 84.16 86.18 84.16 84.16 100.21 78.11 100.21 84.16 100.21 82.15 100.21 100.21 98.19 100.21 114.23 98.19 114.23 114.23 92.14 114.23 114.23 114.23 98.19
433.80 450.00 460.43 464.70 465.00 484.00 469.64 470.00 496.00 511.60 488.70 518.00 499.90 507.00 497.50 518.00 504.40 504.00 519.90 516.00 507.35 518.00 532.70 520.48 562.10 519.70 553.40 531.10 560.40 537.30 535.20 537.20 540.15 544.10 572.10 550.06 553.50 591.70 566.43 561.70 568.87 606.20
32.02 35.16 33.80 40.53 34.45 38.50 33.72 34.45 39.92 45.09 30.80 32.42 31.31 34.45 30.09 32.42 31.21 31.71 32.53 32.73 30.10 32.83 37.90 27.76 48.94 27.36 40.73 29.59 43.47 29.08 28.17 28.37 27.34 25.74 34.75 24.87 25.60 41.14 27.26 25.43 26.95 38.10
282.65 293.21 301.00 303.12 304.31 307.22 309.21 311.72 315,18 322.41 322.89 327.01 331.14 332.61 333.42 335.26 336.43 336.63 340.24 341.03 341.89 342.04 344.96 352.35 353.24 353.65 353.89 354.03 356.13 362.93 365.00 366.79 371.57 372.39 374.08 382.25 382.58 383.78 386.62 390.86 390.87 391.94
0.50083 0.49832 0.51808 0.48665 0.54064 0.52250 0.53980 0.54010 0.53054 0.48976 0.52535 0.55300 0.54034 0.51763 0.57152 0.57550 0.56150 0.57182 0.58384 0.58145 0.59178 0.58121 0.52655 0.58476 0.51501 0.60088 0.51738 0.54581 0.50681 0.58880 0.61366 0.62259 0.64167 0.59916 0.53508 0.65206 0.64105 0.55087 0.60796 0.66519 0.63125 0.53757
0.23673 0.23552 0.25585 0.22311 0.25713 0.24487 0.26893 0.26678 0.24977 0.23749 0.31541 0.32196 0.32653 0.30654 0.33887 0.33142 0.33602 0.33153 0.33730 0.33435 0.35461 0.33571 0.30768 0.40141 0.26661 0.40760 0.30795 0.38927 0.29677 0.41048 0.42296 0.42416 0.44616 0.47327 0.38601 0.50662 0.49763 0.35488 0.48742 0.51935 0.50011 0.39592
112.20 112.22
587.70 566.60
29.69 26.24
392.50 394.43
0.56789 0.66458
0.47634 0.5 1506
(K) PC(bar)
Tb
(K) rn
a0x10-s (cm6 bar mol-‘)
52 TABLE l(b) (continued) Compound
2,2,4,4_Tetramethylpentane 2,2,5Trimethylhexane Octane 2,2,4_Trimethylhexane Ethylcyclohexane 2,ZDimethylheptane 2,4-Dimethylheptane 2,2,3,4_Tetramethylpentane 4,4_Dimethylheptane 2,6_Dimethylheptane 2J-Dimethylheptane 3,5-Dimethylheptane Ethylbenzene 3,3_Dimethylheptane p-Xylene 2,3,4_Trimethylhexane m-Xylene 2,2,3,3_Tetramethylpentane 2,3_Dimethylheptane 3,4_Dimethylheptane 4-Ethylheptane 2,3,3,4_Tetramethylpentane 4-Methyloctane 3-Ethylheptane 2-Methyloctane 3-Methyloctane o-Xylene Styrene 3,3-Diethylpentane Nonane Cyclooctane Isopropylbenzene Isopropylcyclohexane Propylcyclohexane Propylbenzene 2,6_Dimethyloctane 1,3,5Trimethylbenzene wMethylstyrene tert-butylbenzene 1,2,4_Trimethylbenzene Decane 1,2,3_Trimethylbenzene Butylbenzene tram-Decalin ck-Decalin
MW (g mol-‘)
128.26 128.26 114.23 128.26 112.22 128.26 128.26 128.26 . 128.26 128.26 128.26 128.26 106.17 128.26 106.17 128.26 106.17 128.26 128.26 128.26 128.26 128.26 128.26 128.26 128.26 128.26 106.17 104.15 128.26 128.26 112.22 120.20 126.24 126.24 120.20 142.29 120.20 118.18 134.22 120.20 142.29 120.20 134.22 138.25 138.25
T, (K)
PC(bar)
Tb (K) m
uo x10-s km6 bar mol-‘)
574.70 568.00 568.76 574.20 609.00 576.80 581.00 592.70 591.00 576.60 581.00 585.50 617.10 591.00 616.20 597.40 617.00 607.70 591.40 595.80 594.30 607.70 589.80 594.30 587.00 589.80 630.20 647.00 616.00 593.60 647.20 631.00 640.00 639.00 638.30 604.90 637.30 654.00 660.00 649.10 617.65 664.50 660.50 690.00 702.20
24.85 23.30 24.87 23.70 30.30 23.50 23.00 26.02 24.50 22.10 23.00 23.90 36.07 24.50 35.16 25.80 35.46 27.41 24.10 25.10 24.30 27.16 23.40 24.30 23.10 23.40 37.30 39.92 27.30 22.90 35.64 32.12 28.37 28.07 32.02 21.00 31.31 34.05 29.69 32.32 20.95 34.55 28.88 31.41 31.41
395.43 397.23 398.82 399.69 404.93 405.84 406.04 406.17 408.35 408.36 409.15 409.15 409.35 410.16 411.51 412.19 412.27 413.42 413.65 413.75 414.35 414.70 415.57 416.15 416.41 417.36 417.58 418.29 419.32 423.97 424.29 425.56 427.91 429.87 432.39 433.53 437.89 438.65 442.27 442.53 447.30 449.27 456.42 460.42 468.92
0.60910 0.66619 0.68941 0.64194 0.59287 0.66903 0.67894 0.60997 0.66903 0.67894 0.67894 0.67894 0.58996 0.66903 0.59385 0.65111 0.59718 0.59759 0.67894 0.67894 0.70630 0.60465 0.70630 0.70630 0.70630 0.70630 0.59103 0.58810 0.60863 0.73343 0.56110 0.62319 0.61688 0.63506 0.62532 0.72159 0.65731 0.64815 0.65160 0.64105 0.78143 0.62724 0.67120 0.58208 0.59683
0.54702 0.57443 0.54738 0.57573 0.50032 0.59246 0.60713 0.55731 0.58821 0.62779 0.61257 0.59717 0.44257 0.59126 0.45441 0.57477 0.45259 0.55627 0.60571 0.58936 0.60521 0.56255 0.62269 0.60828 0.62840 0.62583 0.44816 0.43357 0.57388 0.65244 0.48512 0.51998 0.58749 0.59519 0.53624 0.73088 0.55376 0.52856 0.60311 0.55426 0.77291 0.54396 0.64064 0.62573 0.64862
53 TABLE l(b) (continued) Compound
Undecane Pentylbenzene Tetralin Dodecane Naphthalene Hexylbenzene Tridecane 2-Methylnaphthalene I-Methylnaphthalene Tetradecane Biphenyl Diphenylmethane Pentadecane 1-Hexadecene Hexadecane Heptadecane 1-Octadecene Octadecane Eicosane
MW (g mol-‘)
T, (K)
156.31 148.25 132.21 170.33 128.17 162.28 184.37 142.20 142.20 198.38 154.21 168.24 212.42 224.43 226.43 240.47 252.49 254.48 282.55
640.66 679.90 719.00 660.54 748.43 700.20 675.80 761.00 772.00 694.72 789.26 770.00 707.00 717.00 722.92 733.37 739.00 746.80 767.78
PC (bar)
Tb (K)
m
a0 x 10-8 (cm6 bar mol-‘)
19.68 26.04 35.16 18.38 40.53 23.81 17.23 35.06 35.60 16.24 38.46 28.60 15.20 13.37 14.54 13.17 11.35 13.16 12.02
469.08 478.55 480.51 489.47 491.09 499.25 508.62 514.20 517.83 526.73 528.15 537.42 543.83 558.02 560.01 575.17 587.97 589.45 616.95
0.82141 0.72100 0.61149 0.86161 0.58011 0.76248 0.90192 0.63057 0.63183 0.93844 0.64847 0.72996 0.98129 1.03604 1.01548 1.05220 1.05000 1.08586 1.15816
0.88703 0.75613 0.61919 1.01180 0.58272 0.87710 1.13780 0.70425 0.70944 1.27310 0.69284 0.88803 1.41600 1.63930 1.54810 1.75240 2.04310 1.83240 2.12970
kij being the binary parameters interaction, and the first term on the r.h.s. of eqn. (5) was shown to be strictly equivalent to these if the second term were zero and Eij was defined by Eij = (Si -
si)”+ 26iSjkij
(7)
Moreover, since that term is Van Laar-like, we use, as did Kehiaian et al. (19711, the equation of Redlich et al. (1959) to express Eij in terms of group contributions E,j = ( -
l/2) f
2 (aiK - ‘yjK)(“iL- “jL)AKLCT)
K=l L=l
(8)
The summation is over all N groups present in solution, aiK is the fraction of molecule i occupied by group K and has the feature of site fraction of the lattice model, but relating to a group, and A,, is a temperature-dependent function. The second term on the right-hand side of eqn. (5) accounts for the differences in the conformation of the molecules. It is a universal temperature-dependent function determined to correlate the heat effects arising from mixing of globular molecules with long-chain molecules. This term is
54 TABLE 2 Parameters
of conformation
for substituted
alkanes a
Compound
lj
licalc
Compound
1,
licalc
Neopentane Isopentane 2,ZDimethylbutane 2,3_Dimethylbutane 2-Methylpentane 3-Methylpentane 2,2_Dimethylpentane 2,CDimethylpentane 2,2,3_Trimethylpentane 2,3_Dimethylpentane 3-Methylhexane Isooctane 2J-Dimethylhexane 2,4-Dimethylhexane 2,3,4_Trimethylpentane 4-Methylheptane 3,4-Dimethylhexane 2,2,4,4_Tetramethylpentane 2,2,5_Trimethylhexane 2,2,4_Trimethylhexane
2.04 4.00 4.28 4.59 4.32 5.11 4.38 2.68 6.09 5.72 5.44 4.48 3.43 3.56 8.99 5.68 4.00 4.82 0.75 5.47
2.04 3.78 4.28 4.59 4.32 5.11 4.38 2.68 6.09 5.72 5.44 3.00 3.43 3.56 6.03 5.68 6.54 5.52 3.13 4.61
2,ZDimethylheptane 2,CDimethylheptane 2,2,3,4_Tetramethylpentane 4,4-Dimethylheptane 2,6_Dimethylheptane 2J-Dimethylheptane 3,5-Dimethylheptane 3,3-Dimethylheptane 2,3,4_Trimethylhexane 2,2,3,4_Tetramethylpentane 2,3_Dimethylheptane 3,CDimethylheptane 4-Ethylheptane 2,3,3,4_Tetramethylpentane 4-Methyloctane 3-Ethylheptane 2-Methyloctane 3-Methyloctane 3,3_Diethylpentane 2,6_Dimethyloctane
4.71 4.67 9.23 7.58 4.89 4.10 4.64 8.49 10.52 12.96 6.87 6.25 6.34 14.39 7.41 6.34 6.87 7.20 12.84 6.09
5.24 5.61 8.01 i.49 5.64 5.61 5.70 8.49 7.45 12.96 7.38 7.35 7.50 11.42 7.41 7.50 6.94 7.41 10.43 6.09
a I,, value used in our calculations; Appendix B).
licak, value estimated
from correlation
found to contribute to the description of vapor-liquid tures of light alkanes with higher ones as well. EChij
=
Eo(Zi-
Zj)‘[
(Bl) (see
equilibria in mix-
( To/T)4 - 2(‘/T,)]
where we have set E, = 2.95 and TO= 298.15 K. The Zi values may be regarded either as parameters of conformation for substituted alkanes and non-alkane hydrocarbons or as chain-length parameters for straight-chain alkanes (in fact they are the numbers of carbon atoms in these). A formula using the compound critical properties is provided (see Appendix B) for the estimation of the parameters Zi for alkanes. In some instances, the parameters obtained in such a way lead to poor results, in which case fitted parameters should be favored. Table 2 compares the parameters Zi actually used in this method with those estimated by the correlation of Appendix B. Table 3 is a collection of all such parameters used in this work.
55
Excess enthalpy
The departure H-H*
enthalpy being given by
= {a[(A -A*)/T]/~(l/T)},,~
+RT(z
- 1)
(10)
we obtain the following for the residual enthalpy: HrFS= [QL/WW/T)l
X=
w/ww~)1
x 141+ Y?7)
(11)
As previously and according to eqn. (5), the binary enthalpy parameter defined by Hij=(-1/2)
E K-l
It L=l
(CYiK-(YjK)(CYiL-LYjL)BKL(T)
is
(12)
BKL is a temperature-dependent
function. A,, (eqn. (8)) and BKL are related to two group interaction parameters, AiL and BiL, per pair of groups corresponding to a reference temperature T,, taken as 298.15 K: A KL = AiL( TO/T)‘“”
AiL = A0LK
B KL = B;,( To/T)‘“”
BgL = BjK
TKL =
B;,/Ao,,
- 1
(134 uw
(134
The parameters B& are essential in excess enthalpy calculations and are therefore determined by that quantity. Such a formulation avoids having to consider a third set of group contributions to describe the the estimation of which would be changes of HE with temperature, difficult given the current shortage of excess heat capacity data. The binary conformation enthalpy is defined by Hchij = 5Eo(Ii - Zj)‘( To/T)4
(14)
Definition of the groups and calculation of their surface fractions
In order to apply the group-contribution formulae (eqns. (8) and (12)), we have to define the nature of the groups and calculate their surface fractions or site fractions. Nature of the groups
It is found that the following groups must be considered to correlate the mixtures of the compounds we are mainly concerned with (i.e. hydrocarbons, carbon dioxide, hydrogen sulfide, nitrogen and water): 1 or Al : alkane, end of chain (CH,-) 2 or A2 : alkane, body of chain (-CH,-)
56 TABLES 3(a) and 3(b) Numbers of sites of each kind of group (numbered from 1 to 17; see text) and parameters of conformation characterizing the different compounds studied a Compound
1
2
3
4
5
6
Propylene Propane Isobutane Isobutene 1-Butene 13Butadiene Butane Neopentane 3-Methyl-1-butene Isopentane 1-Pentene 2-Methyl-1-butene Isoprene Pentane 2-Methyl-Zbutene trans-1,3-Pentadiene Cyclopentane 2,ZDimethylbutane 4-Methyl-1-pentene 2,3-Dimethylbutane 1,5-Hexadiene 2-Methylpentane 2-Methyl-1-pentene 3-Methylpentane 1-Hexene truns-3-Hexene truns-2-Hexene Hexane cis-2-Hexene Methylcyclopentane 2,ZDimethylpentane Benzene 2,4_Dimethylpentane Cyclohexane 2,2,3_Trimethylbutane Cyclohexene 2,3-Dimethylpentane 3-Methylhexane 1-Heptene Heptane Isooctane Methylcyclohexane 2,SDimethylhexane 2,4-Dimethylhexane Toluene
1.0
-
-
-
-
-
2.0 1.0 2.0 1.0 1.0 1.0 1.0 2.0 0.5 1.0
1.0 1.0 2.0 0.5 2.0 0.5 3.0 0.5 0.5 -
4.0 5.0 3.0 3.5 4.5 3.5 6.0 3.5 1.0 4.5 -
-------
-
-
-
-
-
-
5.0 5.0 -
6.0 5.0
1.0 1.0 2.0 1.0 2.8 2.6 2.0 3.0 1.0 1.0 1.0 2.0 1.0 2.0 1.0 1.0 1.0
2.0 1.5 2.5 3.0 3.0 2.2 2.4 4.0 2.0 1.5 1.0 4.0 4.0 5.0 1.0 2.5 -
6.0 7.0 5.0 1.0 8.0 7.0 4.5 -
-6.0 2.0 -
6.0 -
7
13
14
15
Ii
-
3.00 3.00 2.43 2.00 4.00 4.00 4.00 2.04 5.00 4.00 5.00 3.00 2.00 5.00 3.00 5.00 5.00 4.28 5.00 4.59 6.00 4.32 5.00 5.11 6.00 6.00 6.00 6.00 6.00 5.00 4.38 4.00 2.68 4.00 6.09 2.00 5.72 5.44 7.00 7.00 4.48 4.00 3.43 3.56 4.00
---
2.0
-
-
----------
2.0 4.0
4.0 2.0 3.5 4.0 -
---------------1.0
2.0 -
2.0 4.0
4.5 5.0 -
-
2.5 -
2.0 1.0 1.0
-
1.0
2.0
-
-
-
-
-
-
4.0
-
57 TABLE 3(a) (continued) Compound 2,3,4_Trimethylpentane 4-Methylheptane 3,bDimethylhexane Cycloheptane fruns-1,4-Dimethylcyclohexane 1-Octene 2,2,4,4_Tetramethylpentane 2,2,5Trimethylhexane Octane 2,2,4_Trimethylhexane Ethylcyclohexane 2,2_Dimethylheptane 2,CDimethylheptane 2,2,3,4_Tetramethylpentane 4,4-Dimethylheptane 2,6-Dimethylheptane 2,5-Dimethylheptane 3,5_Dimethylheptane Ethylbenzene 3,3_Dimethylheptane p-Xylene 2,3,4_Trimethylhexane m-Xylene 2,2,3,3_Tetramethylpentane 2,3_Dimethylheptane 3,bDimethylheptane 4-Ethylheptane 2,3,3,4_Tetramethylpentane 4-Methyloctane 3-Ethylheptane 2-Methyloctane 3-Methyloctane o-Xylene Styrene 3,3-Diethylpentane Nonane Cyclooctane Isopropylbenzene Isopropylcyclohexane Propylcyclohexane Propylbenzene 2,6-Dimethyloctane 1,3,5_Trimethylbenzene a-Methylstyrene tertButylbenzene 1,2,4_Trimethylbenzene
1234567 2.0 2.0 2.0 1.0 2.0 1.0 1.0 1.0 1.0 2.0 1.0 2.0 1.0 2.0 2.0 1.0 2.0 1.0 1.0 2.0 2.0 2.0 2.0 1.0 2.0 2.0 2.0 1.0 1.0 1.0 3.0 2.0 3.0
5.0 4.0 5.0 1.0 6.0 2.5 1.0 3.5 3.5 3.0 2.0 3.5 5.0 1.0 3.0 1.5 0.5 3.0 5.0 5.0 6.0 5.0 4.5 6.0 7.0 2.0 2.0 4.5 -
8.0 1.0 2.0 9.0 8.0 5.5 4.5 4.5 9.0 4.0 7.0 4.5 2.0 4.0 6.5 7.5 5.0 2.0 2.0 9.0 1.0 2.0 3.5 1.0 9.0 3.0 3.0 4.5 4.0 -
7.0 8.0 -
6.0 _ 6.0 -
-
-
-
-
-
6.0 6.0 -
4.0 5.0 6.0 5.3 3.0 6.0 6.0 3.0
-
-
13
14
15
-
-
_
2.0
-
-
-
-
-
-
-
-
l.O-
-
-
2.0 0.7 3.0 3.0 -
-
li 8.99
5.68 4.00 7.00 4.00 8.00 4.82 0.75 8.00 5.47 4.00 4.71 4.67 9.23 7.58 4.89 4.10 4.64 4.00 8.49 4.00 10.52 4.00 12.96 6.87 6.25 6.34 14.39 7.41 6.34 6.87 7.20 4.00 4.00 12.84 9.00 8.00 4.00 4.00 4.00 4.00 6.09 4.00 4.00 6.00 4.00
58 TABLE 3(a) (continued) Compound
12
Decane 1,2,3_Trimethylbenzene Butylbenzene rrans-Decalin c&Decalin Undecane Pentylbenzene Tetralin Dodecane Naphthalene Hexylbenzene Tridecane 2-Methylnaphthalene 1-Methylnaphthalene Tetradecane Biphenyl Diphenylmethane Pentadecane I-Hexadecene Hexadecane Heptadecane 1-Octadecene Octadecane Eicosane
2.0 3.0 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 2.0 1.0 2.0 2.0 1.0 2.0 2.0
a Zi, parameter
3456 8.0 3.0 9.0 4.0 10.0 5.0 11.0 12.0 1.0 13.0 13.0 14.0 15.0 15.0 16.0 18.0
of conformation
-------
8.0 8.0 4.0 -
2.0 2.0 -
-
-
-
-
-
-
3.0 5.3 5.3 4.0 10.0 5.3 9.0 10.0 7.5 10.6 -
7
13
14
15
Ii
3.0 0.7 0.7 2.0 -
-
-
10.00
0.7 1.0 4.5 1.4 -
2.0 2.0 -
-
-
6.00 4.00 10.00 10.00 11.00 5.00 8.00 12.00 4.00 6.00 13.00 4.00 4.00 14.00 8.00 8.00 15.00 16.00 16.00 17.00 18.00 18.00 20.00
of the compound.
TABLE 3(b) Compound
8
9
Nitrogen Methane Ethylene Ethane Carbon dioxide Hydrogen sulfide Water
1.0
-
-
1.0 -
3orA3: 4 or CO 5 or Cl 6 or BO 7or Bl 8
: : : : :
10
11
12
16
17
li ’
1.0 -
1.0 -
1.0 -
1.0 -
1.0
1.00 1.00 2.00 2.00 2.00 2.00 2.00
substituted alkane, or rigid alkane cyclohexane substituted cycloalkane benzene substituted benzene methane (CH,)
59
9 : ethane (C,H,) : carbon dioxide (CO,) 10 11 : nitrogen (N,) 12 : hydrogen sulfide (H,S) 13 or MO : 1-alkene 14 or DO : substituted alkene 15 or CA : cycloalkene 16 : ethylene (C,H,) 17 : water (H,O) Note that the alkane groups (Al, A2 and A3) are different from the cyclohexane group. All groups interact with each other. This approach differs from that adopted in other group-contribution methods such as UNIFAC and ASOG. Considering the cycloalkanes, for example, the following remarks are made. (1) Cyclohexane is the most frequent representative of this class of compounds. Examination of the binary systems it forms shows that its -CH,groups do not have the same properties as those of an alkane. Consequently, a cyclohexane group is defined. (2) When cyclohexane is substituted, its -CH,- groups seem to be close to those of an alkane. Assuming this, the binary systems such as alkanemethylcyclohexane and aromatic-methylcyclohexane are correctly described. (3) The system cyclohexane-methylcyclohexane is poorly represented under these conditions. A substituted cycloalkane group (Cl) is therefore defined. Although the improvement brought about by the above distinction between alkane groups and cycloalkane groups is better appreciated in excess enthalpy calculations, we include here a comparison of our results with those from the various versions of the UNIFAC method (Fredenslund et al., 1977; Fredenslund and Rasmussen, 1985) for low-pressure vaporliquid equilibria. It focuses on the systems which, according to the UNIFAC method, are free from interactions (see Table 4). The model described above requires a knowledge of the group surface fractions. We examined the possibility of assigning a specific surface area to each group, as is usually done in classical group-contribution methods, but found that procedure to be ineffective if all effects were to be taken into account. So we allotted a unit area to each carbon site with a modification made necessary by substitution and heat effects. The site fraction (yiK of group K on molecule i is defined as the ratio of the number of sites of K-type on molecule i to the total number of sites of that molecule. Some examples will illustrate this point.
60 TABLE 4 Comparison of the relative mean deviations in the bubble-point pressures of the binary systems alkane-alkane, alkane-cyclane and cyclane-cyclane obtained by our method (4) and the various versions of the UNIFAC method: original UNIFAC (0, SUPERFAC (21, modified UNIFAC (3) Binary system Neopentane-pentane Isopentane-hexane Cyclopentane-2,3_dimethylbutane Cyclopentane-cyclopentane Cyclopentane-cyclooctane Cyclopentane- trans-decalin Cyclopentane-cis-decalin Pentane-hexane Pentane-methylcyclopentane Pentane-cyclohexane Pentane-methylcyclohexane 2,2-Dimethylbutane-hexane 2,2_Dimethylbutane_hexadecane 2,3-Dimethylbutane-hexane 2,3-Dimethylbutane-cyclohexane 2,3-Dimethylbutane-hexadecane 2-Methylpentane-hexane 2-Methylpentane-heptane 2-Methylpentane-octane 2-Methylpentane-hexadecane 3-Methylpentane-hexane 3-Methylpentane-heptane 3-Methylpentane-octane 3-Methylpentane-hexadecane Hexane-methylcyclopentane Hexane-cyclohexane Hexane-2,4_dimethylpentane Hexane-heptane Hexane-methylcyclopentane Hexane-octane Hexane-decane Hexane-undecane Hexane-dodecane Hexane-hexadecane Methylcyclopentane-cyclohexane 2,4-Dimethylpentane-octane Cyclohexane-2,2,3_trimethylbutane Cyclohexane-heptane Cyclohexane-isooctane Cyclohexane-methylcyclohexane Cyclohexane-4-methylheptane Cyclohexane-cycloheptane Cyclohexane-octane Cyclohexane-cyclooctane
Np 58 34 10 9 36 12 27 22 46 30 49 11 11 11 15 8 55 44 99 9 60 25 96 9 109 189 52 99 35 13 19 12 12 50 14 60 18 212 117 23 11 13 101 13
(1)
(2)
(3)
(4)
1.22 0.72 3.44 0.25 3.37 2.33 5.88 0.81 0.78 2.34 2.32 0.54 7.36 0.15 2.39 6.25 0.83 0.40 2.74 7.65 0.69 0.12 2.43 6.98 0.34 1.14 1.24 1.45 0.33 1.43 3.70 3.30 4.30 10.66 0.11 4.08 0.76 1.19 1.70 1.26 3.30 0.58 1.87 2.96
1.22 0.58 3.00 0.08 1.54 1.90 2.01 0.85 0.77 2.32 2.05 0.54 1.52 0.15 2.33 0.66 0.83 0.30 2.27 1.45 0.69 0.19 2.01 1.27 0.37 1.10 1.13 1.51 0.33 1.86 2.15 0.87 1.04 1.84 0.11 3.98 0.52 0.98 0.99 1.20 1.77 0.48 1.15 2.19
1.32 0.60 3.67 0.08 1.54 1.42 2.91 0.84 1.76 3.40 3.34 0.46 2.61 0.15 3.14 1.21 0.83 0.31 2.33 1.97 0.69 0.18 2.05 1.05 0.49 1.79 1.15 1.50 1.19 1.80 0.91 1.15 1.41 3.98 0.11 3.98 1.32 1.47 1.86 1.20 2.31 0.48 1.67 2.19
0.97 0.69 4.32 0.34 0.58 2.59 1.09 0.81 0.94 0.72 1.56 0.46 0.68 0.13 0.38 1.54 0.80 0.26 1.91 0.51 0.73 0.14 1.77 2.10 0.32 0.75 1.46 1.64 0.31 1.80 1.69 0.14 0.29 0.81 0.29 3.77 0.26 1.06 0.60 1.21 0.21 0.10 0.89 0.82
61 TABLE 4 (continued) Binary system Cyclohexane-nonane Cyclohexane- cis-decalin Cyclohexane-dodecane Cyclohexane-hexadecane Cyclohexane-eicosane Heptane-isooctane Heptane-methylcyclohexane Heptane-octane Isooctane-methylcyclohexane Isooctane-octane Cycloheptane-cyclooctane Octane-ethylcyclohexane Decane- tmns-decalin Overall average
Np 22 12 36 74 27 13 93 85 35 20 9 53 82 2519
(1)
(2)
(3)
(4)
0.96 2.28 5.56 5.61 7.89 0.22 0.28 1.75 0.26 2.32 0.60 0.73 3.64
1.46 1.39 1.10 2.16 4.60 0.15 0.29 1.71 0.31 2.32 0.51 0.77 3.71
0.79 1.85 1.34 1.89 4.37 0.16 0.54 1.72 0.68 2.28 0.51 0.19 2.30
1.42 1.52 0.88 1.01 0.87 0.39 0.26 1.67 0.78 2.04 1.06 0.64 2.78
2.08
1.37
1.62
1.10
For a normal alkane i of n carbon atoms (except for methane and ethane), the different site numbers are as follows: 2(Al) + (n - 2)(A2) sites, which gives site fractions of aiAl = 2/n and (YiAz= (n - 2)/n; as an example, for n-heptane we have 2(Al) + 5(A2). When i stands for toluene, we have l(A1) + 5(BO) + l(B1) sites, which gives site fractions of aiA, = l/7, (yin,,= 5/7 and a,nr = l/7. For a 1-alkene i of n carbon atoms there are 2(MO) + (n - 3)(A2) + l(A1) sites, giving site fractions of CY,~,,= 2/n, aiA2 = (n - 3)/n and aiAl = l/n. For the alkane side of the 1-alkenes, the rules are the same as those relating to the n-alkanes. It was found, however, that not all behavior could be described if the alkanes were divided into CH,, CH,, CH and C groups. For instance, 2,2_dimethylpropane (neopentane), 2,3-dimethylbutane and 2,2,4_trimethylpentane seem to be equivalent as far as their superficial composition is concerned. These are alkanes substituted enough to have lost their flexibility, and can therefore be regarded as typical of the group A3 (rigid alkane), characterized by cyA3= 1. The rules for the allocation of site numbers for the various types of substitutions vary. Substituted alkane.
Allocation of site numbers in this case use the following rules. (1) Consider the longest straight chain that can be formed by the alkane. (2) Allocate one Al site to each CH, group at the ends of that chain unless it belongs to an isopropyl or a tert-butyl grouping. (3) Allocate one A2 site to each CH, group and to each CH group (unless it belongs to an isopropyl grouping); then substract a half A2 site
62
(OS(A2)) each time that group is in the neighborhood of a quaternary carbon atom or of a ternary carbon atom itself neighboring another ternary carbon atom or quaternary carbon atom. (4) The complement of the total number of carbon atoms of that alkane makes up the number of A3-type sites. Consequently, an alkane molecule of it carbon atoms with nl Al-type sites and n2 A2-type sites has (n - nl - n2) A3-type sites. For example, 2-Methylpentane 3-Methylhexane 2,2,5Trimethylhexane 2,3,4_Trimethylhexane 2,2,4_Trimethylhexane 2,2,3,4_Tetramethylpentane 2,2,3,3_Tetramethylpentane
: : : : : : :
l(A1) 2(Al) l(A2) l(A1) l(A1) 9(A3) l(A1)
+ + + + +
lS(A2) + 3.5(A3) 4(A2) + l(A3) 8(A3) 1.5(A2) + 6.5&S) 2.5(A2) + 5.5(A3)
+ 0.5(A2) + 7.5(A3)
Note that the combination of rules (11, (2) and (3) implies that either isopropyl or tert-butyl groups are of type A3. Substituted benzene.
The benzene ring consists of 5.3(BO) + 0.7(Bl) sites and the same rules as those for the normal alkanes are applied to the alkane side of the substituted benzene. For ethylbenzene and propylbenzene, for example, we have l(Al) + l(A2) + 5.3(BO) + 0.7(Bl) and l(Al) + 2(A2) + 5.3(BO) + 0.7(Bl) sites, respectively. All the number of extent the with some
compounds investigated in this work are reported with the sites of each type in Table 3. It should be noted that to a large assignment of the numbers of sites obeys the foregoing rules adjustment, depending on the host molecule.
The definition of groups as influenced by heat effects
The correct description of heat effects introduces further requirements in the definition of groups. Whereas xylenes from ortho, meta and para substitutions on benzene may be similar as far as vapor-liquid equilibrium calculations are concerned, the excess enthalpies arising from their binary mixtures with a given compound show very different values. The same problem is encountered when comparing cyclopentane and substituted cyclopentane, cyclohexane and substituted cyclohexane. For excess enthalpy calculations involving the compounds mentioned above, we were led to define substitution groups: replacement of 5 by 5’ for cyclopentane and methylcyclopentane and replacement of 7 by 7’, 7” and 7”’ for ortho, meta and para xylenes. Clearly, it means that these substitution groups have specific BzL p arameters (see eqn. (13)) and it should be borne in mind that
63
these are mainly determined by excess enthalpies. Thus the interactions of cyclopentane or methyl~clopentane with other compounds are described by the same A*KL parameters as those used for the interactions of methylcyclohexane with other compounds. The Bi,_ parameters relating to cydlopentane and methylcyclopentane, however, are different from those for methyl~clohexane. The same applies to the parameters for xylenes and for ~om~unds containing the substituted benzene group. These specific parameters occupy the four lowermost lines in Table 5, which displays the group interaction parameters. Determination of group interactionparameters For the 17 groups considered here, we have to estimate 272 plus 64 group interaction parameters. Given the current limited amount and quality of experimental data relating to some systems, it is not possible to determine the whole set of parameters reliably. For this reason, a sequential procedure is used for the determination of group interaction parameters. The parameters are adjusted sequentially, by sets of two, four or six, with those which have been determined previously, and which are necessary for a knowledge of current parameters, being assumed. The order in which the different sets of parameters are fitted depends upon the relative abundance of data available for them. Thus the set of six interaction parameters between the three alkane groups are first estimated, given the extensive experimental data available for such systems. Then, to describe the interactions between benzene and alkanes, for example, which requires 12 parameters, only 6 additional parameters relating to benzene-alkane interactions have to be estimated, those for alkane-alkane interactions being considered known. As the parameters of our model are connected with vapor-liquid equilibria as well as with excess enthalpies, both types of data are taken into account when determining the parameters. In this determination an objective function which is the sum of the relative percentage mean deviations between calculated and experimental bubble-point pressures, S,(P), and l/5 times the sum of the relative percentage mean deviations between calculated and experimental excess enthalpies, 8,(HE), is minimized. The minimization technique of Nelder and Mead (1965) was used. For the adjustment of a given set of parameters, excess enthalpy and vapor-liquid equilibrium (isotherm and isobar alike) data for several binary systems involving that set were taken at different temperatures and used simultaneously. Thus the six alkane-alkane interaction parameters were determined at once from excess enthalpy and vapor-liquid equilibrium data for binary mixtures of alkanes (ranging from pentane to hexadecane) over a wide range of temperature. Those for benzene-alkane interactions
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1
0 77.97 32.68 49.48 35.67 189.01 629.31 45.87 2.21 438.21 314.35 521.62 75.27 188.78 64.00 55.86
0
3
4
5
7.70 0 28.46 22.21 0 14.93 1.84 3.62 0 85.78 113.26 87.04 72.73 177.83 291.25 382.72 281.65 97.44 44.38 112.56 28.70 67.08 0.12 30.79 5.56 408.21 409.24 173.30 368.83 230.35 231.10 307.53 438.02 393.91 401.97 57.85 -51.75 63.73 147.43 31.38 33.06 72.63 38.31 76.93
2
Group interaction parameters AtL (.I cmm3)
TABLE 5(a)
7
8
9 10
11
12
0 250.39 0 149.27 643.39 0 120.92 509.39 7.86 0 277.57 590.94 391.56 397.98 0 636.92 726.92 130.00 284.77 368.80 0 324.35 -31.56 594.10 487.52 419.33 1017.64 0 123.42 67.43 62.76 382.41 45.55 221.88 - 27.12 36.75 67.57 26.31 208.63 295.74 88.25 2289.72 1827.09 2277.16 1617.70
6 14
15
0 -75.31 0 -71.95 111.56 0 10.97 -
13
0 -
16
0
17
a
2
3
parameters
4
5
S& U cme3) 6
7
8
9
10
11
12
13
14
15 16 17
5’ 7’ 7” 7‘*
72.71 26.48 7.07 0 166.31 842.61 323.70 337-27 327.88 403.00 %X3,76192.57 369.55 380.94 327.88 227&l 875.08 120.16 369SO 437,03 338.38 228.35
1074.11 312.32 2847.25 1 0 0 3513-761282.99284725 654,22-3244.85 0 2847.25 _ _
-
.-._I -_- - -
_
$34.36 0 3 49.98 18.75 0 4 129.30 45.74 31.07 It 5 112.91 3.84 0 16.78 -6.69 6 396.15 134.11 188.09 190.54 148.04 0 0 7 634.79 127.76 369.55 423.63 327x@ 310_23 8 - 13#&3 183.62- m.94 49-83 518Sl - 62.23 3513.76 0 9 43Al6 31.30 - 2.66 -14.98 -41&I 130.46 1282,99 24.71 0 10 513.05 789.08 624.54 86.371074.11.24M.16 2847.25565.41 690.93 0 0 11 - 33.09 380.06 356.Q4 472.49 s129.22 654.22 108*98-41.74 523.80 0 12 1082.011 3658.26- ‘3244.85 72.22 567.66 617.95 1359.52 36.71)885.21 493.29 0 13 115.00 238.09- 127.42 122.71 87+49 6Nm 211.90 - 7.58 - 527.49-709,$X 648.01 0 - 24.80 x4 600-32- IS.93 - 52,30 277.94 1.92 15 Ifif. 125.38 34.6m x44 52.38 16 -80.02 91.73 2mUO 329.78 266AS IQ.76 - 0 13.81 l33.67 17 934.75 - - 0 524.45 1068_48 8133
1 2
1
Group interaction
TABLE %&I
66
were obtained from binary mixtures of benzene with alkanes (ranging from pentane to eicosane) in the same way. It should be mentioned, however, that for a given binary system not all the available data sets were used in the estimation procedure. Table 5 shows the group interaction parameters obtained.
RESULTS AND DISCUSSION
Low-pressure vapor-liquid equilibria The deviations in the bubble-point pressure and the vapor-phase composition are used as criteria for the judgement of the quality of the results obtained. In the case of isobaric VLE data, the calculated bubble-point pressures are tested against the specified pressure. The whole set of experimental data is divided into subsets to show the capability and the possible failure of the method to describe the interactions between the different families (alkanes, naphthenes, aromatics, alkenes) of compounds. The results are summarized in Table 6 and the distribution of the deviations is shown in Fig. 1. The data bank used in the investigation consists of data taken from the collections of Maczynski (1976) and Gmehling and co-workers (1980a,b, 1983) for low-pressure VLE, Christensen and co-workers (1982, 1984) for HE, and Knapp et al. (1982) for high-pressure VLE, completed with more recent data from the literature. Given the extensive experimental data
TABLE 6 Relative mean deviations in the bubble-point pressure, al(P), and absolute mean deviations in the vapor-phase composition, 6(y), for the various subsets of low-pressure experimental data defined Subset Alkane-alkane Alkane-naphthene Naphthene-naphthene > Aromatic-alkane Aromatic-naphthene > AromaticFaromatic Alkene-another hydrocarbon (alkene included) Overall average
Data sets
NP
S,(P) (%o)
N,
S(y)
79
987
1.38
482
0.0055
116
1532
0.91
714
0.0038
407
5940
0.98
3935
0.0059
47
607
0.71
298
0.0047
156
1568
1.06
457
0.0069
805
10634
1.00
5886
0.0056
67 n
50
-
-
0
dP/P %
0
Fig. 1. Histogram of the relative percentage mean deviations between calculated and experimental bubble-point pressures for low-pressure vapor-liquid equilibria. 169 binaries were treated. n is the number of binaries, the deviations related to which are in the ranges shown.
investigated, only summaries of the comparisons are reported here. Reference should be made to Abdoul (1987) for details of specific systems, conditions and references. The proposed method gives an average precision of 1% (relative) in the prediction of bubble-point pressure and of 0.005 (absolute) for the vaporphase composition. The ’ outstanding homogeneity of the average results obtained for the various families (see Table 6) should be noted. Figure 1 shows that the cases for which the mean deviation is over 3.5% are exceptional. These results foreshadow the fairly good quality of the predictions that could be made for the binary systems not yet experimentally studied. Some excess function models (NRTL, UNIQUAC, UNIFAC) are derived under the assumption that the excess Helmholtz energy at constant volume AE can be equated with the excess Gibbs energy at constant pressure GE, whereas a proper conversion would require an equation of state. Our model enables one to carry out calculations of both functions and to compare them. Figure 2 shows that if the binary system under investigation is made up of molecules which are similar (e.g. hexanecyclohexane, hexane-heptanel, the two functions have practically equal values. But if that condition is not fulfilled a great difference between AE
68
T
GE,AE/J
0.0
I
\ \ \
-200
-400
I
0.0
\\ I
\\ I
\ \ .-- / / / 1
I
0.5
I
I
I/ /1’ I
x1\
I
1.0
Fig. 2. Comparison of the molar excess Gibbs energy, GE (full lines), with the molar excess Helmholtz energy, AE (dashed lines), both calculated by the method presented, at 298.15 K for binaries whose components are of nearly equal sizes: 6-C6 (A) and 6-7 (B), or of very different sizes: 6-16 (C).
and GE is observed (e.g. hexane-hexadecane). This supports the idea that an equation of state should be used, even for low-pressure calculations, if non-ideality stemming from the difference in size of the molecules is to be properly represented. We regard as failures all the binary systems for which the relative mean deviation in the bubble-point pressure is higher than 3.5%. Thus there are seven failures: deviation 3.77% (Liu and Davison, 1981); (1) 2,4-dimethylpentane-octane, deviation 4.32% (Ewing and Marsh, (2) cyclopentane-2,3_dimethylbutane, 1973); deviation 4.61% (Willman and Teja, 1985); (3) toluene-decane, deviation 4.51% (Messow and Engel, 1977); (4) toluene-hexadecane, deviation 5.4% (Haynes and Van Winkle, (5) naphthalene-tetradecane, 1954; Ward and Van Winkle, 1954); deviation 3.69% (Kudryavtseva et al., 1972); (6) 1-heptene-octane, deviation 4.36% (Laesvskaya et al., 1979). (7) styrene-propylbenzene, These binary systems present a delicate problem in that we do not know whether the disagreements can be ascribed to the inadequacy of the model or to the accuracy of the experiments. In all the cases mentioned above, the systems were determined only once.
69
However, two cases may be resolved. (1) The experimental data on the binary cyclopentane-2,3-dimethylbutane (Ewing and Marsh, 1973) can hardly be questioned. We blame the discrepancy on the inadequacy of the model. (2) The data on the binary naphthalene-tetradecane (Haynes and Van Winkle, 1954; Ward and Van Winkle, 1954) belong to a series of data, provided by a single laboratory, for which the errors in the pure-component vapor pressures make it impossible to reach any correct treatment. The disagreement observed for these binaries can by no means be used to question the model submitted. To solve the problem set by the remaining systems, new experimental data are to be obtained to confirm those already existing. Excess enthalpy comparisons
The values of excess enthalpies vary greatly depending upon the system: they can become negative and in some instances they are nearly zero. In these cases the relative percentage mean deviations 8,(HE> are meaningless. To measure the quality of the predictions by the proposed method, we consider here either relative percentage or absolute mean deviations between calculated and observed excess enthalpies depending on the system under consideration. For binary systems of the type paraffin-paraffin, naphthene-naphthene, aromatic-aromatic or alkene-alkene, the excess enthalpies have small values, and high relative mean deviations do not mean a poor agreement. So for these systems absolute mean deviation 6(HE> is used. The deviations are collected in Table 7 and their distributions shown in Fig. 3(a) and Fig. 3(b). For the binaries for which the heat of mixing is small (always less than 300 J), the average deviation is 12 J for 7009 determinations. For those whose heat of mixing is higher (reaching 1000 J or more) the average deviation is 5.54% for 4299 determinations. The discussion of the causes of the deviations (poor quality of the data or inadequacy of the model) is in numerous cases more difficult with excess enthalpies than it is with vapor-liquid equilibria. Of course, in the case of systems thoroughly determined experimentally, it is possible to pick out aberrant experimental results easily. This is the case, for example, for the binary system hexane-benzene (Fig. 4). When the only two sets of data available are in disagreement, however, the program is more delicate. It should be mentioned that of the 282 binary systems investigated, 19 show deviations which are outside the range of the diagrams. When the difference in size increases in a homologous series, the excess enthalpy increases. Our model is able to follow this trend exemplified by
70 TABLE 7 Relative mean deviations, s,(Z-ZE), or absolute mean deviations, 8(HE), in molar excess enthalpy, for the various subsets of low-pressure excess enthalpy experimental data defined, depending on the subset Subset
Data sets
Alkane-alkane Alkane-naphthene Naphthene-naphthene
N,,
8(HE)
arCHE)
(J mol-‘)
(%) -
149
1549
5.96
223
3728
13.44
270
4299
Aromatic-aromatic Alkene-another hydrocarbon (alkene included)
100
1299
15.88
47
433
9.97
Overall average
270 519
4299 7009
12.02
Aromatic-alkane Aromatic-naphthene
> >
-
5.54
5.54
the binary benzene-alkane systems (Fig. 5(a)) and hexane-higher alkanes (Fig. 5(b)). The b inary system hexane-hexadecane (Fig. 6) shows an interesting phenomenon. This system exhibits a strong decrease in its excess enthalpy with increasing temperature, leading to negative values, first and foremost for mixtures rich in the most volatile component and then over the entire mole fraction range. This equation of state effect is correctly described by the model. The mixture of a long-chain compound with a globular one yields an endothermic effect, which rapidly decreases with increasing temperature. In mixtures of normal alkanes this effect is well described by means of the term xixjEchij (eqn. (5)), Ech,, being readily calculated from chain lengths Zi (eqn. (9)). The extension of that expression to substituted alkanes is no problem as far as light compounds (up to hexane isomers) are concerned. The problem becomes more complicated, however, with isomers of higher alkanes. The nonane isomers were extensively studied by Tancrede et al. (1977a,b) and De St-Romain et al. (1979a,b) who pointed out that the excess enthalpies of their mixtures with hexadecane showed very different behavior depending on the number and the place of the substitutions. The authors quoted above submitted correlations with molecular optical anisotropy and molar volume, but these are far from being satisfactory. We report here a correlation which relates the excess enthalpies of binary mixtures of nonane isomers with hexadecane to their critical temperature. The correlation is obvious but inadequate to lead to a good prediction of excess enthalpies. To improve that correlation the Zi values are estimated
71
T n
dH/joules
Tn
10
dli/joutas
0
M”
5
Fig. 3. Histograms of the absolute mean deviations between calculated and experimental molar excess enthalpies for binaries of the types: (a) a&ane-aikane; 70 binaries were treated; for 7 binaries, the deviations are out of the range of the diagram: Cbl aIkane-cyciane, cyclane-cyclane, aromatic-aromatic, aIkene-alkene, alkene-alkane, alkene-cyclane, alkene-aromatic; 131 binaries were treated; for 7 binaries, the deviations are out of the range of the diagram. IZ is the number of binaries, the deviations related to which are in the ranges shown.
72
T
HE/J
900
600
300
0.0
I 0.0
I
1
I
I
I
0.5
1
I
I
I
1.0
Fig. 4. Comparison of the molar excess enthalpies at 298.15 K from various sources of experimental data: hexane(l)-benzene(2); data from Schnaible et al. (1957) (triangle), Baluja Santos (1970) (circle), Romani and Paz-Andrado (1975) (square). Curves calculated by the present method, full lines.
from eqn. (Bl) (see Appendix B). The results of that correlation for all the isoalkanes investigated in this work are displayed in Table 2 along with the values of Zi,which were fitted so as to obtain better results. The estimate is often correct but sometimes poor, and in that case inadequate for excess enthalpy prediction. There is a score of binary systems for which the excess enthalpies are very poorly predicted and may be regarded as failures of the method. It is important that they should be discussed as they show the directions in which improvements should be sought. They are collected (Table 8) in three families corresponding to three problems for which satisfactory solutions were not found. (1) Binary systems including isoalkanes: as is pointed out above, the changes in conformation which may occur in such systems upon the mixing process prevent a completely satisfactory solution being reached. (2) Binary systems including cycloalkanes: it is difficult to find a satisfactory solution to problems of conformation in cycloalkanes through group contributions. (3) Binary systems including di- and trimethylbenzenes: the isomers of substitution in the benzene ring are also difficult to account for by group contributions.
73
THE/J
0.5
Fig. 5. Evolution of the molar excess enthalpy in a homologu~ series at 298.15 IC (a> Benzene-aikane; Heptane: data from Mmwh (19781 (circle); Undecane: data from DiazPena and Menduina (1974) (star); Pentadecane: data from Diaz-Pena and Menduina (1974) (cross). (b) Hexane-higher alkane; Octane: data from Hamam et al. (1984a) (circle); Decane: data from Marsh et al. (1980) (star); Dodecane: data ~~CXLI Hamam et al. (J984b) (rhombus); Hexadecane: data from Larkin et al. (1966) (cross). Calculated curves, full lines.
74 150
T
HE/J
100
50
0.0
-50 0.0
0.5
1.0
Fig. 6. Temperature dependence of the molar excess enthalpy versus composition curves (solid lines) for the binary hexane(l)-hexadecane(2). Experimental data are taken from Fernandez-Garcia and Boissonass (1966) at 293.15 K (square); from Tancrede et al. (1977a) at 298.15 K (cross); from Holleman (1965) at 313.15 K (rhombus), at 324.15 K (star), at 333.15 K (circle) and at 349.15 K (triangle).
Overall, the results obtained show the superiority of the models involving equations of state for the correlation of thermodynamic properties of mixtures of non-polar and non-associating compounds, even under low pressures. The same conclusion was reached by Hamam et al. (1986) who pointed out that even though the equations of state such as PR and RIG were primarily designed for vapor-liquid equilibrium calculations, their use for the prediction of excess enthalpies could lead to an improvement in the reproduction of these, especially for mixtures of the type normal alkanesubstituted alkane. High-pressure vapor-liquid equilibtium calculations
The correlations determined from low-pressure data are extended to high-pressure vapor-liquid equilibria merely by defining new groups corresponding to supercritical fluids. Thus methane, ethane, carbon dioxide, nitrogen, hydrogen sulfide, ethylene and water are distinct groups. Their interaction parameters with other families of compounds (alkanes, naphthenes, aromatics and alkenes) are estimated as described earlier, i.e. by
75 TABLE 8 Failures of our method for excess enthalpy calculations Family
Binary system
Deviation in HE
Isoalkanes
Hexane-3,3_diethylpentane Octane-3,3-diethylpentane 3,3-Diethylpentane-dodecane 3,3-Diethylpentane-tetradecane 3,3-Diethylpentane-hexadecane Octane-2,3,3,4_tetramethylpentane Neopentane-isooctane Cyclohexane-2,6_dimethyloctane
107.4 J 109.7 J 111.8 J 104.8 J 75.3 J 109.7 J 54.7 J 65.6 J
Cyclanes
Cyclopentane- trans-decalin Cyclopentane-cis-decalin Cyclooctane-tetradecane Heptane-tetralin Octane-tetralin Tetralin-dodecane Tetralin-hexadecane Benzene-tetralin
168.5 J 71.8 J 72.4 J 42.5% 30.1% 47.5% 43.4% 57.6 J
di- and trimethylbenzenes
Benzene-p-xylene Benzene- m-xylene Heptane-1,3,5-trimethylbenzene Toluene-1,2,4_trimethylbenzene 1,3,5-Trimethylbenzene-hexadecane
39.9 J 32.6 J 63.0% 28.8 J 64.4%
assuming as known the interaction parameters within or between those families. The method was tested on a database made up of nearly 12300 VLE data points, of which over 10300 are complete (i.e. having a full set of P, T, x, y data). The methods proposed in the literature for the prediction of high-pressure equilibria of non-polar and non-associating fluids were scarcely tested on so considerable a database, and it is therefore difficult to make an extensive comparison. It can be noted, however, that when the databases reported partly overlap ours (Hamam et al., 1977; Ishikawa et al., 1980; Machat and Boublik, 19851, our results are at least of the same quality. These are summarized in Table 9. The criteria used to measure the quality of the predictions are the deviations from experiment in the bubble-point pressure and in the vaporphase composition. The distribution of errors in bubble-point pressures is displayed in Fig. 7. These results are obviously less accurate than those with low-pressure data, but they remain within the range of experimental accuracy, given the
76 TABLE 9 Relative mean deviations in the bubble-point pressure, 8@‘), and absolute mean deviations in the vapor-phase composition, 6(y), for the various subsets of high-pressure experiment data defined Subset Nitrogen-carbon dioxide Nitrogen-hydrogen sulfide Nitrogen-water Nitrogen-hydrocarbon a ) Hydrogen sulfide-water Hydrogen sulfide-hydrocarbon
Data sets
b>
Carbon dioxide-hydrogen sulfide Carbon dioxide-water 1 Carbon dioxide-hydrocarbon \ b Methane-carbon dioxide Methane-hydrogen sulfide Methane-water Ethylene-carbon dioxide Ethane-carbon dioxide ) Ethane-hydrogen sulfide Hydrocarbon-hydrocarbon ’ Hexane-benzene Benzene-heptane I Overall average
iV,,
6,(P) (%o)
N,
6(Y)
174
1603
4.72
1452
0.0145
73
557
4.19
533
0.0145
246
2328
4.47
1888
0.0109
836
7784
2.39
6418
0.0089
1329
12272
3.17
10291
0.0103
a Including methane, ethylene, ethane, propene, propane, butanes, pentanes and higher hydrocarbons (up to hexadecane). b Methane, ethylene and ethane precluded. ’ Including methane, ethylene, ethane, propene, propane, butanes, butenes, butadiene, pentanes and higher hydrocarbons (up to eicosane).
discrepancies in data between different sources. Robinson et al. (1981) noted differences between their data (Li et al., 1981a; Gupta et al., 1982) and those from Ohgaki and Katayama (1976) for the carbon dioxide-hexane (Fig. 8) and carbon dioxide-benzene systems. High and regular deviations (often over 10%) are observed for systems involving nitrogen. Were it not for the conformation term, x,xiEch,, in eqn. (51, our method would merely be a correlation of the binary interaction parameters kij since we have seen that the energy binary parameters Eij actually used in this method are related to the k,j parameters by eqn. (7). This is the case for binary systems involving components with the Zi parameters not differing by more than three or four unities: the conformation term is not very important. In that case, of course, our method cannot yield better results than the classical application of the Peng-Robinson equation of state with a kij fitted to each binary system can. The main point to be noted here is
77 40
Tn
T
10
0
r 0
2
L
dP/P %
Fig. 7. Histogram of the relative percentage mean deviations between calculated and experimental bubble-point pressures for high-pressure vapor-liquid equilibria. 163 binaries were treated. n is the number of binaries, the deviations related to which are in the ranges shown. T P/bar 80.
60.
Xl *Yl 0.
0.5
1.
Fig. 8. Discrepancies between data from various sources for the vapor-liquid equilibria of the binary system carbon dioxide(l)-hexane(2) at 313.15 K: circle, Ohgaki and Katayama (1976); rhombus, Li et al. (1981a); star, Wagner and Wichterle (1987). Calculated curve, full line.
78
T
P/bar
80.
0.
0.
0.5
1.
Fig. 9. Vapor-liquid equilibria for the binary carbon dioxideWpentane(2). The data are from Besserer and Robinson (1973): rhombus (277.65 K), circle (311.04 K), triangle (344.15 K), star (377.59 K). Calculated curves, full lines.
that the correlation via group contributions does not lead to an appreciable error. This is illustrated by comparison of theory with experiment for the carbon dioxide-pentane system (Fig. 9). When the conformation term reaches values that cannot be neglected, especially for light-heavy binary systems, our method is better than the classical application of the Peng-Robinson equation. Figure 10 shows a very good correlation of the carbon dioxide-hexadecane system. It should be mentioned, however, that for this type of binary, elevated errors are encountered in the vicinity of critical points and when three-phase or liquid-liquid equilibria exist, they are predicted qualitatively but not precisely located. Binary methane-higher
alkane systems
For mixtures of alkanes, the methods which make use of the classical approximation with zero binary interaction parameters yield good results as long as the volatilities of the two components are not greatly different. When the volatilities of the two alkanes are very different, though, for example for binary mixtures of methane with alkanes ranging from hexane up, the zero binary interaction parameter hypothesis leads to the calculation of bubble-point pressures which differ significantly from the experimental values; this all the more so as the size of the alkane increases.
79
25.
Xl,Yl
0. 0.
0.5
1.
Fig. 10. Vapor-liquid equilibria for the binary carbon dioxide(Whexadecane(2); the data are from Sebastian et al. (1980): circle (463.05 K), rhombus (542.85 K), triangle (623.15 K), star (663.75 K). Calculated curves, full lines.
These errors may be over 20%. In such cases, non-zero binary interaction parameters are required as pointed out by Legret et al. (1984). For example, one system is considered here to show the difference between the results obtained by our method (with non-zero implicit binary interaction parameters and chain-length effects taken into account) and those obtained with the translated basic form of the Peng-Robinson equation with zero binary interaction parameters, i.e. the methane-hexadecane (Fig. 11). This curve illustrates that it is not possible to describe both low-pressure equilibrium and the vicinity of the critical point in a completely satisfactory way. Temperature dependence: binary carbon dioxide-alkane systems
Numerous authors (Huron et al., 1978; Kato et al., 1981; Legret et al., 1984) have noticed a temperature dependence of the binary interaction parameters. Since the optimal values were different depending on the source of data used for their determination, it was not possible to set up a simple law or a general correlation. Kato et al. (1981) proposed for mixtures of carbon dioxide with normal alkanes a quadratic function of absolute temperature, the three coefficients of which were correlated as functions of the acentric factor of the n-alkane.
80
T
P/bar
600
0.
0.5
1.
Fig. 11. Vapor-liquid ebuilibria for the binary methane(l)-hexadecane(2) at 350.00 K, circles are experimental points from Glaser et al. (1985). The curves in dashed lines are calculated with the basic form (eqn. (3)) with k,, = 0. The curves in full lines are calculated with the method presented.
The proposed method solves the problem for binary mixtures of carbon dioxide with alkanes for the correlation with respect to either temperature or alkape nature. The binary interaction parameters kij for carbon dioxide-alkane systems given by our model (eqn. (7)) are plotted versus temperature, for alkane ranging from methane to eicosane (Fig. 12). It should be noted that the curves show minima similar to those obtained by Kato et al. (1981), thereby confirming a phenomenon noticed experimentally with tabulated kij adjusted for each temperature. Prediction of multicomponent excess enthalpies
To test the applicability of the foregoing material, the heats of mixing of multicomponent (ternary and higher order) mixtures of hydrocarbons are calculated and compared with the relevant available experimental data. As is clearly shown in Table 10, excellent agreement is obtained between calculated and experimental values. The deviations (relative percentage mean deviations) are quite within the experimental accuracy and are not larger than those obtained for binary mixtures of the same type of compound. The results suggest that the proposed method may be reliable to
81 0.25 /
100
250
400
550
IT
k12
700
Fig. 12. Temperature dependence of the binary interaction parameters derived from eon. (7) for the binary systems carbon dioxide-alkane. The number of carbon atoms of the alkane is indicated at the right of the diagram; it is varied with a step of 1 from methane to decane and with a step of 2 from decane to eicosane. Dashed line: our method; full line: calculated by Kato et al. (1981).
TABLE 10 Relative mean deviations, values
6,( HE>, between calculated
Mixture
and experimental
T
(K) Hexane-benzene-cyclohexane Hexane-cyclohexane-toluene Benzene-cyclohexane-heptane Benzene-cyclohexane-toluene Benzene-heptane-toluene Benzene-heptane-tetradecane Benzene-toluene-tetradecane Benzene-decane-tetradecane Cyclohexane-heptane-toluene Hexane-benzene-cyclohexane-heptane Hexane-cyclohexane-heptane-toluene Hexane-benzene-cyclohexane-heptane-toluene Overall average
293.15 293.15 293.15 293.15 293.15 298.15 298.15 298.15 293.15 293.15 293.15 293.15
excess enthalpy
6,(HE)
Nh
Ref.
(%o)
11 5 31
2.46 4.40 2.63
a a
6
7 19 20 21 6 15 8 18
4.77 2.41 4.52 6.47 3.77 4.96 2.78 4.24 3.82
a a ’ c c a a = =
167
3.86
a Mathieson and Thynne (1956). b Brown et al. (1955). ’ Pfestorf et al. (1978).
b
82 TABLE 11 Relative mean deviations in the bubble-point pressure, d,(P), and absolute mean deviations in the vapor-phase composition, 6(y), between calculated and experimental values, for ternary systems Mixture
Nitrogen-methaneethane
T range K) 140.00 160.00 200.00 220.00 260.00 270.00 280.00 171.43 122.04
P range (bar) 20.0-40.0 20.0-40.0 20.0-120.0 80.0 50.7-76.0 35.4-86.0 50.7-75.9 13.7-27.6 5.7-27.3
Overall average
N,,
&(P) (%I
22 14.2 28 10.5 34 4.5 10 1.7 33 0.9 42 1.9 18 1.6 11 2.0 10 3.8 208
4.7
Ref.
NY S(Y)
22 28 34 10 33 42 18 11 10
Yl
Y2
y3
0.0094 0.0190 0.0108 0.0071 0.0068 0.0097 0.0084 0.0207 0.0042
0.0100 0.0188 0.0049 0.0006 0.0048 0.0053 0.0067 0.0219 0.0046
0.0010 0.0007 0.0059 0.0068 0.0098 0.0133 0.0146 0.0018 0.0007
a = = a b b b ’ ’
208 0.0107 0.0082 0.0071
Nitrogen-methane Carbon dioxide
270.00
45.6-111.4
45
1.9
45 0.0110 0.0090 0.0199 d
Nitrogen-methanepropane
313.70 114.10
89.0-95.8 1.14-17.5
4 11
2.2 4.6
e - 11 0.0094 0.0094 0.0000 f
Overall average
15
3.9
11 0.0094 0.0094 0.0000
Nitrogen-ethanepropane
200.00
36 14.4
36 0.0061 0.0104 0.0010 a
Nitrogen-butanedecane
352.60 275.0
17 14.6
17 0.0459 0.0400 0.0069 g
Nitrogen-heptane methylcyclohexane
Methane-ethanecarbon dioxide Methane-ethanepropane
20.0-120.0
453.15 235.0-329.0 472.15 168.0-246.0 497.15 129.0-198.0
9 9 9
7.6 8.2 7.0
Overall average
27
7.6
27 0.0107 0.0062 0.0048
250.00
21.3-30.4
15
2.3
15 0.0205 0.0126 0.0263 i
158.15 172.04 185.93 199.82 213.71 313.70 144.26 172.04 199.82 227.59 255.37 283.15
2.2-6.9 2.2-20.7 2.2-37.9 2.7-50.0 6.9-60.3 57.7-71.5 6.9 6.9-13.8 6.9-41.4 6.9-68.9 6.9-89.6 6.9-82.7
2 7 9 17 13 4 2 6 12 24 39 31
2.2 2.0 6.3 2.6 1.1 4.4 4.5 2.3 1.9 2.3 2.0 2.1
2 7 9 17 13 2 6 12 24 39 31
166
2.4
Overall average
9 0.0106 0.0072 0.0035 h 9 0.0087 0.0063 0.0027 h 9 0.0129 0.0051 0.0081 h
0.0006 0.0011 0.0064 0.0035 0.0037 0.0022 0.0063 0.0031 0.0062 0.0104 0.0121
0.0007 0.0011 0.0053 0.0030 0.0031
0.0001 0.0001 0.0012 0.0008 0.0007
j j j j j e
0.0009 0.0051 0.0032 0.0049 0.0082 0.0063
0.0006 0.0012 0.0002 0.0023 0.0042 0.0070
k k k k k k
162 0.0073 0.0053 0.0031
83 TABLE 11 (continued) Mixture
T
P
range
range (bar)
(K) Methaneethaneheptane
Methanecarbon dioxide -hydrogen sulfide
Methanecarbon dioxide -butane
Np gr(P) A',
Methanepropanebutane
Methanepropane-pentane
Methanepropanepentane
Yl
Y2
Y3
2.9 3.5 3.0
9 11 12
0.0008 0.0242 0.0142
0.0008 0.0240 0.0139
0.0000 0.0002 0.0003
Overall average
32
3.2
32
0.0139
0.0137
0.0002
310.93 277.59 344.26
16 13 7
5.4 10.3 4.8
16 13 7
0.0318 0.0573 0.0196
0.0289 0.0320 0.0220
0.0427 0.0386 0.0299
Overall average
36
7.1
36.
0.0386
0.0286
0.0387
177.59 210.93 244.26 277.59 310.93
3 7 22 27 23
5.3 3.5 8.5 4.7 5.6
3 7 22 27 23
0.0189 0.0972 0.0841 0.0426 0.0351
0.0147 0.0889 0.0868 0.0379 0.0344
0.0042 0.0082 0.0059 0.0162 0.0132
82
5.9
82
0.0554
0.0536
0.0115
8 5 5
1.8 2.0 3.7
8 5 5
0.0107 0.0120 0.0492
0.0120 0.0052 0.0079
0.0103 0.0167 0.0569
18
2.4
18
0.0218
0.0090
0.0250
3 5 6 5
1.4 1.3 0.9 1.3
3 5 6 5
0.0098 0.0135 0.0027 0.0077
0.0073 0.0130 0.0028 0.0068
0.0037 0.0005 0.0014 0.0066
Overall average
19
1.2
19
0.0080
0.0073
0.0029
277.59 310.93 344.26 377.59 344.26
20 20 13 5 40
1.7 0.7 1.6 2.7 1.6
20 20 13 5 40
0.0050 0.0085 0.0101 0.0091 0.0200
0.0042 0.0054 0.0066 0.0059 0.0088
0.0014 0.0035 0.0065 0.0130 0.0116
Overall average
98
1.5
98
0.0127
0.0067
0.0073
344.26 377.59
17 13
1.8 3.2
17 13
0.0073 0.0276
0.0089 0.0083
0.0046 0.0061
Overall average
30
2.4
30
0.0161
0.0086
0.0053
244.26 233.15 222.04 210.93 199.82 244.26 233.15
14 15 13 7 4 6 6
3.1 2.8 3.5 3.8 1.9 8.3 12.5
14 15 13 7 4 6 6
0.0010 0.0007 0.0004 0.0004 0.0008 0.0019 0.0038
0.0013 0.0249 0.0004
0.0003 0.0001 0.0000
0.0019 0.0036
0.0003 0.0003
65
4.5
65
0.0011
0.0067
0.0002
6.9-55.2 6.9-68.9 6.9-68.9
41.3-124.1 27.6-110.3 68.9-110.3
27.58 27.6-68.9 27.6-113.6 27.6-117.2 27.6-117.2
342.20 343.00 342.60
85.10 115.20 140.80
Overall average Methanecarbon dioxide -heptane
Ref.
6(y)
9 11 12
222.04 233.15 244.26
Overall average Methanecarbon dioxide -pentane
(%)
341.80 343.30 343.20 343.40
101.60 101.00 130.50 160.60
13.8-123.9 13.8-117.2 13.8-103.4 34.5-68.9 13.8-103.9
34.5-137.9 34.5-103.4
6.9-68.9 6.9-68.9 6.9-55.1 6.9-41.4 6.9-27.6 7.4-68.5 6.7-68.3
Overall average
’ ’ ’
m ” ”
o o o o ’
p p p
p p P p
q q q q q
r r
’ ’ ’ 0.0004 o.oooo ’ 0.0008 0.0000 ’ S
’
84 TABLE 11 (continued) Mixture
Methanepropanetoluene Methanepropanedecane
T range (K)
P range (bar)
N,
S,(P) (%o)
A’,,
Ref.
6(y) Yl
Y2
Y3
233.15
6.9-68.9
16
11.4
16 0.0041
0.0040
0.0001
’
277.59 310.93 410.93 410.93 477.59 510.93 294.26 277.59 255.37 244.26
27.6-275.8 27.6-275.8 27.6-275.8 27.6-275.8 27.6-206.8 27.6-137.9 2.8-68.8 2.9-70.5 2.3-69.6 3.1-69.5
17 17 13 14 7 5 51 51 30 20
3.1 1.6 1.9 2.3 3.3 4.9 3.2 2.0 3.2 4.8
17 0.0107 17 0.0062 13 0.0205 14 0.0179 7 0.0201 5 0.0142 51 0.0018 51 0.0020 30 0.0020 20 0.0015
0.0044 0.0044 0.0145 0.0113 0.0158 0.0159 0.0019 0.0020 0.0022 0.0015
0.0062 0.0020 0.0085 0.0068 0.0086 0.0101 0.0002 0.0001 o.oooo o.oooo
q q q q q q s s s s
225
2.8
225
0.0058
0.0044
0.0070
4
3.4
4
0.0075
0.0028
0.0070
g
Overall average Methanebutanedecane
352.60
Methaneneopentanepentane
344.26
34.7-146.2
8
2.0
8
0.0130
0.0027
0.0105
t
Methaneisopentanepentane
344.26 377.59 410.93
34.7-156.4 34.9-141.1 37.2-107.9
7 7 9
4.9 1.6 1.6
7 7 9
0.0322 0.0137 0.0449
0.0079 0.0041 0.0112
0.0244 0.0096 0.0338
t t t
Overall average
23
2.6
23
0.0316
0.0080
0.0236
260.00
5.0-25.0
28
5.1
28
0.0484
0.0572
0.0106
’
304.34 305.67 307.04 304.62 305.45 306.50 260.00 270.00 280.00
45.0-47.2 46.1-48.4 47.4-49.2 6.6-47.8 6.7-48.6 6.9-49.2 5.0-15.0 5.0-15.0 5.0-15.0
14 16 13 23 22 14 7 10 7
0.2 0.2 0.3 1.3 1.2 1.5 5.7 5.4 5.4
14 16 13 23 22 14 7 10 7
0.0009 0.0021 0.0027 0.0049 0.0035 0.0027 0.0177 0.0194 0.0210
0.0003 0.0010 0.0011 0.0079 0.0030 0.0060 0.0182 0.0185 0.0230
0.0006 0.0011 0.0016 0.0045 0.0016 0.0061 0.0018 0.0055 0.0057
” ” ” w w w ” ” ”
126
1.7
102
0.0061
0.0066
0.0030
28 21 21 20
3.6 4.7 4.5 1.8
28 21 21 20
0.0130 0.0244 0.0360 0.0080
0.0051 0.0139 0.0079 0.0060
0.0001 0.0110 0.0294 0.0033
90
3.7
90
0.0199
0.0080
0.0127
Ethanecarbon dioxide -butane Ethanepropanebutane
275.80
Overall average Ethanebutanepentane
366.48 394.26 422.04 338.70
37.0-64.9 31.1-56.9 31.8-55.6 34.9-60.6
Overall average
x x ’ y
85 TABLE 11 (continued) Mixture
T range (K)
P range (bar)
Np
S,(P) (%o)
N,,
Ref.
S(Y) Yl
Y2
y3
Ethane-butanehexane
322-404
28.3-67.9
25
2.9
25
0.0193
0.0107
0.0126
’
Ethane-butaneheptane
422.04 449.82
35.3-83.1 36.4-79.5
30 22
4.3 4.5
30 22
0.0263 0.0467
0.0129 0.0144
0.0134 0.0369
’ ’
Overall average
52
4.4
52
0.0349
0.0135
0.0233
263.15 293.15
28 4
0.4 1.1
28 4
0.0027 0.0115
0.0027 0.0021
0.0042 0.0127
Overall average
32
0.5
32
0.0038
0.0026
0.0053
253.12 263.14 273.18 278.15 283.15 290.75
11 12 15 8 16 21
4.5 4.7 4.7 4.2 4.8 3.8
11 12 15 8 16 21
0.0088 0.0096 0.0074 0.0168 0.0089 0.0081
0.0085 0.0090 0.0063 0.0171 0.0079 0.0067
0.0006 0.0009 0.0171 0.0012 0.0017 0.0018
83
4.4
83
0.0093
0.0084
0.0042
8 8 4
2.6 3.0 2.1
8 8 4
0.0026 0.0071 0.0127
0.0023 0.0043 0.0041
0.0033 0.0028 0.0088
Overall average
20
2.6
20
0.0064
0.0034
0.0042
303.15 313.15 323.15
18 14 15
5.9 6.1 3.9
18 14 15
0.0050 0.0058 0.0025
0.0024 0.0032 0.0013
0.0026 0.0026 0.0012
47
5.3
47
0.0044
0.0023
0.0022
9 9
11.9 12.9
9 9
0.0194 0.0434
0.0190 0.0288
0.0012 0.0146
dd dd
18
12.4
18
0.0314
0.0239
0.0079
dd
21.6-130.80 5 26.0-127.90 4
4.4 15.9
5 4
0.0003 0.0011
0.0005 0.0005
0.0004 0.0007
ee ee
Carbon dioxide -ethene-ethane
Carbon dioxide -ethanepentane
23.3-32.4 45.6-60.9
6.9-21.2 6.9-28.7 6.9-37.1 6.9-40.1 6.9-45.9 6.9-54.6
Overall average Carbon dioxide -pentaneheptane
Carbon dioxide -l-hexenehexane
342.90 343.60 343.50
78.50 90.90 101.50
20.3-65.0 18.6-67.9 20.3-85.9
Overall average Carbon dioxide -benzenetetralin
416.50 519.50
50.0-148.0 50.0-148.1
Overall average Carbon dioxide 353.15 -toluene-l413.15 methylnaphthalene
Carbon dioxide -decane-tetralin
Propanepropene-tetralin
Overall average
9
9.5
9
0.0007
0.0005
0.0006
343.30 520.00
9 9
10.2 9.9
9 9
0.0021 0.0170
0.0017 0.0094
0.0009 0.0076
Overall average
18
10.0
18
0.0096
0.0056
0.0042
273.15 293.15
16 12
3.2 4.2
-
28
3.6
-
50.0-118.6 46.8-120.9
1.5-4.7 2.1-5.1
Overall average
=* aa
bb bb bb bb bb bb
p P P
‘= Cc cc
dd dd ff ff
-
-
-
86 TABLE 11 (continued) Mixture
Isobutanel-butene1,3-butadiene
T range (K)
P range (bar)
277.67 294.36 310.70 327.70 344.03
1.6-1.7 2.8-2.9 4.5-4.8 7.0-7.3 10.3-10.5
Np
S,(P) (o/o)
NY
Ref.
6(y) Yl
y2
y3
8 8 8 8 8
10.3 8.9 7.7 6.6 5.6
_ -
-
Overall average 310.93 3.87-4.18 324.82 5.52-5.91 338.71 7.91-8.46
40 5 5 5
7.8 0.4 1.9 0.9
-
-
Overall average
15
1.1
-
-
1-Hexene hexaneoctane
328.15
0.20-0.67
12
1.1
-
-
Hexanemethylcyclopentanebenzene
333.15 342-350 342-349 344-346
0.60-0.76 1.013 1.013 1.013
103 54 10 11
1.1 0.3 0.3 0.9
103 54 10 11
0.0030 0.0026 0.0032 0.0027
0.0022 0.0026 0.0028 0.0009
0.0035 0.0029 0.0029 0.0027
Overall average
178
0.3
178
0.0029
0.0023
0.0032
283.15 288.15 298.15 343.15 343.15 342-352
10 10 12 6 21 108
1.1 0.7 0.2 0.2 0.5 0.8
10 10 12 6 21 108
0.0022 0.0044 0.0025 0.0023 0.0041 0.0047
0.0019 0.0025 0.0032 0.0016 0.0069 0.0034
0.0024 0.0032 0.0038 0.0017 0.0049 0.0047
167
0.7
167
0.0042
0.0036
0.0043
Butanel-butene13-butadiene
Hexanebenzenecyclohexane
0.08-0.09 0.09-0.12 0.16-0.19 0.82-0.99 0.78-0.98 1.013
Overall average Hexanebenzenetoluene
ge u -
BB BB BB
hh hh hh
-
-
ii
i kk ” mm
“” “” “’ “II O” pp
348-376
1.013
11
2.0
11
0.0127
0.0081
0.0051
qq
345-368
1.013
26
0.3
26
0.0055
0.0018
0.0049
**
Overall average
37
0.8
37
0.0077
0.0036
0.0050
343-382
1.013
38
0.5
38
0.0088
0.0033
0.0069
ss
Methylcyclopentane -benzenecyclohexane 346-354
1.013
39
0.3
39
0.0040
0.0040 0.0055
tt
Benzenecyclohexanetoluene
353-367
1.013
24
10.1
24
0.0285
0.0216
0.0231
“”
Benzenetolueneethylbenzene
362-399
1.013
6
2.7
6
0.0889
0.0207
0.0833
w
Hexanebenzene-
p-xylene
87 TABLE 11 (continued) Mixture
cyclohexaneheptanetoluene
T
P
range (K)
range (bar)
298.15 361-381 358-379
0.05-0.12 1.013 1.013
Overall average
Np
S,(P) (%o)
16 1.5 93 0.4 35 0.5
NY
S(Y) Yl
Ref. y2
Y3
16 0.0041 0.0025 0.0029 93 0.0035 0.0029 0.0029 35 0.0023 0.0030 0.0027
144 0.5
144 0.0033 0.0029
0.0029
ww xx YY
1-Hepteneheptaneoctane
328.15
0.16-0.25
12 2.8
12 0.0057 0.0043
0.0054
72
Heptanemethylcyclohexane -toluene
372-382
1.013
78 0.6
78 0.0059
0.0041
0.0055
ma
Tolueneoctaneethylbenzene
383-405
1.013
20 0.7
20 0.0083 0.0075
0.0093
bbb
370-377 437-444
0.114 0.964
28 2.1 28 1.4
-
-
56
1.8
-
-
0.114 0.951
18 0.4 16 0.4
-
-
Overall average
34 0.4
-
-
371-377 440-441
0.114 0.964
21 3.0 19 2.3
-
-
Overall average
40 2.7
-
-
373-376 441-445
0.114 0.964
20 3.4 20 2.4
-
-
Overall average
40 2.9
-
-
Overall average
2741 3.1
1,3,5_Trimethylbenzene-1,2,4trimethylbenzenedecane
Overall average 1,3,5_Trimethylbenzene- 1,2,4369-373 trimethylbenzene337-443 1,2,3_trimethylbenzene
1,3,5-Trimethylbenzene-decane1,2,3-trimethylbenzene
1,2,4-Trimethylbenzene-decane1,2,3-Trimethylbenzene
Ternary systems
2468 0.0118 0.0089
CCC ccc
ccc ccc
ccc ccc
ccc ccc
0.0068
88
predict multicomponent data.
heats of mixing, given the current scarcity of such
Prediction of multicomponent vapor-liquid equilibria
Multicomponent vapor-liquid equilibrium calculations are performed on mixtures containing several (up to 12) normal compounds (i.e. hydrocarbons, nitrogen, carbon dioxide, hydrogen sulfide). The experimental data against which the investigation is carried out are those reported in the open literature and cover practically all the types of compounds pertaining to the realm of applicability of the method, i.e. weakly polar and non-associating fluids and low to high pressure range. The criteria used for the comparison are the bubble-point pressures and the vapor-phase compositions. The extensive results presented in Tables 11 and 12, show good agreement between experimental and calculated values.
CONCLUSIONS
The main purpose in this work was to test the applicability of the formalism of excess functions-equations of state we outlined through the Guggenheim quasi-lattice model to the prediction of high-pressure vaporliquid equilibria of weakly polar and non-associating systems. Thus we took advantage of the attributes of the model we described previously (PCneloux
Notes to Table 11: a Trappehl and Knapp (1987). b Gupta et al. (1980). ’ Chang and Lu (1967).d Somait and Kidnay (1978). e Yarborough and Smith (1970). f Poon and Lu (1974). g Llave and Chung (1988). h Brunner et al. (1974). i Davalos et al. (1976). j Wichterle and Kobayashi (1972). k Price and Kobayashi (1959). ’ Van Ham and Kobayashi (1967). m Robinson and Bailey (1957). ” Robinson et al. (1959). ’ Wang and McKetta (1964). p Nagahama et al. (1985). q Wiese et al. (1970b). ’ Dourson et al. (1942). ’ Koonce and Kobayashi (1964). t Prodany and Williams (1971). ” Clark and Stead (1988). ” Uchytil and Wichterle (1983). w Lhotak and Wichterle (1983). x Mehra and Thodos (1963). y Herlihy and Thodos (1962). ’ Dingrani and Thodos (1978). aa Fredenslund et al. (1976). bb Hong and Kobayashi (1983). cc Wagner and Wichterle (1987). dd Inomata et al. (1987). ee Morris and Donohue (1985). “Noda et al. (1982). 88 Steele et al. (1976). hh Laurence and Swift (1974). ii Kirss et al. (1975). i Beyer et al. (1965). kk Belknap and Weber (1961). ” Rowan and Weber (1961). mmYoung and Weber (1974). “” Li et al. (1974). M,Susarev and Chen (1963). pp Ridgway and Butler (1967). qq Avai et al. (1969). ” Saito (1969). ” Michishita et al. (1971). tt SUSarev and Lyzlova (1962). “” Draiko et al. (1973). w Makh and Azarova(1946).- Katayama et al. (1965). xx Myers (1957). * Black (1959). u Kudryatseva et al. (1972). aaa Wisniak and Tamir (1977). bbb Badina et al. (1977). cc’ Malanowski (1980).
89
et al., 1989) to coin a zeroth approximation group-contribution method which was combined with the translated basic form of the Peng-Robinson equation of state (PCneloux et al., 1982; Rauzy, 1982). This goal is met to a large extent as the predictive power of group contributions is brought to high-pressure calculation for the correlation of the binary interaction parameters required in the equation of state mixing rules, without damaging the quality of the results with respect to the classical application of the Peng-Robinson method with these parameters fitted. To do so, the temperature dependence of group-contribution correlations were generated from low-pressure excess enthalpy and vapor-liquid equilibrium data and then extended to the prediction of high-pressure vapor-liquid equilibria. This was an occasion to show the improvements that can be brought about by the use of equations of state in the low-pressure calculation of excess functions (AE, GE and HE>.
LIST OF SYMBOLS
1 A A[; b B BEL,
E(T, Ezj
Echij ECI H Hij HChij kij li ; Nh Np % P P P,
x>
equation of state energy parameter Helmholtz energy group interaction parameter between groups K and L parameter A,, at a reference temperature covolume; equation of state size parameter group interaction parameter between groups K and L parameter B,, at a reference temperature residual excess function binary energy parameter binary parameter of conformation constant in the term of conformation molar enthalpy binary enthalpy parameter binary enthalpy parameter of conformation binary interaction parameter in the classical mixing rules parameter of conformation of component i number of moles in a mixture number of groups in a solution number of determinations of excess enthalpy number of determinations of bubble-point pressure number of determinations of vapor-phase composition number of components in a mixture pressure vapor pressure of component i
Hexane-methylcyclopentane-benzene-cyclohexane-toluene Nitrogen-methane-carbon dioxide-ethane-hydrogen
Methane-ethane-propane-butane-pentane Methane-ethane-propane-toluene-l-methylnaphthalene
sulfide-propane
313.70 352.60 260.00 346.75 354.80 354.03 355.76,356.39 357.30 410.62-426.75 410.52-423.83 372.05-376.70 322.04 313.70 310.93 377.59 410.93 44.26 477.59 348.86-357.56 171.95 177.00 185.95 199.80 200.20 213.75 213.70 227.60 235.85 79.1, 89.4 275.80 10.00 1.013 1.013 1.013 1.013 1.013 1.013 1.013 0.114 73.2- 92.1 54.4-106.2 35.6-119.7 13.8-137.9 16.5-140.8 14.1-136.4 14.3-136.5 1.013 19.70 21.90 27.40 37.90 41.30 49.40 55.10 62.10, 68.90 80.40
(bar) 2 8 7 1 1 1 2 1 3 5 45 10 4 5 9 9 9 9 10 1 1 1 1 1 1 1 2 1
N, 4.3 5.7 8.0 0.1 0.5 0.5 0.6 0.5 4.6 1.4 2.4 2.9 3.6 2.3 14.9 8.4 4.9 6.1 1.0 3.1 4.9 4.1 5.1 2.8 3.9 0.7 1.2 0.1
6,(P) (o/o)
values, for quaternary
P range
and experimental
Nitrogen-methane-ethane-propane Nitrogen-methane-butane-decane Methane-ethane-carbon dioxide-butane Hexane-methylcyclopentane-benzene-cyclohexane Hexane-methylcyciopentane-benzene-toluene Hexane-methylcyclopentane-cyclohexane-toluene Hexane-benzene-cyclohexane-toluene Methylcyclopentane-benzene-cyclohexane-toluene Benzene-ethylbenzene-isopropylbenzene-a-methyls~rene Toluene-ethylbenzene-isopropylbenzene-a-methylstyrene 1,3,5-Trimethylbenzene-l,2,4-trimethylbenzene-decane-1,2,3-trimethylbenzene Nitrogen-methane-carbon dioxide-ethane-propane
d,(P), between calculated T range (RI
pressure,
Mixture
Relative mean deviations in the bubble-point order systems
TABLE 12
i
I
I
I
I
f
i
I
I
d
h
h
h
h
g
a
a
f
e
e
d
d
d
d
d
c
b
a
Ref.
and higher
ZS
and higher order systems overall average
310.70 210.93 188.70 172.04 313.70 253.12 263.14 273.18 283.17 293.15 273.18 283.17 293.15 243.14 253.12 263.14 273.18 283.17 172.04 185.93 199.82 338.70 322.00 338.70
104.9, 108.6 68.9-110.3 48,3- 82.7 27.6- 58.6 68.2, 75.9 6.9- 17.1 6.9- 23.0 6,9- 30.7 6.9- 38.4 6.9- 45.4 6.9- 28.0 6.9- 37.9 6.9- 44.6 6.9- 25.2 6.9- 31.0 6.9- 42.5 6.9- 53.8 6.9- 53.8 5.2- 27.6 13.81 41.4 27.6- 55.2 41.6-130.6 82.7-111.6 110.6, 111.6 278
2 3 3 3 2 4 6 6 7 8 6 7 9 5 6 8 8 9 3 3 3 7 7 2 4.6
0.7 8.0 9.7 8.0 5.9 4.5 4.1 3.6 3.8 4.3 10.6 3.9 5.6 4.5 4.5 4.4 3.8 3.9 6.5 6.0 5.0 5.7 3.4 10.7 m
m
m
j
j
i
J
1
I
I
1
k
k
k
k
k
k
k
k
a
* j j j
a Yarborough and Smith (1970). b Llave and Chung (1988). ’ Clark and Stead (1988). d Weatherford and Van Winkle (1970). e Beregovykh et al, (1979). f Malanowski (1980). g Hanson and Brown (1945). h Li et al, (198lb). i Kalva and Robinson (1979). j Cunningham et al. (1980). k Hong and Kobayashi (1986). ’ Hong and Kobayashi (1987). m Turek et al. (1984).
Quatemary
Carbon dioxide-methane-ethane-propane-butane-pentane-h~x~~~heptane-octane-decane-tetradecane
Nitrogen-carbon dio~de-methane-ethane-pro~ane-butane-~cntanehexane-heptane-octane
dioxide-ethane-hydrogen-s~lf~de-propane-bMtane-pent~e
sulfide-propane-butane-toluene
Carbon dioxide-ethane-hydrogen
Methane-carbon
su~~de-propane-bu~e-~~t~e
Carbon dio~d~-ethane-hydrogen
Nitrogen-methane-carbon dioxide-ethane-propane-butane Nitrogen-methane-ethane-propane-butane-pen&me
92
R T
G 4 w V ‘i Yi
gas constant temperature normal boiling temperature cubic equation of state parameters molar volume mole fraction of component i mole fraction of i in the vapor phase
Greek Letters aik
s’
surface area fraction or site fraction of group K in molecule i parameter of the “basic form” of a cubic equation of state deviation between calculated and experimental values of a property solubility parameter of compone_nt i packing fraction (7 = b/v, 17 = b/u’) function of the equation of state energy parameter a and of the packing fraction q, in the Helmholtz energy at constant packing fraction acentric factor reduced parameter of an equation of state reduced parameter of an equation of state
Subscripts C
cal exp . . i,lL
r res
critical property calculated experimental referring to components i, j referring to groups K, L reduced property residual
Superscripts a
E ‘KL
*
in the temperature dependence of the energy parameter the equation of state, depending on Ln, excess property in the temperature dependence of A,, and BKL ideal gas property pseudo-property
a(T) of
93
The financial support provided PCtroles is greatly appreciated.
by TOTAL-Compagnie
Fraqaise
des
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APPENDIX
A: BASIC FORM OF THE PENG-ROBINSON
EQUATION
OF STATE
The properties of each component are defined by the parameters o or T,, PC, Tb, m, depending on its class.
T,, PC,
P = RT,‘( t; - i;) - a/q v’+ ri;)
(Al)
where 6 = &,RT,/P,,
CM)
a, = fl,( RTC)*/PC and y = 4.82843
(a) Fun&~~~n a(T) for the commends of class (11, de~ned by T,, PC, Tb and m if TIT,
a=a
m if T,sTrT,
a=AZ:,+(l--X)a
where X= (T - TJ3/[(T
(W with q=0.9TC
- TJ3 + (q - T)~]
if T 2 Ti
a=a S
VW
(fi) (A61
a, is calculated by a, =a,(1 +m,[l
- (T/Tb)*‘“] -m,(l
-T/T,))
w?
with m, = 7.1553Om - 1.98255 m2 = 25776Sm - 0.99127
(W
where m is the parameter adjusted to the compound vapor-pressure data, and a, is the value of a(T) at the normal boiling temperature T,, (see Table 1 for approximate values of a,). a, is calculated by a, = a,(1 + m,[ 1 - (T/TJG]}’
(Y= 3.6 - 6.9i-&
(A91
101
with m, = [ ( a,/a,)l’z
- I]/[
1 - WT,)~]
(*lo)
(b) Function a(T) for the compounds of class (2), defined by T,, PC and o
a = a,(1 + m[l - (T/T,)“]}’
(A14)
m = 6.8130 (1.12753 + 0.517250 - 0.00374~11~)~‘~ - I]
W5)
[
APPENDIX B: CORRELATION OF SUBSTITUTED ALKANES
FOR THE PARAMETER
OF “CONFORMATION”
I = E, + 0.023437( T - T&Z,, + 0.002589( T,2/P, - T&&,) +2.966&z, + 0.55605n,
Pl)
I is the parameter of a substituted alkane isomer of the normal alkane whose parameter is 1, and critical constants are T, and P,,, n3 is the number of ternary carbon atoms, and n4 the number of quaternary carbon atoms. The results of correlation (Bl) for all the isoalkanes considered in this study are reported in Table 2 where they are compared with the parameters Zi eventually adopted, which were adjusted so as to yield better results. The estimates from that correlation, though often correct, are sometimes inadequate for the prediction of excess enthalpies.
APPENDIX C
Relative percentage mean deviation between experimental and calculated properties (e.g. bubble-point pressure and excess enthalpy):
‘r(‘) = (loo/&)
31‘exp,i-
i=l
‘cal,i
1/‘exp,i
102
Absolute mean deviation between experimental and calculated properties (e.g. vapor-phase composition and excess enthalpy): 6(Y) = (1/N_) Z
1 Yexp,i -Ycal,i
1
i-l
6(HE)
=
(1/Nh>
2
i=l
1HeEp,i
-Hctl,i
1
47
Group-contribution equation of state for correlating and predicting thermodynamic properties of weakly polar and non-associating mixtures. Binary and multicomponent systems W. Abdoul, E. Rauzy and A. PCneloux Laboratoire de Chimie Physique, Faculte’ des Sciences de Luminy, 13288 Marseille G5de.x 9 (France) (Received November 14, 1990; accepted in final form June 12, 1991)
ABSTRACT Abdoul, W., Rauzy, E. and PCneloux, A., 1991. Group-contribution equation of state for correlating and predicting thermodynamic properties of weakly polar and non-associating mixtures. Binary and multicomponent systems. Fluid Phase Equilibria, 68: 47-102. The applicability of the “Guggenheim quasi-lattice excess function equation of state” formalism is tested through a zeroth approximation group-contribution model and a Peng-Robinson-type equation of state. The temperature-dependent correlations of the group-contribution model are derived from low-pressure excess enthalpy (iYE) and vaporliquid equilibrium (VLE) binary data and are then extended to the prediction of high-pressure vapor-liquid equilibria of mixtures of weakly polar and non-associating compounds (i.e. hydrocarbons, CO,, H,S, N,). The numbers of binary systems, data sets and determinations used in the development and testing of the method are 169, 805 and 10634 for low-pressure VLE, 282, 789 and 11308 for HE; and 163, 1329 and 12272 for high-pressure VLE, respectively. The average relative deviation with experiment in bubble-point pressure is 1% and the average absolute deviation in vapor-phase composition is 0.005 for low-pressure VLE data. Heat effects are well reproduced with average deviations of 12 J for 7009 determinations of binaries with HE less than 300 J and 5.5% for 4299 determinations of binaries with higher HE. Owing to a term accounting for chain-length effects, the accuracy of the prediction of high-pressure VLE is not deteriorated by the group-contribution approach (average deviations of 3% in bubble-point pressure, 0.0103 in vapor-phase composition). Thus, in addition to its predictive power, the method is found to be an improvement over the classical Peng-Robinson procedure with constant binary interaction parameters, especially for mixtures of molecules of different sizes.
INTRODUCTION
The most common way to describe high-pressure fluid phase equilibria is from equations of state. The works of Vidal (19781, Huron and Vidal 0378-3812/91/$03.50
0 1991 Elsevier Science Publishers B.V. All rights reserved
48
(1979), Mollerup (1981) and Whiting and Prausnitz (1982) have shown the connection between equation of state (EOS) mixing rules and excess functions. As examples, several pairs may be mentioned: Redlich-KwongSoave EOS + NRTL (Huron and Vidal, 1979), Hole Theory EOS + NRTL (Takishima and Saito, 1985), Ishikawa-Chung-Lu, Redlich-Kwong-Soave, Peng-Robinson, Martin EOS + ASOG (Tochigi and Kojima, 1984; Tochigi et al., 19851, Simonet-Behar-Rauzy EOS + UNIQUAC (Lermite and Vidal, 1986), van der Waals EOS + UNIFAC (Gupte et al., 1986) and Perturbed Soft Chain Theory EOS + a group-contribution model (Jin et al., 1986). In a preliminary paper (Peneloux et al., 1989) we have shown such connections between the zeroth approximation of Guggenheim’s quasilattice model and various equations of state. Here, we apply the principles we reported previously to a specific excess function model and equation of state, namely a zeroth approximation group-contribution model coupled with a translated Peng-Robinson-like (PR) equation of state (PCneloux et al., 1982; Rauzy, 1982). Many temperature-dependent correlations included in the method stem empirically from correlating low-pressure vapor-liquid equilibrium and excess enthalpy data and are then extended to high-pressure vapor-liquid equilibria. The range of applicability of this method is restricted to mixtures of weakly polar and non-associating compounds such as hydrocarbons, hydrogen sulfide, carbon dioxide and nitrogen.
DESCRIPTION OF THE MODEL
As was reported previously (Peneloux et al., 1989) the model used is characterized by the two following conditions. (1) The pure-component Helmholtz energies are calculated from equations of state. (2) The excess functions are defined at constant packing fraction, the latter described by v’/u, u” being the close-packed volume. We assume that it is possible to define a covolume bi proportional to u”, which enables us to measure the packing fraction by 77= b/u. We choose the same packing fraction for pure components as for the mixture. Those assumptions lead us to write the molar Helmholtz energy of a mixture as follows: A(T, q, n) =A*(T,
7, n) - nRT ln(1 - 77)- 2 (niai/bi)*l(v)
+ ml:,
i=l (1)
The various working equations which are used in conjunction (1) are reported in a previous paper (PCneloux et al., 1989).
with eqn.
49
Pure-component equations of state
For the components’ equation of state we have chosen a translated Peng-Robinson cubic equation of the form P=RT/(u-b)-a(T)/(u2+ubu+wb2)
(2)
Harmens and Knapp (1980), Schmidt and Wenzel (1980) and Pate1 and Teja (1981) proposed correlations for the parameters u, w, a and b for components obeying the law of corresponding states. We use this equation according to a technique which was originally applied to the RedlichKwong-Soave equation (RKS), the general principles of which were given in previous papers (PCneloux et al., 1982; Rauzy, 1982; Rauzy and PCneloux, 1986). In particular, it was observed that a translation in linear dependence on component mole fraction along the volume axis leaves the results T, P, x of vapor-liquid equilibrium calculations unchanged. Thus it is possible to express eqn. (2) in terms of a translated volume 5, the pseudo-volume: P = RT/( fi - 6) - a(T)/fi(
i; + 76)
We call this the basic form of eqn. (21, which is defined by a parameter y which is characteristic of the equation of state under consideration (e.g. y=OforvdW,y=lforRKS,y = 4.82843 for PR). In this work y = 4.82843 is retained, i.e. we use the translated basic form of the Peng-Robinson (1976) EOS. It is well known that a good correlation of the pure-component vapor pressures is a requirement for accurate prediction of mixture vapor-liquid equilibria. The success obtained with the Soave (1972) equation of state is a good example. For low reduced temperatures, however, Soave’s function a(T) leads to relative deviations with respect to experimental vapor pressures well above 1%. These shortcomings may be overcome in two ways. (1) a(T) is adjusted at each temperature to reproduce pure-component vapor pressure well (Goral et al., 1981). The value obtained is then used in mixing rules for the description of mixtures. A major drawback with this procedure is that it does not allow excess enthalpy calculations. (2) A temperature-dependent correlation is designed, the parameters of which are fitted to provide good agreement with experimental pure-component vapor pressure data (Vidal, 1983; Gibbons and Laughton, 1984). Since we are concerned with low-pressure vapor-liquid equilibrium and excess enthalpy calculations, we are led to adopt the latter procedure and to define two classes of compounds. (1) For components which are likely to be encountered at very low pressures in our calculations, we resort to the Carrier-Rogalski-PCneloux correlation for a(T) (Carrier et al., 1988; Appendix A(a)>. The properties
50
of those components are defined by their critical temperature and pressure, their normal boiling temperature and a fourth parameter m fitted to the experimental pure-component vapor-pressure data. (2) For the remaining compounds, a(T) is still calculated by the procedure defined by Rauzy (1982) which makes use of a Soave-like expression (see Appendix A(b)). A list of the compounds considered in this study and the parameters defining their properties are given in Table 1. Residual excess Helmholtz energy A:,, For AFe, we choose a function in which the composition fraction variables are separated %7) = ln(l + Yrl)/Y to be a sum of two terms:
4% = E(T, -++7) E(T, X) is postulated E = (l/2)
and packing (4
i i x~&x~~~,!Z~~(T)/~+ (l/2) i i xixjEchij(T) ix1 j=l i-1 j=l
(5)
The first term on the right-hand side of eqn. (5), a Van Laar-like excess function, is related to the classical van der Waals mixing rules linear in b and quadratic in a: b = i xibi i=l
U= ~ i=l
~
XiXj(
aiaj)1’2(
1-
kij) (6)
j=l
TABLE l(a) Parameters of the compounds of class 2, defined by T,, PC, w Compound
MW
T, (K)
PC(bar)
o
Tb (K)
126.26 190.55 282.40 305.43 304.21 373.20 364.85 369.82 408.14 417.90 419.60 425.37 425.16 647.37
33.97 46.01 50.39 48.75 73.79 89.40 46.01 42.68 36.55 40.02 40.23 43.27 37.96 221.20
0.0400 0.0111 0.0850 0.0970 0.2250 0.1090 0.1440 0.1536 0.1846 0.1900 0.1870 0.1950 0.2008 0.3440
77.40 111.64 169.44 184.57 194.70 212.80 225.45 231.08 261.42 266.25 266.89 268.74 272.65 373.15
(g mol-‘I
Nitrogen Methane Ethylene Ethane Carbon dioxide Hydrogen sulfide Propylene Propane Isobutane Isobutene 1-Butene 1,3-Butadiene Butane Water
28.01 16.04 28.05 30.07 44.01 34.08 42.08 44.10 58.12 56.11 56.11 54.09 58.12 18.02
51 TABLE l(b) Parameters of the compounds of class 1, defined by Compound
Neopentane 3-Methyl-1-butene Isopentane I-Pentene 2-Methyl-1-butene Isoprene Pentane 2-Methyl-2-butene trans-13Pentadiene Cyclopentane 2,ZDimethylbutane 4-Methyl-1-pentene 2,3_Dimethylbutane 1,SHexadiene 2-Methylpentane 2-Methyl-1-pentene 3-Methylpentane 1-Hexene truns-3-Hexene trans-2-Hexene Hexane cis-2-Hexene Methylcyclopentane 2,ZDimethylpentane Benzene 2,4-Dimethylpentane Cyclohexane 2,2,3_Trimethylbutane Cyclohexene 2,3_Dimethylpentane 3-Methylhexane 1-Heptene Heptane Isooctane Methylcyclohexane 2,SDimethylhexane 2,CDimethylhexane Toluene 2,3,4_Trimethylpentane 4-Methylheptane 3,4_Dimethylhexane Cycloheptane truw1,4,-Dimethylcyclohexane 1-Octene
T,,PC,Tbrm
MW (g mol-‘I
T,
72.15 70.14 72.15 70.14 70.14 68.12 72.15 70.14 68.12 70.14 86.18 84.16 86.18 82.15 86.18 84.16 86.18 84.16 84.16 84.16 86.18 84.16 84.16 100.21 78.11 100.21 84.16 100.21 82.15 100.21 100.21 98.19 100.21 114.23 98.19 114.23 114.23 92.14 114.23 114.23 114.23 98.19
433.80 450.00 460.43 464.70 465.00 484.00 469.64 470.00 496.00 511.60 488.70 518.00 499.90 507.00 497.50 518.00 504.40 504.00 519.90 516.00 507.35 518.00 532.70 520.48 562.10 519.70 553.40 531.10 560.40 537.30 535.20 537.20 540.15 544.10 572.10 550.06 553.50 591.70 566.43 561.70 568.87 606.20
32.02 35.16 33.80 40.53 34.45 38.50 33.72 34.45 39.92 45.09 30.80 32.42 31.31 34.45 30.09 32.42 31.21 31.71 32.53 32.73 30.10 32.83 37.90 27.76 48.94 27.36 40.73 29.59 43.47 29.08 28.17 28.37 27.34 25.74 34.75 24.87 25.60 41.14 27.26 25.43 26.95 38.10
282.65 293.21 301.00 303.12 304.31 307.22 309.21 311.72 315,18 322.41 322.89 327.01 331.14 332.61 333.42 335.26 336.43 336.63 340.24 341.03 341.89 342.04 344.96 352.35 353.24 353.65 353.89 354.03 356.13 362.93 365.00 366.79 371.57 372.39 374.08 382.25 382.58 383.78 386.62 390.86 390.87 391.94
0.50083 0.49832 0.51808 0.48665 0.54064 0.52250 0.53980 0.54010 0.53054 0.48976 0.52535 0.55300 0.54034 0.51763 0.57152 0.57550 0.56150 0.57182 0.58384 0.58145 0.59178 0.58121 0.52655 0.58476 0.51501 0.60088 0.51738 0.54581 0.50681 0.58880 0.61366 0.62259 0.64167 0.59916 0.53508 0.65206 0.64105 0.55087 0.60796 0.66519 0.63125 0.53757
0.23673 0.23552 0.25585 0.22311 0.25713 0.24487 0.26893 0.26678 0.24977 0.23749 0.31541 0.32196 0.32653 0.30654 0.33887 0.33142 0.33602 0.33153 0.33730 0.33435 0.35461 0.33571 0.30768 0.40141 0.26661 0.40760 0.30795 0.38927 0.29677 0.41048 0.42296 0.42416 0.44616 0.47327 0.38601 0.50662 0.49763 0.35488 0.48742 0.51935 0.50011 0.39592
112.20 112.22
587.70 566.60
29.69 26.24
392.50 394.43
0.56789 0.66458
0.47634 0.5 1506
(K) PC(bar)
Tb
(K) rn
a0x10-s (cm6 bar mol-‘)
52 TABLE l(b) (continued) Compound
2,2,4,4_Tetramethylpentane 2,2,5Trimethylhexane Octane 2,2,4_Trimethylhexane Ethylcyclohexane 2,ZDimethylheptane 2,4-Dimethylheptane 2,2,3,4_Tetramethylpentane 4,4_Dimethylheptane 2,6_Dimethylheptane 2J-Dimethylheptane 3,5-Dimethylheptane Ethylbenzene 3,3_Dimethylheptane p-Xylene 2,3,4_Trimethylhexane m-Xylene 2,2,3,3_Tetramethylpentane 2,3_Dimethylheptane 3,4_Dimethylheptane 4-Ethylheptane 2,3,3,4_Tetramethylpentane 4-Methyloctane 3-Ethylheptane 2-Methyloctane 3-Methyloctane o-Xylene Styrene 3,3-Diethylpentane Nonane Cyclooctane Isopropylbenzene Isopropylcyclohexane Propylcyclohexane Propylbenzene 2,6_Dimethyloctane 1,3,5Trimethylbenzene wMethylstyrene tert-butylbenzene 1,2,4_Trimethylbenzene Decane 1,2,3_Trimethylbenzene Butylbenzene tram-Decalin ck-Decalin
MW (g mol-‘)
128.26 128.26 114.23 128.26 112.22 128.26 128.26 128.26 . 128.26 128.26 128.26 128.26 106.17 128.26 106.17 128.26 106.17 128.26 128.26 128.26 128.26 128.26 128.26 128.26 128.26 128.26 106.17 104.15 128.26 128.26 112.22 120.20 126.24 126.24 120.20 142.29 120.20 118.18 134.22 120.20 142.29 120.20 134.22 138.25 138.25
T, (K)
PC(bar)
Tb (K) m
uo x10-s km6 bar mol-‘)
574.70 568.00 568.76 574.20 609.00 576.80 581.00 592.70 591.00 576.60 581.00 585.50 617.10 591.00 616.20 597.40 617.00 607.70 591.40 595.80 594.30 607.70 589.80 594.30 587.00 589.80 630.20 647.00 616.00 593.60 647.20 631.00 640.00 639.00 638.30 604.90 637.30 654.00 660.00 649.10 617.65 664.50 660.50 690.00 702.20
24.85 23.30 24.87 23.70 30.30 23.50 23.00 26.02 24.50 22.10 23.00 23.90 36.07 24.50 35.16 25.80 35.46 27.41 24.10 25.10 24.30 27.16 23.40 24.30 23.10 23.40 37.30 39.92 27.30 22.90 35.64 32.12 28.37 28.07 32.02 21.00 31.31 34.05 29.69 32.32 20.95 34.55 28.88 31.41 31.41
395.43 397.23 398.82 399.69 404.93 405.84 406.04 406.17 408.35 408.36 409.15 409.15 409.35 410.16 411.51 412.19 412.27 413.42 413.65 413.75 414.35 414.70 415.57 416.15 416.41 417.36 417.58 418.29 419.32 423.97 424.29 425.56 427.91 429.87 432.39 433.53 437.89 438.65 442.27 442.53 447.30 449.27 456.42 460.42 468.92
0.60910 0.66619 0.68941 0.64194 0.59287 0.66903 0.67894 0.60997 0.66903 0.67894 0.67894 0.67894 0.58996 0.66903 0.59385 0.65111 0.59718 0.59759 0.67894 0.67894 0.70630 0.60465 0.70630 0.70630 0.70630 0.70630 0.59103 0.58810 0.60863 0.73343 0.56110 0.62319 0.61688 0.63506 0.62532 0.72159 0.65731 0.64815 0.65160 0.64105 0.78143 0.62724 0.67120 0.58208 0.59683
0.54702 0.57443 0.54738 0.57573 0.50032 0.59246 0.60713 0.55731 0.58821 0.62779 0.61257 0.59717 0.44257 0.59126 0.45441 0.57477 0.45259 0.55627 0.60571 0.58936 0.60521 0.56255 0.62269 0.60828 0.62840 0.62583 0.44816 0.43357 0.57388 0.65244 0.48512 0.51998 0.58749 0.59519 0.53624 0.73088 0.55376 0.52856 0.60311 0.55426 0.77291 0.54396 0.64064 0.62573 0.64862
53 TABLE l(b) (continued) Compound
Undecane Pentylbenzene Tetralin Dodecane Naphthalene Hexylbenzene Tridecane 2-Methylnaphthalene I-Methylnaphthalene Tetradecane Biphenyl Diphenylmethane Pentadecane 1-Hexadecene Hexadecane Heptadecane 1-Octadecene Octadecane Eicosane
MW (g mol-‘)
T, (K)
156.31 148.25 132.21 170.33 128.17 162.28 184.37 142.20 142.20 198.38 154.21 168.24 212.42 224.43 226.43 240.47 252.49 254.48 282.55
640.66 679.90 719.00 660.54 748.43 700.20 675.80 761.00 772.00 694.72 789.26 770.00 707.00 717.00 722.92 733.37 739.00 746.80 767.78
PC (bar)
Tb (K)
m
a0 x 10-8 (cm6 bar mol-‘)
19.68 26.04 35.16 18.38 40.53 23.81 17.23 35.06 35.60 16.24 38.46 28.60 15.20 13.37 14.54 13.17 11.35 13.16 12.02
469.08 478.55 480.51 489.47 491.09 499.25 508.62 514.20 517.83 526.73 528.15 537.42 543.83 558.02 560.01 575.17 587.97 589.45 616.95
0.82141 0.72100 0.61149 0.86161 0.58011 0.76248 0.90192 0.63057 0.63183 0.93844 0.64847 0.72996 0.98129 1.03604 1.01548 1.05220 1.05000 1.08586 1.15816
0.88703 0.75613 0.61919 1.01180 0.58272 0.87710 1.13780 0.70425 0.70944 1.27310 0.69284 0.88803 1.41600 1.63930 1.54810 1.75240 2.04310 1.83240 2.12970
kij being the binary parameters interaction, and the first term on the r.h.s. of eqn. (5) was shown to be strictly equivalent to these if the second term were zero and Eij was defined by Eij = (Si -
si)”+ 26iSjkij
(7)
Moreover, since that term is Van Laar-like, we use, as did Kehiaian et al. (19711, the equation of Redlich et al. (1959) to express Eij in terms of group contributions E,j = ( -
l/2) f
2 (aiK - ‘yjK)(“iL- “jL)AKLCT)
K=l L=l
(8)
The summation is over all N groups present in solution, aiK is the fraction of molecule i occupied by group K and has the feature of site fraction of the lattice model, but relating to a group, and A,, is a temperature-dependent function. The second term on the right-hand side of eqn. (5) accounts for the differences in the conformation of the molecules. It is a universal temperature-dependent function determined to correlate the heat effects arising from mixing of globular molecules with long-chain molecules. This term is
54 TABLE 2 Parameters
of conformation
for substituted
alkanes a
Compound
lj
licalc
Compound
1,
licalc
Neopentane Isopentane 2,ZDimethylbutane 2,3_Dimethylbutane 2-Methylpentane 3-Methylpentane 2,2_Dimethylpentane 2,CDimethylpentane 2,2,3_Trimethylpentane 2,3_Dimethylpentane 3-Methylhexane Isooctane 2J-Dimethylhexane 2,4-Dimethylhexane 2,3,4_Trimethylpentane 4-Methylheptane 3,4-Dimethylhexane 2,2,4,4_Tetramethylpentane 2,2,5_Trimethylhexane 2,2,4_Trimethylhexane
2.04 4.00 4.28 4.59 4.32 5.11 4.38 2.68 6.09 5.72 5.44 4.48 3.43 3.56 8.99 5.68 4.00 4.82 0.75 5.47
2.04 3.78 4.28 4.59 4.32 5.11 4.38 2.68 6.09 5.72 5.44 3.00 3.43 3.56 6.03 5.68 6.54 5.52 3.13 4.61
2,ZDimethylheptane 2,CDimethylheptane 2,2,3,4_Tetramethylpentane 4,4-Dimethylheptane 2,6_Dimethylheptane 2J-Dimethylheptane 3,5-Dimethylheptane 3,3-Dimethylheptane 2,3,4_Trimethylhexane 2,2,3,4_Tetramethylpentane 2,3_Dimethylheptane 3,CDimethylheptane 4-Ethylheptane 2,3,3,4_Tetramethylpentane 4-Methyloctane 3-Ethylheptane 2-Methyloctane 3-Methyloctane 3,3_Diethylpentane 2,6_Dimethyloctane
4.71 4.67 9.23 7.58 4.89 4.10 4.64 8.49 10.52 12.96 6.87 6.25 6.34 14.39 7.41 6.34 6.87 7.20 12.84 6.09
5.24 5.61 8.01 i.49 5.64 5.61 5.70 8.49 7.45 12.96 7.38 7.35 7.50 11.42 7.41 7.50 6.94 7.41 10.43 6.09
a I,, value used in our calculations; Appendix B).
licak, value estimated
from correlation
found to contribute to the description of vapor-liquid tures of light alkanes with higher ones as well. EChij
=
Eo(Zi-
Zj)‘[
(Bl) (see
equilibria in mix-
( To/T)4 - 2(‘/T,)]
where we have set E, = 2.95 and TO= 298.15 K. The Zi values may be regarded either as parameters of conformation for substituted alkanes and non-alkane hydrocarbons or as chain-length parameters for straight-chain alkanes (in fact they are the numbers of carbon atoms in these). A formula using the compound critical properties is provided (see Appendix B) for the estimation of the parameters Zi for alkanes. In some instances, the parameters obtained in such a way lead to poor results, in which case fitted parameters should be favored. Table 2 compares the parameters Zi actually used in this method with those estimated by the correlation of Appendix B. Table 3 is a collection of all such parameters used in this work.
55
Excess enthalpy
The departure H-H*
enthalpy being given by
= {a[(A -A*)/T]/~(l/T)},,~
+RT(z
- 1)
(10)
we obtain the following for the residual enthalpy: HrFS= [QL/WW/T)l
X=
w/ww~)1
x 141+ Y?7)
(11)
As previously and according to eqn. (5), the binary enthalpy parameter defined by Hij=(-1/2)
E K-l
It L=l
(CYiK-(YjK)(CYiL-LYjL)BKL(T)
is
(12)
BKL is a temperature-dependent
function. A,, (eqn. (8)) and BKL are related to two group interaction parameters, AiL and BiL, per pair of groups corresponding to a reference temperature T,, taken as 298.15 K: A KL = AiL( TO/T)‘“”
AiL = A0LK
B KL = B;,( To/T)‘“”
BgL = BjK
TKL =
B;,/Ao,,
- 1
(134 uw
(134
The parameters B& are essential in excess enthalpy calculations and are therefore determined by that quantity. Such a formulation avoids having to consider a third set of group contributions to describe the the estimation of which would be changes of HE with temperature, difficult given the current shortage of excess heat capacity data. The binary conformation enthalpy is defined by Hchij = 5Eo(Ii - Zj)‘( To/T)4
(14)
Definition of the groups and calculation of their surface fractions
In order to apply the group-contribution formulae (eqns. (8) and (12)), we have to define the nature of the groups and calculate their surface fractions or site fractions. Nature of the groups
It is found that the following groups must be considered to correlate the mixtures of the compounds we are mainly concerned with (i.e. hydrocarbons, carbon dioxide, hydrogen sulfide, nitrogen and water): 1 or Al : alkane, end of chain (CH,-) 2 or A2 : alkane, body of chain (-CH,-)
56 TABLES 3(a) and 3(b) Numbers of sites of each kind of group (numbered from 1 to 17; see text) and parameters of conformation characterizing the different compounds studied a Compound
1
2
3
4
5
6
Propylene Propane Isobutane Isobutene 1-Butene 13Butadiene Butane Neopentane 3-Methyl-1-butene Isopentane 1-Pentene 2-Methyl-1-butene Isoprene Pentane 2-Methyl-Zbutene trans-1,3-Pentadiene Cyclopentane 2,ZDimethylbutane 4-Methyl-1-pentene 2,3-Dimethylbutane 1,5-Hexadiene 2-Methylpentane 2-Methyl-1-pentene 3-Methylpentane 1-Hexene truns-3-Hexene truns-2-Hexene Hexane cis-2-Hexene Methylcyclopentane 2,ZDimethylpentane Benzene 2,4_Dimethylpentane Cyclohexane 2,2,3_Trimethylbutane Cyclohexene 2,3-Dimethylpentane 3-Methylhexane 1-Heptene Heptane Isooctane Methylcyclohexane 2,SDimethylhexane 2,4-Dimethylhexane Toluene
1.0
-
-
-
-
-
2.0 1.0 2.0 1.0 1.0 1.0 1.0 2.0 0.5 1.0
1.0 1.0 2.0 0.5 2.0 0.5 3.0 0.5 0.5 -
4.0 5.0 3.0 3.5 4.5 3.5 6.0 3.5 1.0 4.5 -
-------
-
-
-
-
-
-
5.0 5.0 -
6.0 5.0
1.0 1.0 2.0 1.0 2.8 2.6 2.0 3.0 1.0 1.0 1.0 2.0 1.0 2.0 1.0 1.0 1.0
2.0 1.5 2.5 3.0 3.0 2.2 2.4 4.0 2.0 1.5 1.0 4.0 4.0 5.0 1.0 2.5 -
6.0 7.0 5.0 1.0 8.0 7.0 4.5 -
-6.0 2.0 -
6.0 -
7
13
14
15
Ii
-
3.00 3.00 2.43 2.00 4.00 4.00 4.00 2.04 5.00 4.00 5.00 3.00 2.00 5.00 3.00 5.00 5.00 4.28 5.00 4.59 6.00 4.32 5.00 5.11 6.00 6.00 6.00 6.00 6.00 5.00 4.38 4.00 2.68 4.00 6.09 2.00 5.72 5.44 7.00 7.00 4.48 4.00 3.43 3.56 4.00
---
2.0
-
-
----------
2.0 4.0
4.0 2.0 3.5 4.0 -
---------------1.0
2.0 -
2.0 4.0
4.5 5.0 -
-
2.5 -
2.0 1.0 1.0
-
1.0
2.0
-
-
-
-
-
-
4.0
-
57 TABLE 3(a) (continued) Compound 2,3,4_Trimethylpentane 4-Methylheptane 3,bDimethylhexane Cycloheptane fruns-1,4-Dimethylcyclohexane 1-Octene 2,2,4,4_Tetramethylpentane 2,2,5Trimethylhexane Octane 2,2,4_Trimethylhexane Ethylcyclohexane 2,2_Dimethylheptane 2,CDimethylheptane 2,2,3,4_Tetramethylpentane 4,4-Dimethylheptane 2,6-Dimethylheptane 2,5-Dimethylheptane 3,5_Dimethylheptane Ethylbenzene 3,3_Dimethylheptane p-Xylene 2,3,4_Trimethylhexane m-Xylene 2,2,3,3_Tetramethylpentane 2,3_Dimethylheptane 3,bDimethylheptane 4-Ethylheptane 2,3,3,4_Tetramethylpentane 4-Methyloctane 3-Ethylheptane 2-Methyloctane 3-Methyloctane o-Xylene Styrene 3,3-Diethylpentane Nonane Cyclooctane Isopropylbenzene Isopropylcyclohexane Propylcyclohexane Propylbenzene 2,6-Dimethyloctane 1,3,5_Trimethylbenzene a-Methylstyrene tertButylbenzene 1,2,4_Trimethylbenzene
1234567 2.0 2.0 2.0 1.0 2.0 1.0 1.0 1.0 1.0 2.0 1.0 2.0 1.0 2.0 2.0 1.0 2.0 1.0 1.0 2.0 2.0 2.0 2.0 1.0 2.0 2.0 2.0 1.0 1.0 1.0 3.0 2.0 3.0
5.0 4.0 5.0 1.0 6.0 2.5 1.0 3.5 3.5 3.0 2.0 3.5 5.0 1.0 3.0 1.5 0.5 3.0 5.0 5.0 6.0 5.0 4.5 6.0 7.0 2.0 2.0 4.5 -
8.0 1.0 2.0 9.0 8.0 5.5 4.5 4.5 9.0 4.0 7.0 4.5 2.0 4.0 6.5 7.5 5.0 2.0 2.0 9.0 1.0 2.0 3.5 1.0 9.0 3.0 3.0 4.5 4.0 -
7.0 8.0 -
6.0 _ 6.0 -
-
-
-
-
-
6.0 6.0 -
4.0 5.0 6.0 5.3 3.0 6.0 6.0 3.0
-
-
13
14
15
-
-
_
2.0
-
-
-
-
-
-
-
-
l.O-
-
-
2.0 0.7 3.0 3.0 -
-
li 8.99
5.68 4.00 7.00 4.00 8.00 4.82 0.75 8.00 5.47 4.00 4.71 4.67 9.23 7.58 4.89 4.10 4.64 4.00 8.49 4.00 10.52 4.00 12.96 6.87 6.25 6.34 14.39 7.41 6.34 6.87 7.20 4.00 4.00 12.84 9.00 8.00 4.00 4.00 4.00 4.00 6.09 4.00 4.00 6.00 4.00
58 TABLE 3(a) (continued) Compound
12
Decane 1,2,3_Trimethylbenzene Butylbenzene rrans-Decalin c&Decalin Undecane Pentylbenzene Tetralin Dodecane Naphthalene Hexylbenzene Tridecane 2-Methylnaphthalene 1-Methylnaphthalene Tetradecane Biphenyl Diphenylmethane Pentadecane I-Hexadecene Hexadecane Heptadecane 1-Octadecene Octadecane Eicosane
2.0 3.0 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 2.0 1.0 2.0 2.0 1.0 2.0 2.0
a Zi, parameter
3456 8.0 3.0 9.0 4.0 10.0 5.0 11.0 12.0 1.0 13.0 13.0 14.0 15.0 15.0 16.0 18.0
of conformation
-------
8.0 8.0 4.0 -
2.0 2.0 -
-
-
-
-
-
-
3.0 5.3 5.3 4.0 10.0 5.3 9.0 10.0 7.5 10.6 -
7
13
14
15
Ii
3.0 0.7 0.7 2.0 -
-
-
10.00
0.7 1.0 4.5 1.4 -
2.0 2.0 -
-
-
6.00 4.00 10.00 10.00 11.00 5.00 8.00 12.00 4.00 6.00 13.00 4.00 4.00 14.00 8.00 8.00 15.00 16.00 16.00 17.00 18.00 18.00 20.00
of the compound.
TABLE 3(b) Compound
8
9
Nitrogen Methane Ethylene Ethane Carbon dioxide Hydrogen sulfide Water
1.0
-
-
1.0 -
3orA3: 4 or CO 5 or Cl 6 or BO 7or Bl 8
: : : : :
10
11
12
16
17
li ’
1.0 -
1.0 -
1.0 -
1.0 -
1.0
1.00 1.00 2.00 2.00 2.00 2.00 2.00
substituted alkane, or rigid alkane cyclohexane substituted cycloalkane benzene substituted benzene methane (CH,)
59
9 : ethane (C,H,) : carbon dioxide (CO,) 10 11 : nitrogen (N,) 12 : hydrogen sulfide (H,S) 13 or MO : 1-alkene 14 or DO : substituted alkene 15 or CA : cycloalkene 16 : ethylene (C,H,) 17 : water (H,O) Note that the alkane groups (Al, A2 and A3) are different from the cyclohexane group. All groups interact with each other. This approach differs from that adopted in other group-contribution methods such as UNIFAC and ASOG. Considering the cycloalkanes, for example, the following remarks are made. (1) Cyclohexane is the most frequent representative of this class of compounds. Examination of the binary systems it forms shows that its -CH,groups do not have the same properties as those of an alkane. Consequently, a cyclohexane group is defined. (2) When cyclohexane is substituted, its -CH,- groups seem to be close to those of an alkane. Assuming this, the binary systems such as alkanemethylcyclohexane and aromatic-methylcyclohexane are correctly described. (3) The system cyclohexane-methylcyclohexane is poorly represented under these conditions. A substituted cycloalkane group (Cl) is therefore defined. Although the improvement brought about by the above distinction between alkane groups and cycloalkane groups is better appreciated in excess enthalpy calculations, we include here a comparison of our results with those from the various versions of the UNIFAC method (Fredenslund et al., 1977; Fredenslund and Rasmussen, 1985) for low-pressure vaporliquid equilibria. It focuses on the systems which, according to the UNIFAC method, are free from interactions (see Table 4). The model described above requires a knowledge of the group surface fractions. We examined the possibility of assigning a specific surface area to each group, as is usually done in classical group-contribution methods, but found that procedure to be ineffective if all effects were to be taken into account. So we allotted a unit area to each carbon site with a modification made necessary by substitution and heat effects. The site fraction (yiK of group K on molecule i is defined as the ratio of the number of sites of K-type on molecule i to the total number of sites of that molecule. Some examples will illustrate this point.
60 TABLE 4 Comparison of the relative mean deviations in the bubble-point pressures of the binary systems alkane-alkane, alkane-cyclane and cyclane-cyclane obtained by our method (4) and the various versions of the UNIFAC method: original UNIFAC (0, SUPERFAC (21, modified UNIFAC (3) Binary system Neopentane-pentane Isopentane-hexane Cyclopentane-2,3_dimethylbutane Cyclopentane-cyclopentane Cyclopentane-cyclooctane Cyclopentane- trans-decalin Cyclopentane-cis-decalin Pentane-hexane Pentane-methylcyclopentane Pentane-cyclohexane Pentane-methylcyclohexane 2,2-Dimethylbutane-hexane 2,2_Dimethylbutane_hexadecane 2,3-Dimethylbutane-hexane 2,3-Dimethylbutane-cyclohexane 2,3-Dimethylbutane-hexadecane 2-Methylpentane-hexane 2-Methylpentane-heptane 2-Methylpentane-octane 2-Methylpentane-hexadecane 3-Methylpentane-hexane 3-Methylpentane-heptane 3-Methylpentane-octane 3-Methylpentane-hexadecane Hexane-methylcyclopentane Hexane-cyclohexane Hexane-2,4_dimethylpentane Hexane-heptane Hexane-methylcyclopentane Hexane-octane Hexane-decane Hexane-undecane Hexane-dodecane Hexane-hexadecane Methylcyclopentane-cyclohexane 2,4-Dimethylpentane-octane Cyclohexane-2,2,3_trimethylbutane Cyclohexane-heptane Cyclohexane-isooctane Cyclohexane-methylcyclohexane Cyclohexane-4-methylheptane Cyclohexane-cycloheptane Cyclohexane-octane Cyclohexane-cyclooctane
Np 58 34 10 9 36 12 27 22 46 30 49 11 11 11 15 8 55 44 99 9 60 25 96 9 109 189 52 99 35 13 19 12 12 50 14 60 18 212 117 23 11 13 101 13
(1)
(2)
(3)
(4)
1.22 0.72 3.44 0.25 3.37 2.33 5.88 0.81 0.78 2.34 2.32 0.54 7.36 0.15 2.39 6.25 0.83 0.40 2.74 7.65 0.69 0.12 2.43 6.98 0.34 1.14 1.24 1.45 0.33 1.43 3.70 3.30 4.30 10.66 0.11 4.08 0.76 1.19 1.70 1.26 3.30 0.58 1.87 2.96
1.22 0.58 3.00 0.08 1.54 1.90 2.01 0.85 0.77 2.32 2.05 0.54 1.52 0.15 2.33 0.66 0.83 0.30 2.27 1.45 0.69 0.19 2.01 1.27 0.37 1.10 1.13 1.51 0.33 1.86 2.15 0.87 1.04 1.84 0.11 3.98 0.52 0.98 0.99 1.20 1.77 0.48 1.15 2.19
1.32 0.60 3.67 0.08 1.54 1.42 2.91 0.84 1.76 3.40 3.34 0.46 2.61 0.15 3.14 1.21 0.83 0.31 2.33 1.97 0.69 0.18 2.05 1.05 0.49 1.79 1.15 1.50 1.19 1.80 0.91 1.15 1.41 3.98 0.11 3.98 1.32 1.47 1.86 1.20 2.31 0.48 1.67 2.19
0.97 0.69 4.32 0.34 0.58 2.59 1.09 0.81 0.94 0.72 1.56 0.46 0.68 0.13 0.38 1.54 0.80 0.26 1.91 0.51 0.73 0.14 1.77 2.10 0.32 0.75 1.46 1.64 0.31 1.80 1.69 0.14 0.29 0.81 0.29 3.77 0.26 1.06 0.60 1.21 0.21 0.10 0.89 0.82
61 TABLE 4 (continued) Binary system Cyclohexane-nonane Cyclohexane- cis-decalin Cyclohexane-dodecane Cyclohexane-hexadecane Cyclohexane-eicosane Heptane-isooctane Heptane-methylcyclohexane Heptane-octane Isooctane-methylcyclohexane Isooctane-octane Cycloheptane-cyclooctane Octane-ethylcyclohexane Decane- tmns-decalin Overall average
Np 22 12 36 74 27 13 93 85 35 20 9 53 82 2519
(1)
(2)
(3)
(4)
0.96 2.28 5.56 5.61 7.89 0.22 0.28 1.75 0.26 2.32 0.60 0.73 3.64
1.46 1.39 1.10 2.16 4.60 0.15 0.29 1.71 0.31 2.32 0.51 0.77 3.71
0.79 1.85 1.34 1.89 4.37 0.16 0.54 1.72 0.68 2.28 0.51 0.19 2.30
1.42 1.52 0.88 1.01 0.87 0.39 0.26 1.67 0.78 2.04 1.06 0.64 2.78
2.08
1.37
1.62
1.10
For a normal alkane i of n carbon atoms (except for methane and ethane), the different site numbers are as follows: 2(Al) + (n - 2)(A2) sites, which gives site fractions of aiAl = 2/n and (YiAz= (n - 2)/n; as an example, for n-heptane we have 2(Al) + 5(A2). When i stands for toluene, we have l(A1) + 5(BO) + l(B1) sites, which gives site fractions of aiA, = l/7, (yin,,= 5/7 and a,nr = l/7. For a 1-alkene i of n carbon atoms there are 2(MO) + (n - 3)(A2) + l(A1) sites, giving site fractions of CY,~,,= 2/n, aiA2 = (n - 3)/n and aiAl = l/n. For the alkane side of the 1-alkenes, the rules are the same as those relating to the n-alkanes. It was found, however, that not all behavior could be described if the alkanes were divided into CH,, CH,, CH and C groups. For instance, 2,2_dimethylpropane (neopentane), 2,3-dimethylbutane and 2,2,4_trimethylpentane seem to be equivalent as far as their superficial composition is concerned. These are alkanes substituted enough to have lost their flexibility, and can therefore be regarded as typical of the group A3 (rigid alkane), characterized by cyA3= 1. The rules for the allocation of site numbers for the various types of substitutions vary. Substituted alkane.
Allocation of site numbers in this case use the following rules. (1) Consider the longest straight chain that can be formed by the alkane. (2) Allocate one Al site to each CH, group at the ends of that chain unless it belongs to an isopropyl or a tert-butyl grouping. (3) Allocate one A2 site to each CH, group and to each CH group (unless it belongs to an isopropyl grouping); then substract a half A2 site
62
(OS(A2)) each time that group is in the neighborhood of a quaternary carbon atom or of a ternary carbon atom itself neighboring another ternary carbon atom or quaternary carbon atom. (4) The complement of the total number of carbon atoms of that alkane makes up the number of A3-type sites. Consequently, an alkane molecule of it carbon atoms with nl Al-type sites and n2 A2-type sites has (n - nl - n2) A3-type sites. For example, 2-Methylpentane 3-Methylhexane 2,2,5Trimethylhexane 2,3,4_Trimethylhexane 2,2,4_Trimethylhexane 2,2,3,4_Tetramethylpentane 2,2,3,3_Tetramethylpentane
: : : : : : :
l(A1) 2(Al) l(A2) l(A1) l(A1) 9(A3) l(A1)
+ + + + +
lS(A2) + 3.5(A3) 4(A2) + l(A3) 8(A3) 1.5(A2) + 6.5&S) 2.5(A2) + 5.5(A3)
+ 0.5(A2) + 7.5(A3)
Note that the combination of rules (11, (2) and (3) implies that either isopropyl or tert-butyl groups are of type A3. Substituted benzene.
The benzene ring consists of 5.3(BO) + 0.7(Bl) sites and the same rules as those for the normal alkanes are applied to the alkane side of the substituted benzene. For ethylbenzene and propylbenzene, for example, we have l(Al) + l(A2) + 5.3(BO) + 0.7(Bl) and l(Al) + 2(A2) + 5.3(BO) + 0.7(Bl) sites, respectively. All the number of extent the with some
compounds investigated in this work are reported with the sites of each type in Table 3. It should be noted that to a large assignment of the numbers of sites obeys the foregoing rules adjustment, depending on the host molecule.
The definition of groups as influenced by heat effects
The correct description of heat effects introduces further requirements in the definition of groups. Whereas xylenes from ortho, meta and para substitutions on benzene may be similar as far as vapor-liquid equilibrium calculations are concerned, the excess enthalpies arising from their binary mixtures with a given compound show very different values. The same problem is encountered when comparing cyclopentane and substituted cyclopentane, cyclohexane and substituted cyclohexane. For excess enthalpy calculations involving the compounds mentioned above, we were led to define substitution groups: replacement of 5 by 5’ for cyclopentane and methylcyclopentane and replacement of 7 by 7’, 7” and 7”’ for ortho, meta and para xylenes. Clearly, it means that these substitution groups have specific BzL p arameters (see eqn. (13)) and it should be borne in mind that
63
these are mainly determined by excess enthalpies. Thus the interactions of cyclopentane or methyl~clopentane with other compounds are described by the same A*KL parameters as those used for the interactions of methylcyclohexane with other compounds. The Bi,_ parameters relating to cydlopentane and methylcyclopentane, however, are different from those for methyl~clohexane. The same applies to the parameters for xylenes and for ~om~unds containing the substituted benzene group. These specific parameters occupy the four lowermost lines in Table 5, which displays the group interaction parameters. Determination of group interactionparameters For the 17 groups considered here, we have to estimate 272 plus 64 group interaction parameters. Given the current limited amount and quality of experimental data relating to some systems, it is not possible to determine the whole set of parameters reliably. For this reason, a sequential procedure is used for the determination of group interaction parameters. The parameters are adjusted sequentially, by sets of two, four or six, with those which have been determined previously, and which are necessary for a knowledge of current parameters, being assumed. The order in which the different sets of parameters are fitted depends upon the relative abundance of data available for them. Thus the set of six interaction parameters between the three alkane groups are first estimated, given the extensive experimental data available for such systems. Then, to describe the interactions between benzene and alkanes, for example, which requires 12 parameters, only 6 additional parameters relating to benzene-alkane interactions have to be estimated, those for alkane-alkane interactions being considered known. As the parameters of our model are connected with vapor-liquid equilibria as well as with excess enthalpies, both types of data are taken into account when determining the parameters. In this determination an objective function which is the sum of the relative percentage mean deviations between calculated and experimental bubble-point pressures, S,(P), and l/5 times the sum of the relative percentage mean deviations between calculated and experimental excess enthalpies, 8,(HE), is minimized. The minimization technique of Nelder and Mead (1965) was used. For the adjustment of a given set of parameters, excess enthalpy and vapor-liquid equilibrium (isotherm and isobar alike) data for several binary systems involving that set were taken at different temperatures and used simultaneously. Thus the six alkane-alkane interaction parameters were determined at once from excess enthalpy and vapor-liquid equilibrium data for binary mixtures of alkanes (ranging from pentane to hexadecane) over a wide range of temperature. Those for benzene-alkane interactions
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1
0 77.97 32.68 49.48 35.67 189.01 629.31 45.87 2.21 438.21 314.35 521.62 75.27 188.78 64.00 55.86
0
3
4
5
7.70 0 28.46 22.21 0 14.93 1.84 3.62 0 85.78 113.26 87.04 72.73 177.83 291.25 382.72 281.65 97.44 44.38 112.56 28.70 67.08 0.12 30.79 5.56 408.21 409.24 173.30 368.83 230.35 231.10 307.53 438.02 393.91 401.97 57.85 -51.75 63.73 147.43 31.38 33.06 72.63 38.31 76.93
2
Group interaction parameters AtL (.I cmm3)
TABLE 5(a)
7
8
9 10
11
12
0 250.39 0 149.27 643.39 0 120.92 509.39 7.86 0 277.57 590.94 391.56 397.98 0 636.92 726.92 130.00 284.77 368.80 0 324.35 -31.56 594.10 487.52 419.33 1017.64 0 123.42 67.43 62.76 382.41 45.55 221.88 - 27.12 36.75 67.57 26.31 208.63 295.74 88.25 2289.72 1827.09 2277.16 1617.70
6 14
15
0 -75.31 0 -71.95 111.56 0 10.97 -
13
0 -
16
0
17
a
2
3
parameters
4
5
S& U cme3) 6
7
8
9
10
11
12
13
14
15 16 17
5’ 7’ 7” 7‘*
72.71 26.48 7.07 0 166.31 842.61 323.70 337-27 327.88 403.00 %X3,76192.57 369.55 380.94 327.88 227&l 875.08 120.16 369SO 437,03 338.38 228.35
1074.11 312.32 2847.25 1 0 0 3513-761282.99284725 654,22-3244.85 0 2847.25 _ _
-
.-._I -_- - -
_
$34.36 0 3 49.98 18.75 0 4 129.30 45.74 31.07 It 5 112.91 3.84 0 16.78 -6.69 6 396.15 134.11 188.09 190.54 148.04 0 0 7 634.79 127.76 369.55 423.63 327x@ 310_23 8 - 13#&3 183.62- m.94 49-83 518Sl - 62.23 3513.76 0 9 43Al6 31.30 - 2.66 -14.98 -41&I 130.46 1282,99 24.71 0 10 513.05 789.08 624.54 86.371074.11.24M.16 2847.25565.41 690.93 0 0 11 - 33.09 380.06 356.Q4 472.49 s129.22 654.22 108*98-41.74 523.80 0 12 1082.011 3658.26- ‘3244.85 72.22 567.66 617.95 1359.52 36.71)885.21 493.29 0 13 115.00 238.09- 127.42 122.71 87+49 6Nm 211.90 - 7.58 - 527.49-709,$X 648.01 0 - 24.80 x4 600-32- IS.93 - 52,30 277.94 1.92 15 Ifif. 125.38 34.6m x44 52.38 16 -80.02 91.73 2mUO 329.78 266AS IQ.76 - 0 13.81 l33.67 17 934.75 - - 0 524.45 1068_48 8133
1 2
1
Group interaction
TABLE %&I
66
were obtained from binary mixtures of benzene with alkanes (ranging from pentane to eicosane) in the same way. It should be mentioned, however, that for a given binary system not all the available data sets were used in the estimation procedure. Table 5 shows the group interaction parameters obtained.
RESULTS AND DISCUSSION
Low-pressure vapor-liquid equilibria The deviations in the bubble-point pressure and the vapor-phase composition are used as criteria for the judgement of the quality of the results obtained. In the case of isobaric VLE data, the calculated bubble-point pressures are tested against the specified pressure. The whole set of experimental data is divided into subsets to show the capability and the possible failure of the method to describe the interactions between the different families (alkanes, naphthenes, aromatics, alkenes) of compounds. The results are summarized in Table 6 and the distribution of the deviations is shown in Fig. 1. The data bank used in the investigation consists of data taken from the collections of Maczynski (1976) and Gmehling and co-workers (1980a,b, 1983) for low-pressure VLE, Christensen and co-workers (1982, 1984) for HE, and Knapp et al. (1982) for high-pressure VLE, completed with more recent data from the literature. Given the extensive experimental data
TABLE 6 Relative mean deviations in the bubble-point pressure, al(P), and absolute mean deviations in the vapor-phase composition, 6(y), for the various subsets of low-pressure experimental data defined Subset Alkane-alkane Alkane-naphthene Naphthene-naphthene > Aromatic-alkane Aromatic-naphthene > AromaticFaromatic Alkene-another hydrocarbon (alkene included) Overall average
Data sets
NP
S,(P) (%o)
N,
S(y)
79
987
1.38
482
0.0055
116
1532
0.91
714
0.0038
407
5940
0.98
3935
0.0059
47
607
0.71
298
0.0047
156
1568
1.06
457
0.0069
805
10634
1.00
5886
0.0056
67 n
50
-
-
0
dP/P %
0
Fig. 1. Histogram of the relative percentage mean deviations between calculated and experimental bubble-point pressures for low-pressure vapor-liquid equilibria. 169 binaries were treated. n is the number of binaries, the deviations related to which are in the ranges shown.
investigated, only summaries of the comparisons are reported here. Reference should be made to Abdoul (1987) for details of specific systems, conditions and references. The proposed method gives an average precision of 1% (relative) in the prediction of bubble-point pressure and of 0.005 (absolute) for the vaporphase composition. The ’ outstanding homogeneity of the average results obtained for the various families (see Table 6) should be noted. Figure 1 shows that the cases for which the mean deviation is over 3.5% are exceptional. These results foreshadow the fairly good quality of the predictions that could be made for the binary systems not yet experimentally studied. Some excess function models (NRTL, UNIQUAC, UNIFAC) are derived under the assumption that the excess Helmholtz energy at constant volume AE can be equated with the excess Gibbs energy at constant pressure GE, whereas a proper conversion would require an equation of state. Our model enables one to carry out calculations of both functions and to compare them. Figure 2 shows that if the binary system under investigation is made up of molecules which are similar (e.g. hexanecyclohexane, hexane-heptanel, the two functions have practically equal values. But if that condition is not fulfilled a great difference between AE
68
T
GE,AE/J
0.0
I
\ \ \
-200
-400
I
0.0
\\ I
\\ I
\ \ .-- / / / 1
I
0.5
I
I
I/ /1’ I
x1\
I
1.0
Fig. 2. Comparison of the molar excess Gibbs energy, GE (full lines), with the molar excess Helmholtz energy, AE (dashed lines), both calculated by the method presented, at 298.15 K for binaries whose components are of nearly equal sizes: 6-C6 (A) and 6-7 (B), or of very different sizes: 6-16 (C).
and GE is observed (e.g. hexane-hexadecane). This supports the idea that an equation of state should be used, even for low-pressure calculations, if non-ideality stemming from the difference in size of the molecules is to be properly represented. We regard as failures all the binary systems for which the relative mean deviation in the bubble-point pressure is higher than 3.5%. Thus there are seven failures: deviation 3.77% (Liu and Davison, 1981); (1) 2,4-dimethylpentane-octane, deviation 4.32% (Ewing and Marsh, (2) cyclopentane-2,3_dimethylbutane, 1973); deviation 4.61% (Willman and Teja, 1985); (3) toluene-decane, deviation 4.51% (Messow and Engel, 1977); (4) toluene-hexadecane, deviation 5.4% (Haynes and Van Winkle, (5) naphthalene-tetradecane, 1954; Ward and Van Winkle, 1954); deviation 3.69% (Kudryavtseva et al., 1972); (6) 1-heptene-octane, deviation 4.36% (Laesvskaya et al., 1979). (7) styrene-propylbenzene, These binary systems present a delicate problem in that we do not know whether the disagreements can be ascribed to the inadequacy of the model or to the accuracy of the experiments. In all the cases mentioned above, the systems were determined only once.
69
However, two cases may be resolved. (1) The experimental data on the binary cyclopentane-2,3-dimethylbutane (Ewing and Marsh, 1973) can hardly be questioned. We blame the discrepancy on the inadequacy of the model. (2) The data on the binary naphthalene-tetradecane (Haynes and Van Winkle, 1954; Ward and Van Winkle, 1954) belong to a series of data, provided by a single laboratory, for which the errors in the pure-component vapor pressures make it impossible to reach any correct treatment. The disagreement observed for these binaries can by no means be used to question the model submitted. To solve the problem set by the remaining systems, new experimental data are to be obtained to confirm those already existing. Excess enthalpy comparisons
The values of excess enthalpies vary greatly depending upon the system: they can become negative and in some instances they are nearly zero. In these cases the relative percentage mean deviations 8,(HE> are meaningless. To measure the quality of the predictions by the proposed method, we consider here either relative percentage or absolute mean deviations between calculated and observed excess enthalpies depending on the system under consideration. For binary systems of the type paraffin-paraffin, naphthene-naphthene, aromatic-aromatic or alkene-alkene, the excess enthalpies have small values, and high relative mean deviations do not mean a poor agreement. So for these systems absolute mean deviation 6(HE> is used. The deviations are collected in Table 7 and their distributions shown in Fig. 3(a) and Fig. 3(b). For the binaries for which the heat of mixing is small (always less than 300 J), the average deviation is 12 J for 7009 determinations. For those whose heat of mixing is higher (reaching 1000 J or more) the average deviation is 5.54% for 4299 determinations. The discussion of the causes of the deviations (poor quality of the data or inadequacy of the model) is in numerous cases more difficult with excess enthalpies than it is with vapor-liquid equilibria. Of course, in the case of systems thoroughly determined experimentally, it is possible to pick out aberrant experimental results easily. This is the case, for example, for the binary system hexane-benzene (Fig. 4). When the only two sets of data available are in disagreement, however, the program is more delicate. It should be mentioned that of the 282 binary systems investigated, 19 show deviations which are outside the range of the diagrams. When the difference in size increases in a homologous series, the excess enthalpy increases. Our model is able to follow this trend exemplified by
70 TABLE 7 Relative mean deviations, s,(Z-ZE), or absolute mean deviations, 8(HE), in molar excess enthalpy, for the various subsets of low-pressure excess enthalpy experimental data defined, depending on the subset Subset
Data sets
Alkane-alkane Alkane-naphthene Naphthene-naphthene
N,,
8(HE)
arCHE)
(J mol-‘)
(%) -
149
1549
5.96
223
3728
13.44
270
4299
Aromatic-aromatic Alkene-another hydrocarbon (alkene included)
100
1299
15.88
47
433
9.97
Overall average
270 519
4299 7009
12.02
Aromatic-alkane Aromatic-naphthene
> >
-
5.54
5.54
the binary benzene-alkane systems (Fig. 5(a)) and hexane-higher alkanes (Fig. 5(b)). The b inary system hexane-hexadecane (Fig. 6) shows an interesting phenomenon. This system exhibits a strong decrease in its excess enthalpy with increasing temperature, leading to negative values, first and foremost for mixtures rich in the most volatile component and then over the entire mole fraction range. This equation of state effect is correctly described by the model. The mixture of a long-chain compound with a globular one yields an endothermic effect, which rapidly decreases with increasing temperature. In mixtures of normal alkanes this effect is well described by means of the term xixjEchij (eqn. (5)), Ech,, being readily calculated from chain lengths Zi (eqn. (9)). The extension of that expression to substituted alkanes is no problem as far as light compounds (up to hexane isomers) are concerned. The problem becomes more complicated, however, with isomers of higher alkanes. The nonane isomers were extensively studied by Tancrede et al. (1977a,b) and De St-Romain et al. (1979a,b) who pointed out that the excess enthalpies of their mixtures with hexadecane showed very different behavior depending on the number and the place of the substitutions. The authors quoted above submitted correlations with molecular optical anisotropy and molar volume, but these are far from being satisfactory. We report here a correlation which relates the excess enthalpies of binary mixtures of nonane isomers with hexadecane to their critical temperature. The correlation is obvious but inadequate to lead to a good prediction of excess enthalpies. To improve that correlation the Zi values are estimated
71
T n
dH/joules
Tn
10
dli/joutas
0
M”
5
Fig. 3. Histograms of the absolute mean deviations between calculated and experimental molar excess enthalpies for binaries of the types: (a) a&ane-aikane; 70 binaries were treated; for 7 binaries, the deviations are out of the range of the diagram: Cbl aIkane-cyciane, cyclane-cyclane, aromatic-aromatic, aIkene-alkene, alkene-alkane, alkene-cyclane, alkene-aromatic; 131 binaries were treated; for 7 binaries, the deviations are out of the range of the diagram. IZ is the number of binaries, the deviations related to which are in the ranges shown.
72
T
HE/J
900
600
300
0.0
I 0.0
I
1
I
I
I
0.5
1
I
I
I
1.0
Fig. 4. Comparison of the molar excess enthalpies at 298.15 K from various sources of experimental data: hexane(l)-benzene(2); data from Schnaible et al. (1957) (triangle), Baluja Santos (1970) (circle), Romani and Paz-Andrado (1975) (square). Curves calculated by the present method, full lines.
from eqn. (Bl) (see Appendix B). The results of that correlation for all the isoalkanes investigated in this work are displayed in Table 2 along with the values of Zi,which were fitted so as to obtain better results. The estimate is often correct but sometimes poor, and in that case inadequate for excess enthalpy prediction. There is a score of binary systems for which the excess enthalpies are very poorly predicted and may be regarded as failures of the method. It is important that they should be discussed as they show the directions in which improvements should be sought. They are collected (Table 8) in three families corresponding to three problems for which satisfactory solutions were not found. (1) Binary systems including isoalkanes: as is pointed out above, the changes in conformation which may occur in such systems upon the mixing process prevent a completely satisfactory solution being reached. (2) Binary systems including cycloalkanes: it is difficult to find a satisfactory solution to problems of conformation in cycloalkanes through group contributions. (3) Binary systems including di- and trimethylbenzenes: the isomers of substitution in the benzene ring are also difficult to account for by group contributions.
73
THE/J
0.5
Fig. 5. Evolution of the molar excess enthalpy in a homologu~ series at 298.15 IC (a> Benzene-aikane; Heptane: data from Mmwh (19781 (circle); Undecane: data from DiazPena and Menduina (1974) (star); Pentadecane: data from Diaz-Pena and Menduina (1974) (cross). (b) Hexane-higher alkane; Octane: data from Hamam et al. (1984a) (circle); Decane: data from Marsh et al. (1980) (star); Dodecane: data ~~CXLI Hamam et al. (J984b) (rhombus); Hexadecane: data from Larkin et al. (1966) (cross). Calculated curves, full lines.
74 150
T
HE/J
100
50
0.0
-50 0.0
0.5
1.0
Fig. 6. Temperature dependence of the molar excess enthalpy versus composition curves (solid lines) for the binary hexane(l)-hexadecane(2). Experimental data are taken from Fernandez-Garcia and Boissonass (1966) at 293.15 K (square); from Tancrede et al. (1977a) at 298.15 K (cross); from Holleman (1965) at 313.15 K (rhombus), at 324.15 K (star), at 333.15 K (circle) and at 349.15 K (triangle).
Overall, the results obtained show the superiority of the models involving equations of state for the correlation of thermodynamic properties of mixtures of non-polar and non-associating compounds, even under low pressures. The same conclusion was reached by Hamam et al. (1986) who pointed out that even though the equations of state such as PR and RIG were primarily designed for vapor-liquid equilibrium calculations, their use for the prediction of excess enthalpies could lead to an improvement in the reproduction of these, especially for mixtures of the type normal alkanesubstituted alkane. High-pressure vapor-liquid equilibtium calculations
The correlations determined from low-pressure data are extended to high-pressure vapor-liquid equilibria merely by defining new groups corresponding to supercritical fluids. Thus methane, ethane, carbon dioxide, nitrogen, hydrogen sulfide, ethylene and water are distinct groups. Their interaction parameters with other families of compounds (alkanes, naphthenes, aromatics and alkenes) are estimated as described earlier, i.e. by
75 TABLE 8 Failures of our method for excess enthalpy calculations Family
Binary system
Deviation in HE
Isoalkanes
Hexane-3,3_diethylpentane Octane-3,3-diethylpentane 3,3-Diethylpentane-dodecane 3,3-Diethylpentane-tetradecane 3,3-Diethylpentane-hexadecane Octane-2,3,3,4_tetramethylpentane Neopentane-isooctane Cyclohexane-2,6_dimethyloctane
107.4 J 109.7 J 111.8 J 104.8 J 75.3 J 109.7 J 54.7 J 65.6 J
Cyclanes
Cyclopentane- trans-decalin Cyclopentane-cis-decalin Cyclooctane-tetradecane Heptane-tetralin Octane-tetralin Tetralin-dodecane Tetralin-hexadecane Benzene-tetralin
168.5 J 71.8 J 72.4 J 42.5% 30.1% 47.5% 43.4% 57.6 J
di- and trimethylbenzenes
Benzene-p-xylene Benzene- m-xylene Heptane-1,3,5-trimethylbenzene Toluene-1,2,4_trimethylbenzene 1,3,5-Trimethylbenzene-hexadecane
39.9 J 32.6 J 63.0% 28.8 J 64.4%
assuming as known the interaction parameters within or between those families. The method was tested on a database made up of nearly 12300 VLE data points, of which over 10300 are complete (i.e. having a full set of P, T, x, y data). The methods proposed in the literature for the prediction of high-pressure equilibria of non-polar and non-associating fluids were scarcely tested on so considerable a database, and it is therefore difficult to make an extensive comparison. It can be noted, however, that when the databases reported partly overlap ours (Hamam et al., 1977; Ishikawa et al., 1980; Machat and Boublik, 19851, our results are at least of the same quality. These are summarized in Table 9. The criteria used to measure the quality of the predictions are the deviations from experiment in the bubble-point pressure and in the vaporphase composition. The distribution of errors in bubble-point pressures is displayed in Fig. 7. These results are obviously less accurate than those with low-pressure data, but they remain within the range of experimental accuracy, given the
76 TABLE 9 Relative mean deviations in the bubble-point pressure, 8@‘), and absolute mean deviations in the vapor-phase composition, 6(y), for the various subsets of high-pressure experiment data defined Subset Nitrogen-carbon dioxide Nitrogen-hydrogen sulfide Nitrogen-water Nitrogen-hydrocarbon a ) Hydrogen sulfide-water Hydrogen sulfide-hydrocarbon
Data sets
b>
Carbon dioxide-hydrogen sulfide Carbon dioxide-water 1 Carbon dioxide-hydrocarbon \ b Methane-carbon dioxide Methane-hydrogen sulfide Methane-water Ethylene-carbon dioxide Ethane-carbon dioxide ) Ethane-hydrogen sulfide Hydrocarbon-hydrocarbon ’ Hexane-benzene Benzene-heptane I Overall average
iV,,
6,(P) (%o)
N,
6(Y)
174
1603
4.72
1452
0.0145
73
557
4.19
533
0.0145
246
2328
4.47
1888
0.0109
836
7784
2.39
6418
0.0089
1329
12272
3.17
10291
0.0103
a Including methane, ethylene, ethane, propene, propane, butanes, pentanes and higher hydrocarbons (up to hexadecane). b Methane, ethylene and ethane precluded. ’ Including methane, ethylene, ethane, propene, propane, butanes, butenes, butadiene, pentanes and higher hydrocarbons (up to eicosane).
discrepancies in data between different sources. Robinson et al. (1981) noted differences between their data (Li et al., 1981a; Gupta et al., 1982) and those from Ohgaki and Katayama (1976) for the carbon dioxide-hexane (Fig. 8) and carbon dioxide-benzene systems. High and regular deviations (often over 10%) are observed for systems involving nitrogen. Were it not for the conformation term, x,xiEch,, in eqn. (51, our method would merely be a correlation of the binary interaction parameters kij since we have seen that the energy binary parameters Eij actually used in this method are related to the k,j parameters by eqn. (7). This is the case for binary systems involving components with the Zi parameters not differing by more than three or four unities: the conformation term is not very important. In that case, of course, our method cannot yield better results than the classical application of the Peng-Robinson equation of state with a kij fitted to each binary system can. The main point to be noted here is
77 40
Tn
T
10
0
r 0
2
L
dP/P %
Fig. 7. Histogram of the relative percentage mean deviations between calculated and experimental bubble-point pressures for high-pressure vapor-liquid equilibria. 163 binaries were treated. n is the number of binaries, the deviations related to which are in the ranges shown. T P/bar 80.
60.
Xl *Yl 0.
0.5
1.
Fig. 8. Discrepancies between data from various sources for the vapor-liquid equilibria of the binary system carbon dioxide(l)-hexane(2) at 313.15 K: circle, Ohgaki and Katayama (1976); rhombus, Li et al. (1981a); star, Wagner and Wichterle (1987). Calculated curve, full line.
78
T
P/bar
80.
0.
0.
0.5
1.
Fig. 9. Vapor-liquid equilibria for the binary carbon dioxideWpentane(2). The data are from Besserer and Robinson (1973): rhombus (277.65 K), circle (311.04 K), triangle (344.15 K), star (377.59 K). Calculated curves, full lines.
that the correlation via group contributions does not lead to an appreciable error. This is illustrated by comparison of theory with experiment for the carbon dioxide-pentane system (Fig. 9). When the conformation term reaches values that cannot be neglected, especially for light-heavy binary systems, our method is better than the classical application of the Peng-Robinson equation. Figure 10 shows a very good correlation of the carbon dioxide-hexadecane system. It should be mentioned, however, that for this type of binary, elevated errors are encountered in the vicinity of critical points and when three-phase or liquid-liquid equilibria exist, they are predicted qualitatively but not precisely located. Binary methane-higher
alkane systems
For mixtures of alkanes, the methods which make use of the classical approximation with zero binary interaction parameters yield good results as long as the volatilities of the two components are not greatly different. When the volatilities of the two alkanes are very different, though, for example for binary mixtures of methane with alkanes ranging from hexane up, the zero binary interaction parameter hypothesis leads to the calculation of bubble-point pressures which differ significantly from the experimental values; this all the more so as the size of the alkane increases.
79
25.
Xl,Yl
0. 0.
0.5
1.
Fig. 10. Vapor-liquid equilibria for the binary carbon dioxide(Whexadecane(2); the data are from Sebastian et al. (1980): circle (463.05 K), rhombus (542.85 K), triangle (623.15 K), star (663.75 K). Calculated curves, full lines.
These errors may be over 20%. In such cases, non-zero binary interaction parameters are required as pointed out by Legret et al. (1984). For example, one system is considered here to show the difference between the results obtained by our method (with non-zero implicit binary interaction parameters and chain-length effects taken into account) and those obtained with the translated basic form of the Peng-Robinson equation with zero binary interaction parameters, i.e. the methane-hexadecane (Fig. 11). This curve illustrates that it is not possible to describe both low-pressure equilibrium and the vicinity of the critical point in a completely satisfactory way. Temperature dependence: binary carbon dioxide-alkane systems
Numerous authors (Huron et al., 1978; Kato et al., 1981; Legret et al., 1984) have noticed a temperature dependence of the binary interaction parameters. Since the optimal values were different depending on the source of data used for their determination, it was not possible to set up a simple law or a general correlation. Kato et al. (1981) proposed for mixtures of carbon dioxide with normal alkanes a quadratic function of absolute temperature, the three coefficients of which were correlated as functions of the acentric factor of the n-alkane.
80
T
P/bar
600
0.
0.5
1.
Fig. 11. Vapor-liquid ebuilibria for the binary methane(l)-hexadecane(2) at 350.00 K, circles are experimental points from Glaser et al. (1985). The curves in dashed lines are calculated with the basic form (eqn. (3)) with k,, = 0. The curves in full lines are calculated with the method presented.
The proposed method solves the problem for binary mixtures of carbon dioxide with alkanes for the correlation with respect to either temperature or alkape nature. The binary interaction parameters kij for carbon dioxide-alkane systems given by our model (eqn. (7)) are plotted versus temperature, for alkane ranging from methane to eicosane (Fig. 12). It should be noted that the curves show minima similar to those obtained by Kato et al. (1981), thereby confirming a phenomenon noticed experimentally with tabulated kij adjusted for each temperature. Prediction of multicomponent excess enthalpies
To test the applicability of the foregoing material, the heats of mixing of multicomponent (ternary and higher order) mixtures of hydrocarbons are calculated and compared with the relevant available experimental data. As is clearly shown in Table 10, excellent agreement is obtained between calculated and experimental values. The deviations (relative percentage mean deviations) are quite within the experimental accuracy and are not larger than those obtained for binary mixtures of the same type of compound. The results suggest that the proposed method may be reliable to
81 0.25 /
100
250
400
550
IT
k12
700
Fig. 12. Temperature dependence of the binary interaction parameters derived from eon. (7) for the binary systems carbon dioxide-alkane. The number of carbon atoms of the alkane is indicated at the right of the diagram; it is varied with a step of 1 from methane to decane and with a step of 2 from decane to eicosane. Dashed line: our method; full line: calculated by Kato et al. (1981).
TABLE 10 Relative mean deviations, values
6,( HE>, between calculated
Mixture
and experimental
T
(K) Hexane-benzene-cyclohexane Hexane-cyclohexane-toluene Benzene-cyclohexane-heptane Benzene-cyclohexane-toluene Benzene-heptane-toluene Benzene-heptane-tetradecane Benzene-toluene-tetradecane Benzene-decane-tetradecane Cyclohexane-heptane-toluene Hexane-benzene-cyclohexane-heptane Hexane-cyclohexane-heptane-toluene Hexane-benzene-cyclohexane-heptane-toluene Overall average
293.15 293.15 293.15 293.15 293.15 298.15 298.15 298.15 293.15 293.15 293.15 293.15
excess enthalpy
6,(HE)
Nh
Ref.
(%o)
11 5 31
2.46 4.40 2.63
a a
6
7 19 20 21 6 15 8 18
4.77 2.41 4.52 6.47 3.77 4.96 2.78 4.24 3.82
a a ’ c c a a = =
167
3.86
a Mathieson and Thynne (1956). b Brown et al. (1955). ’ Pfestorf et al. (1978).
b
82 TABLE 11 Relative mean deviations in the bubble-point pressure, d,(P), and absolute mean deviations in the vapor-phase composition, 6(y), between calculated and experimental values, for ternary systems Mixture
Nitrogen-methaneethane
T range K) 140.00 160.00 200.00 220.00 260.00 270.00 280.00 171.43 122.04
P range (bar) 20.0-40.0 20.0-40.0 20.0-120.0 80.0 50.7-76.0 35.4-86.0 50.7-75.9 13.7-27.6 5.7-27.3
Overall average
N,,
&(P) (%I
22 14.2 28 10.5 34 4.5 10 1.7 33 0.9 42 1.9 18 1.6 11 2.0 10 3.8 208
4.7
Ref.
NY S(Y)
22 28 34 10 33 42 18 11 10
Yl
Y2
y3
0.0094 0.0190 0.0108 0.0071 0.0068 0.0097 0.0084 0.0207 0.0042
0.0100 0.0188 0.0049 0.0006 0.0048 0.0053 0.0067 0.0219 0.0046
0.0010 0.0007 0.0059 0.0068 0.0098 0.0133 0.0146 0.0018 0.0007
a = = a b b b ’ ’
208 0.0107 0.0082 0.0071
Nitrogen-methane Carbon dioxide
270.00
45.6-111.4
45
1.9
45 0.0110 0.0090 0.0199 d
Nitrogen-methanepropane
313.70 114.10
89.0-95.8 1.14-17.5
4 11
2.2 4.6
e - 11 0.0094 0.0094 0.0000 f
Overall average
15
3.9
11 0.0094 0.0094 0.0000
Nitrogen-ethanepropane
200.00
36 14.4
36 0.0061 0.0104 0.0010 a
Nitrogen-butanedecane
352.60 275.0
17 14.6
17 0.0459 0.0400 0.0069 g
Nitrogen-heptane methylcyclohexane
Methane-ethanecarbon dioxide Methane-ethanepropane
20.0-120.0
453.15 235.0-329.0 472.15 168.0-246.0 497.15 129.0-198.0
9 9 9
7.6 8.2 7.0
Overall average
27
7.6
27 0.0107 0.0062 0.0048
250.00
21.3-30.4
15
2.3
15 0.0205 0.0126 0.0263 i
158.15 172.04 185.93 199.82 213.71 313.70 144.26 172.04 199.82 227.59 255.37 283.15
2.2-6.9 2.2-20.7 2.2-37.9 2.7-50.0 6.9-60.3 57.7-71.5 6.9 6.9-13.8 6.9-41.4 6.9-68.9 6.9-89.6 6.9-82.7
2 7 9 17 13 4 2 6 12 24 39 31
2.2 2.0 6.3 2.6 1.1 4.4 4.5 2.3 1.9 2.3 2.0 2.1
2 7 9 17 13 2 6 12 24 39 31
166
2.4
Overall average
9 0.0106 0.0072 0.0035 h 9 0.0087 0.0063 0.0027 h 9 0.0129 0.0051 0.0081 h
0.0006 0.0011 0.0064 0.0035 0.0037 0.0022 0.0063 0.0031 0.0062 0.0104 0.0121
0.0007 0.0011 0.0053 0.0030 0.0031
0.0001 0.0001 0.0012 0.0008 0.0007
j j j j j e
0.0009 0.0051 0.0032 0.0049 0.0082 0.0063
0.0006 0.0012 0.0002 0.0023 0.0042 0.0070
k k k k k k
162 0.0073 0.0053 0.0031
83 TABLE 11 (continued) Mixture
T
P
range
range (bar)
(K) Methaneethaneheptane
Methanecarbon dioxide -hydrogen sulfide
Methanecarbon dioxide -butane
Np gr(P) A',
Methanepropanebutane
Methanepropane-pentane
Methanepropanepentane
Yl
Y2
Y3
2.9 3.5 3.0
9 11 12
0.0008 0.0242 0.0142
0.0008 0.0240 0.0139
0.0000 0.0002 0.0003
Overall average
32
3.2
32
0.0139
0.0137
0.0002
310.93 277.59 344.26
16 13 7
5.4 10.3 4.8
16 13 7
0.0318 0.0573 0.0196
0.0289 0.0320 0.0220
0.0427 0.0386 0.0299
Overall average
36
7.1
36.
0.0386
0.0286
0.0387
177.59 210.93 244.26 277.59 310.93
3 7 22 27 23
5.3 3.5 8.5 4.7 5.6
3 7 22 27 23
0.0189 0.0972 0.0841 0.0426 0.0351
0.0147 0.0889 0.0868 0.0379 0.0344
0.0042 0.0082 0.0059 0.0162 0.0132
82
5.9
82
0.0554
0.0536
0.0115
8 5 5
1.8 2.0 3.7
8 5 5
0.0107 0.0120 0.0492
0.0120 0.0052 0.0079
0.0103 0.0167 0.0569
18
2.4
18
0.0218
0.0090
0.0250
3 5 6 5
1.4 1.3 0.9 1.3
3 5 6 5
0.0098 0.0135 0.0027 0.0077
0.0073 0.0130 0.0028 0.0068
0.0037 0.0005 0.0014 0.0066
Overall average
19
1.2
19
0.0080
0.0073
0.0029
277.59 310.93 344.26 377.59 344.26
20 20 13 5 40
1.7 0.7 1.6 2.7 1.6
20 20 13 5 40
0.0050 0.0085 0.0101 0.0091 0.0200
0.0042 0.0054 0.0066 0.0059 0.0088
0.0014 0.0035 0.0065 0.0130 0.0116
Overall average
98
1.5
98
0.0127
0.0067
0.0073
344.26 377.59
17 13
1.8 3.2
17 13
0.0073 0.0276
0.0089 0.0083
0.0046 0.0061
Overall average
30
2.4
30
0.0161
0.0086
0.0053
244.26 233.15 222.04 210.93 199.82 244.26 233.15
14 15 13 7 4 6 6
3.1 2.8 3.5 3.8 1.9 8.3 12.5
14 15 13 7 4 6 6
0.0010 0.0007 0.0004 0.0004 0.0008 0.0019 0.0038
0.0013 0.0249 0.0004
0.0003 0.0001 0.0000
0.0019 0.0036
0.0003 0.0003
65
4.5
65
0.0011
0.0067
0.0002
6.9-55.2 6.9-68.9 6.9-68.9
41.3-124.1 27.6-110.3 68.9-110.3
27.58 27.6-68.9 27.6-113.6 27.6-117.2 27.6-117.2
342.20 343.00 342.60
85.10 115.20 140.80
Overall average Methanecarbon dioxide -heptane
Ref.
6(y)
9 11 12
222.04 233.15 244.26
Overall average Methanecarbon dioxide -pentane
(%)
341.80 343.30 343.20 343.40
101.60 101.00 130.50 160.60
13.8-123.9 13.8-117.2 13.8-103.4 34.5-68.9 13.8-103.9
34.5-137.9 34.5-103.4
6.9-68.9 6.9-68.9 6.9-55.1 6.9-41.4 6.9-27.6 7.4-68.5 6.7-68.3
Overall average
’ ’ ’
m ” ”
o o o o ’
p p p
p p P p
q q q q q
r r
’ ’ ’ 0.0004 o.oooo ’ 0.0008 0.0000 ’ S
’
84 TABLE 11 (continued) Mixture
Methanepropanetoluene Methanepropanedecane
T range (K)
P range (bar)
N,
S,(P) (%o)
A’,,
Ref.
6(y) Yl
Y2
Y3
233.15
6.9-68.9
16
11.4
16 0.0041
0.0040
0.0001
’
277.59 310.93 410.93 410.93 477.59 510.93 294.26 277.59 255.37 244.26
27.6-275.8 27.6-275.8 27.6-275.8 27.6-275.8 27.6-206.8 27.6-137.9 2.8-68.8 2.9-70.5 2.3-69.6 3.1-69.5
17 17 13 14 7 5 51 51 30 20
3.1 1.6 1.9 2.3 3.3 4.9 3.2 2.0 3.2 4.8
17 0.0107 17 0.0062 13 0.0205 14 0.0179 7 0.0201 5 0.0142 51 0.0018 51 0.0020 30 0.0020 20 0.0015
0.0044 0.0044 0.0145 0.0113 0.0158 0.0159 0.0019 0.0020 0.0022 0.0015
0.0062 0.0020 0.0085 0.0068 0.0086 0.0101 0.0002 0.0001 o.oooo o.oooo
q q q q q q s s s s
225
2.8
225
0.0058
0.0044
0.0070
4
3.4
4
0.0075
0.0028
0.0070
g
Overall average Methanebutanedecane
352.60
Methaneneopentanepentane
344.26
34.7-146.2
8
2.0
8
0.0130
0.0027
0.0105
t
Methaneisopentanepentane
344.26 377.59 410.93
34.7-156.4 34.9-141.1 37.2-107.9
7 7 9
4.9 1.6 1.6
7 7 9
0.0322 0.0137 0.0449
0.0079 0.0041 0.0112
0.0244 0.0096 0.0338
t t t
Overall average
23
2.6
23
0.0316
0.0080
0.0236
260.00
5.0-25.0
28
5.1
28
0.0484
0.0572
0.0106
’
304.34 305.67 307.04 304.62 305.45 306.50 260.00 270.00 280.00
45.0-47.2 46.1-48.4 47.4-49.2 6.6-47.8 6.7-48.6 6.9-49.2 5.0-15.0 5.0-15.0 5.0-15.0
14 16 13 23 22 14 7 10 7
0.2 0.2 0.3 1.3 1.2 1.5 5.7 5.4 5.4
14 16 13 23 22 14 7 10 7
0.0009 0.0021 0.0027 0.0049 0.0035 0.0027 0.0177 0.0194 0.0210
0.0003 0.0010 0.0011 0.0079 0.0030 0.0060 0.0182 0.0185 0.0230
0.0006 0.0011 0.0016 0.0045 0.0016 0.0061 0.0018 0.0055 0.0057
” ” ” w w w ” ” ”
126
1.7
102
0.0061
0.0066
0.0030
28 21 21 20
3.6 4.7 4.5 1.8
28 21 21 20
0.0130 0.0244 0.0360 0.0080
0.0051 0.0139 0.0079 0.0060
0.0001 0.0110 0.0294 0.0033
90
3.7
90
0.0199
0.0080
0.0127
Ethanecarbon dioxide -butane Ethanepropanebutane
275.80
Overall average Ethanebutanepentane
366.48 394.26 422.04 338.70
37.0-64.9 31.1-56.9 31.8-55.6 34.9-60.6
Overall average
x x ’ y
85 TABLE 11 (continued) Mixture
T range (K)
P range (bar)
Np
S,(P) (%o)
N,,
Ref.
S(Y) Yl
Y2
y3
Ethane-butanehexane
322-404
28.3-67.9
25
2.9
25
0.0193
0.0107
0.0126
’
Ethane-butaneheptane
422.04 449.82
35.3-83.1 36.4-79.5
30 22
4.3 4.5
30 22
0.0263 0.0467
0.0129 0.0144
0.0134 0.0369
’ ’
Overall average
52
4.4
52
0.0349
0.0135
0.0233
263.15 293.15
28 4
0.4 1.1
28 4
0.0027 0.0115
0.0027 0.0021
0.0042 0.0127
Overall average
32
0.5
32
0.0038
0.0026
0.0053
253.12 263.14 273.18 278.15 283.15 290.75
11 12 15 8 16 21
4.5 4.7 4.7 4.2 4.8 3.8
11 12 15 8 16 21
0.0088 0.0096 0.0074 0.0168 0.0089 0.0081
0.0085 0.0090 0.0063 0.0171 0.0079 0.0067
0.0006 0.0009 0.0171 0.0012 0.0017 0.0018
83
4.4
83
0.0093
0.0084
0.0042
8 8 4
2.6 3.0 2.1
8 8 4
0.0026 0.0071 0.0127
0.0023 0.0043 0.0041
0.0033 0.0028 0.0088
Overall average
20
2.6
20
0.0064
0.0034
0.0042
303.15 313.15 323.15
18 14 15
5.9 6.1 3.9
18 14 15
0.0050 0.0058 0.0025
0.0024 0.0032 0.0013
0.0026 0.0026 0.0012
47
5.3
47
0.0044
0.0023
0.0022
9 9
11.9 12.9
9 9
0.0194 0.0434
0.0190 0.0288
0.0012 0.0146
dd dd
18
12.4
18
0.0314
0.0239
0.0079
dd
21.6-130.80 5 26.0-127.90 4
4.4 15.9
5 4
0.0003 0.0011
0.0005 0.0005
0.0004 0.0007
ee ee
Carbon dioxide -ethene-ethane
Carbon dioxide -ethanepentane
23.3-32.4 45.6-60.9
6.9-21.2 6.9-28.7 6.9-37.1 6.9-40.1 6.9-45.9 6.9-54.6
Overall average Carbon dioxide -pentaneheptane
Carbon dioxide -l-hexenehexane
342.90 343.60 343.50
78.50 90.90 101.50
20.3-65.0 18.6-67.9 20.3-85.9
Overall average Carbon dioxide -benzenetetralin
416.50 519.50
50.0-148.0 50.0-148.1
Overall average Carbon dioxide 353.15 -toluene-l413.15 methylnaphthalene
Carbon dioxide -decane-tetralin
Propanepropene-tetralin
Overall average
9
9.5
9
0.0007
0.0005
0.0006
343.30 520.00
9 9
10.2 9.9
9 9
0.0021 0.0170
0.0017 0.0094
0.0009 0.0076
Overall average
18
10.0
18
0.0096
0.0056
0.0042
273.15 293.15
16 12
3.2 4.2
-
28
3.6
-
50.0-118.6 46.8-120.9
1.5-4.7 2.1-5.1
Overall average
=* aa
bb bb bb bb bb bb
p P P
‘= Cc cc
dd dd ff ff
-
-
-
86 TABLE 11 (continued) Mixture
Isobutanel-butene1,3-butadiene
T range (K)
P range (bar)
277.67 294.36 310.70 327.70 344.03
1.6-1.7 2.8-2.9 4.5-4.8 7.0-7.3 10.3-10.5
Np
S,(P) (o/o)
NY
Ref.
6(y) Yl
y2
y3
8 8 8 8 8
10.3 8.9 7.7 6.6 5.6
_ -
-
Overall average 310.93 3.87-4.18 324.82 5.52-5.91 338.71 7.91-8.46
40 5 5 5
7.8 0.4 1.9 0.9
-
-
Overall average
15
1.1
-
-
1-Hexene hexaneoctane
328.15
0.20-0.67
12
1.1
-
-
Hexanemethylcyclopentanebenzene
333.15 342-350 342-349 344-346
0.60-0.76 1.013 1.013 1.013
103 54 10 11
1.1 0.3 0.3 0.9
103 54 10 11
0.0030 0.0026 0.0032 0.0027
0.0022 0.0026 0.0028 0.0009
0.0035 0.0029 0.0029 0.0027
Overall average
178
0.3
178
0.0029
0.0023
0.0032
283.15 288.15 298.15 343.15 343.15 342-352
10 10 12 6 21 108
1.1 0.7 0.2 0.2 0.5 0.8
10 10 12 6 21 108
0.0022 0.0044 0.0025 0.0023 0.0041 0.0047
0.0019 0.0025 0.0032 0.0016 0.0069 0.0034
0.0024 0.0032 0.0038 0.0017 0.0049 0.0047
167
0.7
167
0.0042
0.0036
0.0043
Butanel-butene13-butadiene
Hexanebenzenecyclohexane
0.08-0.09 0.09-0.12 0.16-0.19 0.82-0.99 0.78-0.98 1.013
Overall average Hexanebenzenetoluene
ge u -
BB BB BB
hh hh hh
-
-
ii
i kk ” mm
“” “” “’ “II O” pp
348-376
1.013
11
2.0
11
0.0127
0.0081
0.0051
345-368
1.013
26
0.3
26
0.0055
0.0018
0.0049
**
Overall average
37
0.8
37
0.0077
0.0036
0.0050
343-382
1.013
38
0.5
38
0.0088
0.0033
0.0069
ss
Methylcyclopentane -benzenecyclohexane 346-354
1.013
39
0.3
39
0.0040
0.0040 0.0055
tt
Benzenecyclohexanetoluene
353-367
1.013
24
10.1
24
0.0285
0.0216
0.0231
“”
Benzenetolueneethylbenzene
362-399
1.013
6
2.7
6
0.0889
0.0207
0.0833
w
Hexanebenzene-
p-xylene
87 TABLE 11 (continued) Mixture
cyclohexaneheptanetoluene
T
P
range (K)
range (bar)
298.15 361-381 358-379
0.05-0.12 1.013 1.013
Overall average
Np
S,(P) (%o)
16 1.5 93 0.4 35 0.5
NY
S(Y) Yl
Ref. y2
Y3
16 0.0041 0.0025 0.0029 93 0.0035 0.0029 0.0029 35 0.0023 0.0030 0.0027
144 0.5
144 0.0033 0.0029
0.0029
ww xx YY
1-Hepteneheptaneoctane
328.15
0.16-0.25
12 2.8
12 0.0057 0.0043
0.0054
72
Heptanemethylcyclohexane -toluene
372-382
1.013
78 0.6
78 0.0059
0.0041
0.0055
ma
Tolueneoctaneethylbenzene
383-405
1.013
20 0.7
20 0.0083 0.0075
0.0093
bbb
370-377 437-444
0.114 0.964
28 2.1 28 1.4
-
-
56
1.8
-
-
0.114 0.951
18 0.4 16 0.4
-
-
Overall average
34 0.4
-
-
371-377 440-441
0.114 0.964
21 3.0 19 2.3
-
-
Overall average
40 2.7
-
-
373-376 441-445
0.114 0.964
20 3.4 20 2.4
-
-
Overall average
40 2.9
-
-
Overall average
2741 3.1
1,3,5_Trimethylbenzene-1,2,4trimethylbenzenedecane
Overall average 1,3,5_Trimethylbenzene- 1,2,4369-373 trimethylbenzene337-443 1,2,3_trimethylbenzene
1,3,5-Trimethylbenzene-decane1,2,3-trimethylbenzene
1,2,4-Trimethylbenzene-decane1,2,3-Trimethylbenzene
Ternary systems
2468 0.0118 0.0089
CCC ccc
ccc ccc
ccc ccc
ccc ccc
0.0068
88
predict multicomponent data.
heats of mixing, given the current scarcity of such
Prediction of multicomponent vapor-liquid equilibria
Multicomponent vapor-liquid equilibrium calculations are performed on mixtures containing several (up to 12) normal compounds (i.e. hydrocarbons, nitrogen, carbon dioxide, hydrogen sulfide). The experimental data against which the investigation is carried out are those reported in the open literature and cover practically all the types of compounds pertaining to the realm of applicability of the method, i.e. weakly polar and non-associating fluids and low to high pressure range. The criteria used for the comparison are the bubble-point pressures and the vapor-phase compositions. The extensive results presented in Tables 11 and 12, show good agreement between experimental and calculated values.
CONCLUSIONS
The main purpose in this work was to test the applicability of the formalism of excess functions-equations of state we outlined through the Guggenheim quasi-lattice model to the prediction of high-pressure vaporliquid equilibria of weakly polar and non-associating systems. Thus we took advantage of the attributes of the model we described previously (PCneloux
Notes to Table 11: a Trappehl and Knapp (1987). b Gupta et al. (1980). ’ Chang and Lu (1967).d Somait and Kidnay (1978). e Yarborough and Smith (1970). f Poon and Lu (1974). g Llave and Chung (1988). h Brunner et al. (1974). i Davalos et al. (1976). j Wichterle and Kobayashi (1972). k Price and Kobayashi (1959). ’ Van Ham and Kobayashi (1967). m Robinson and Bailey (1957). ” Robinson et al. (1959). ’ Wang and McKetta (1964). p Nagahama et al. (1985). q Wiese et al. (1970b). ’ Dourson et al. (1942). ’ Koonce and Kobayashi (1964). t Prodany and Williams (1971). ” Clark and Stead (1988). ” Uchytil and Wichterle (1983). w Lhotak and Wichterle (1983). x Mehra and Thodos (1963). y Herlihy and Thodos (1962). ’ Dingrani and Thodos (1978). aa Fredenslund et al. (1976). bb Hong and Kobayashi (1983). cc Wagner and Wichterle (1987). dd Inomata et al. (1987). ee Morris and Donohue (1985). “Noda et al. (1982). 88 Steele et al. (1976). hh Laurence and Swift (1974). ii Kirss et al. (1975). i Beyer et al. (1965). kk Belknap and Weber (1961). ” Rowan and Weber (1961). mmYoung and Weber (1974). “” Li et al. (1974). M,Susarev and Chen (1963). pp Ridgway and Butler (1967). qq Avai et al. (1969). ” Saito (1969). ” Michishita et al. (1971). tt SUSarev and Lyzlova (1962). “” Draiko et al. (1973). w Makh and Azarova(1946).- Katayama et al. (1965). xx Myers (1957). * Black (1959). u Kudryatseva et al. (1972). aaa Wisniak and Tamir (1977). bbb Badina et al. (1977). cc’ Malanowski (1980).
89
et al., 1989) to coin a zeroth approximation group-contribution method which was combined with the translated basic form of the Peng-Robinson equation of state (PCneloux et al., 1982; Rauzy, 1982). This goal is met to a large extent as the predictive power of group contributions is brought to high-pressure calculation for the correlation of the binary interaction parameters required in the equation of state mixing rules, without damaging the quality of the results with respect to the classical application of the Peng-Robinson method with these parameters fitted. To do so, the temperature dependence of group-contribution correlations were generated from low-pressure excess enthalpy and vapor-liquid equilibrium data and then extended to the prediction of high-pressure vapor-liquid equilibria. This was an occasion to show the improvements that can be brought about by the use of equations of state in the low-pressure calculation of excess functions (AE, GE and HE>.
LIST OF SYMBOLS
1 A A[; b B BEL,
E(T, Ezj
Echij ECI H Hij HChij kij li ; Nh Np % P P P,
x>
equation of state energy parameter Helmholtz energy group interaction parameter between groups K and L parameter A,, at a reference temperature covolume; equation of state size parameter group interaction parameter between groups K and L parameter B,, at a reference temperature residual excess function binary energy parameter binary parameter of conformation constant in the term of conformation molar enthalpy binary enthalpy parameter binary enthalpy parameter of conformation binary interaction parameter in the classical mixing rules parameter of conformation of component i number of moles in a mixture number of groups in a solution number of determinations of excess enthalpy number of determinations of bubble-point pressure number of determinations of vapor-phase composition number of components in a mixture pressure vapor pressure of component i
Hexane-methylcyclopentane-benzene-cyclohexane-toluene Nitrogen-methane-carbon dioxide-ethane-hydrogen
Methane-ethane-propane-butane-pentane Methane-ethane-propane-toluene-l-methylnaphthalene
sulfide-propane
313.70 352.60 260.00 346.75 354.80 354.03 355.76,356.39 357.30 410.62-426.75 410.52-423.83 372.05-376.70 322.04 313.70 310.93 377.59 410.93 44.26 477.59 348.86-357.56 171.95 177.00 185.95 199.80 200.20 213.75 213.70 227.60 235.85 79.1, 89.4 275.80 10.00 1.013 1.013 1.013 1.013 1.013 1.013 1.013 0.114 73.2- 92.1 54.4-106.2 35.6-119.7 13.8-137.9 16.5-140.8 14.1-136.4 14.3-136.5 1.013 19.70 21.90 27.40 37.90 41.30 49.40 55.10 62.10, 68.90 80.40
(bar) 2 8 7 1 1 1 2 1 3 5 45 10 4 5 9 9 9 9 10 1 1 1 1 1 1 1 2 1
N, 4.3 5.7 8.0 0.1 0.5 0.5 0.6 0.5 4.6 1.4 2.4 2.9 3.6 2.3 14.9 8.4 4.9 6.1 1.0 3.1 4.9 4.1 5.1 2.8 3.9 0.7 1.2 0.1
6,(P) (o/o)
values, for quaternary
P range
and experimental
Nitrogen-methane-ethane-propane Nitrogen-methane-butane-decane Methane-ethane-carbon dioxide-butane Hexane-methylcyclopentane-benzene-cyclohexane Hexane-methylcyciopentane-benzene-toluene Hexane-methylcyclopentane-cyclohexane-toluene Hexane-benzene-cyclohexane-toluene Methylcyclopentane-benzene-cyclohexane-toluene Benzene-ethylbenzene-isopropylbenzene-a-methyls~rene Toluene-ethylbenzene-isopropylbenzene-a-methylstyrene 1,3,5-Trimethylbenzene-l,2,4-trimethylbenzene-decane-1,2,3-trimethylbenzene Nitrogen-methane-carbon dioxide-ethane-propane
d,(P), between calculated T range (RI
pressure,
Mixture
Relative mean deviations in the bubble-point order systems
TABLE 12
i
I
I
I
I
f
i
I
I
d
h
h
h
h
g
a
a
f
e
e
d
d
d
d
d
c
b
a
Ref.
and higher
ZS
and higher order systems overall average
310.70 210.93 188.70 172.04 313.70 253.12 263.14 273.18 283.17 293.15 273.18 283.17 293.15 243.14 253.12 263.14 273.18 283.17 172.04 185.93 199.82 338.70 322.00 338.70
104.9, 108.6 68.9-110.3 48,3- 82.7 27.6- 58.6 68.2, 75.9 6.9- 17.1 6.9- 23.0 6,9- 30.7 6.9- 38.4 6.9- 45.4 6.9- 28.0 6.9- 37.9 6.9- 44.6 6.9- 25.2 6.9- 31.0 6.9- 42.5 6.9- 53.8 6.9- 53.8 5.2- 27.6 13.81 41.4 27.6- 55.2 41.6-130.6 82.7-111.6 110.6, 111.6 278
2 3 3 3 2 4 6 6 7 8 6 7 9 5 6 8 8 9 3 3 3 7 7 2 4.6
0.7 8.0 9.7 8.0 5.9 4.5 4.1 3.6 3.8 4.3 10.6 3.9 5.6 4.5 4.5 4.4 3.8 3.9 6.5 6.0 5.0 5.7 3.4 10.7 m
m
m
j
j
i
J
1
I
I
1
k
k
k
k
k
k
k
k
a
* j j j
a Yarborough and Smith (1970). b Llave and Chung (1988). ’ Clark and Stead (1988). d Weatherford and Van Winkle (1970). e Beregovykh et al, (1979). f Malanowski (1980). g Hanson and Brown (1945). h Li et al, (198lb). i Kalva and Robinson (1979). j Cunningham et al. (1980). k Hong and Kobayashi (1986). ’ Hong and Kobayashi (1987). m Turek et al. (1984).
Quatemary
Carbon dioxide-methane-ethane-propane-butane-pentane-h~x~~~heptane-octane-decane-tetradecane
Nitrogen-carbon dio~de-methane-ethane-pro~ane-butane-~cntanehexane-heptane-octane
dioxide-ethane-hydrogen-s~lf~de-propane-bMtane-pent~e
sulfide-propane-butane-toluene
Carbon dioxide-ethane-hydrogen
Methane-carbon
su~~de-propane-bu~e-~~t~e
Carbon dio~d~-ethane-hydrogen
Nitrogen-methane-carbon dioxide-ethane-propane-butane Nitrogen-methane-ethane-propane-butane-pen&me
92
R T
G 4 w V ‘i Yi
gas constant temperature normal boiling temperature cubic equation of state parameters molar volume mole fraction of component i mole fraction of i in the vapor phase
Greek Letters aik
s’
surface area fraction or site fraction of group K in molecule i parameter of the “basic form” of a cubic equation of state deviation between calculated and experimental values of a property solubility parameter of compone_nt i packing fraction (7 = b/v, 17 = b/u’) function of the equation of state energy parameter a and of the packing fraction q, in the Helmholtz energy at constant packing fraction acentric factor reduced parameter of an equation of state reduced parameter of an equation of state
Subscripts C
cal exp . . i,lL
r res
critical property calculated experimental referring to components i, j referring to groups K, L reduced property residual
Superscripts a
E ‘KL
*
in the temperature dependence of the energy parameter the equation of state, depending on Ln, excess property in the temperature dependence of A,, and BKL ideal gas property pseudo-property
a(T) of
93
The financial support provided PCtroles is greatly appreciated.
by TOTAL-Compagnie
Fraqaise
des
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APPENDIX
A: BASIC FORM OF THE PENG-ROBINSON
EQUATION
OF STATE
The properties of each component are defined by the parameters o or T,, PC, Tb, m, depending on its class.
T,, PC,
P = RT,‘( t; - i;) - a/q v’+ ri;)
(Al)
where 6 = &,RT,/P,,
CM)
a, = fl,( RTC)*/PC and y = 4.82843
(a) Fun&~~~n a(T) for the commends of class (11, de~ned by T,, PC, Tb and m if TIT,
a=a
m if T,sTrT,
a=AZ:,+(l--X)a
where X= (T - TJ3/[(T
(W with q=0.9TC
- TJ3 + (q - T)~]
if T 2 Ti
a=a S
VW
(fi) (A61
a, is calculated by a, =a,(1 +m,[l
- (T/Tb)*‘“] -m,(l
-T/T,))
w?
with m, = 7.1553Om - 1.98255 m2 = 25776Sm - 0.99127
(W
where m is the parameter adjusted to the compound vapor-pressure data, and a, is the value of a(T) at the normal boiling temperature T,, (see Table 1 for approximate values of a,). a, is calculated by a, = a,(1 + m,[ 1 - (T/TJG]}’
(Y= 3.6 - 6.9i-&
(A91
101
with m, = [ ( a,/a,)l’z
- I]/[
1 - WT,)~]
(*lo)
(b) Function a(T) for the compounds of class (2), defined by T,, PC and o
a = a,(1 + m[l - (T/T,)“]}’
(A14)
m = 6.8130 (1.12753 + 0.517250 - 0.00374~11~)~‘~ - I]
W5)
[
APPENDIX B: CORRELATION OF SUBSTITUTED ALKANES
FOR THE PARAMETER
OF “CONFORMATION”
I = E, + 0.023437( T - T&Z,, + 0.002589( T,2/P, - T&&,) +2.966&z, + 0.55605n,
Pl)
I is the parameter of a substituted alkane isomer of the normal alkane whose parameter is 1, and critical constants are T, and P,,, n3 is the number of ternary carbon atoms, and n4 the number of quaternary carbon atoms. The results of correlation (Bl) for all the isoalkanes considered in this study are reported in Table 2 where they are compared with the parameters Zi eventually adopted, which were adjusted so as to yield better results. The estimates from that correlation, though often correct, are sometimes inadequate for the prediction of excess enthalpies.
APPENDIX C
Relative percentage mean deviation between experimental and calculated properties (e.g. bubble-point pressure and excess enthalpy):
‘r(‘) = (loo/&)
31‘exp,i-
i=l
‘cal,i
1/‘exp,i
102
Absolute mean deviation between experimental and calculated properties (e.g. vapor-phase composition and excess enthalpy): 6(Y) = (1/N_) Z
1 Yexp,i -Ycal,i
1
i-l
6(HE)
=
(1/Nh>
2
i=l
1HeEp,i
-Hctl,i
1