4008
Ind. Eng. Chem. Res. 1997, 36, 4008-4012
Method for Estimating Critical Properties of Heavy Compounds Suitable for Cubic Equations of State and Its Application to the Prediction of Vapor Pressures Georgios M. Kontogeorgis,*,† Ioannis Smirlis, Iakovos V. Yakoumis, Vassilis Harismiadis, and Dimitrios P. Tassios Department of Chemical Engineering, National Technical University of Athens, Heroon Polytechniou 9, Zographos 15780, Athens, Greece
Cubic equations of state (EoS) are often used for correlating and predicting phase equilibria. Before extending any EoS to mixtures, reliable vapor-pressure prediction is essential. This requires experimental, if possible, critical temperatures Tc, pressures Pc, and acentric factor ω or extensive pure-compound vapor-pressure data which, for heavy and/or complex compounds, are often not available. This work presents a method for estimating Tc, Pc, and ω values for heavy compounds (typically with MW > 130) suitable for vapor-pressure calculations with generalized cubic EoS. The proposed scheme employs a recent group-contribution method (Constantinou et al. Fluid Phase Equilib. 1995, 103 (1), 11) for estimating the acentric factor. The two critical properties are estimated via a generalized correlation for the ratio Tc/Pc (with the van der Waals surface area) and the cubic EoS at a single experimental vapor-pressure point (e.g., the normal boiling point). We have employed a modified version of the Peng-Robinson EoS, but we have verified that any cubic EoS yields similar results at least for n-alkanes up to n-octacosane (MW ) 394). The method is applied to the prediction of vapor pressures for several nonpolar and slightly polar heavy compounds with very satisfactory results, essentially independent of the experimental point used. Furthermore, the method yields critical properties for heavy alkanes (Nc > 20) and other compounds which are in very good agreement with recent available experimental data. 1. Introduction Accurate prediction of pure-component vapor pressures is essential for phase-equilibrium calculations in multicomponent mixtures. Such predictions with generalized three-parameter cubic EoS depend on the critical pressure, the acentric factor, and, especially, the critical temperature of the pure compounds (Voulgaris et al., 1991). Critical constants can be accurately measured only for low molecular weight compounds. For most heavy substances, with molecular weight higher than 150, the critical properties are not known experimentally and estimation procedures must be used. Among the most recent and successful estimation methods for Tc and Pc are those based on the groupcontribution (GC) principle (Joback, Ambrose; see Reid et al., 1987; Constantinou and Gani, 1994). GC methods for the acentric factor have been also proposed recently (Han and Peng, 1993; Constantinou et al., 1995). All these methods provide accurate predictions for low (and possibly also medium) molecular weight compounds, but their performance for heavy and structurally complex substances is obscure and in many cases problematic (see Kontogeorgis et al., 1997). The purpose of this work is to suggest a method for estimating Tc, Pc, and ω for medium/heavy and structurally complex (nonpolar/slightly polar) compounds. The values of these properties should be particularly suitable for vapor-pressure calculations with cubic equations of state. The method is based on the groupcontribution principle (requiring thus knowledge of the structural form of the compound) and a single experimental vapor-pressure point and is described in the next * Corresponding author. † Present address: Department of Environmental Engineering, Demokrition University of Thrace, Xanthi, Greece. S0888-5885(96)00497-6 CCC: $14.00
Figure 1. Ratio of the critical temperature to the critical pressure (experimental and obtained by eq 1).
section. Characteristic results are shown in the Results and Discussion section followed by our conclusions. 2. Proposed Method In a recent work (Kontogeorgis et al., 1997) we have developed, based on theoretical considerations, a method for estimating the ratio Tc/Pc for high molecular weight and complex compounds (with molecular weight above 130):
Tc/Pc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) © 1997 American Chemical Society
(1)
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 4009 Table 1. Critical Properties and Acentric Factors As Obtained from the Proposed Method experimental point compound n-docosane n-tetracosane n-octacosanea n-octacosanec 1-chlorotetradecane 1-chlorohexadecane n-dodecylbenzene 2,2,4,4,6,8,8-heptamethylnonane 4-methylpentanone-2 2-(1,2-dimethylpropyl)-5,6-dimethylheptenal 2-pentylnonenal C28H56d n-decylcyclohexane vid xixd xivd
Tc (K)
Pc (bar)
ω
T (K)
P (kPa)
pred.
exp.
pred.
exp.
pred.
exp.
453.1 573.1 Tb ()641.8) 453.1 588.1 483.1 588.1 453.1 575.3 Tb ()704.8) 414.9 570.2 438.8 599.9 395.9 423.6 545.4 Tb ()519.5) 309.7 415.8 Tb ()389.4) 481.3 432.9 553.3 393.15 333.5 382.1 349.1 421.8 439.4 582.5 379.2 472.9 Tb ()520.8)
0.243 20.72 101.33 0.0901 17.46 0.0821 6.00 0.0131 3.88 101.33 0.693 101.89 0.680 102.21 0.0665 7.09 171.92 101.33 4.92 206.36 101.33 260.37 2.67 110.16 0.00741 0.00498 0.142 0.120 4.16 0.730 101.33 0.570 24.30 101.33
784.4 784.4 784.9 801.28 800.48 826.63 827.04 829.7 828.1 827.9 741.5 744.8 763.9 767.2 783.9 700.6 691.2 693.1 566.8 567.8 567.6 692.9 707.1 711.4 764.3 781.3 776.8 698.5 705.2 696.2 701.9 696.6 697.5 689.1
785.6e
9.78 9.78 9.79 8.71 8.70 7.00 7.00 7.03 7.01 7.01 16.52 16.60 14.28 14.34 16.60 14.00 13.80 13.85 31.54 31.60 31.60 16.35 16.64 16.75 6.38 20.9 20.8 20.9 21.2 9.87 9.95 18.05 18.07 17.85
9.82e
0.9648 0.9648 0.9648 1.0416 1.0416 1.1901 1.1901 1.1901 1.1901 1.1901 0.7182 0.7182 0.8005 0.8005 0.7378 0.4593 0.4593 0.4593 0.3863 0.3863 0.3863 0.7346 0.8088 0.8088 0.9912 0.5779 0.5779 0.6517 0.6517 1.194 1.194 0.6680 0.6680 0.6680
0.962
799.8e 829.9b 829.9b
692.0 574.6
8.66e 7.28b 7.28b
15.70 32.70
1.183b 1.194b
0.351
a Data from Morgan and Kobayashi, 1994. b Estimated values using the reliable method of Magoulas and Tassios (1990). c Data from Chirico et al., 1989. d C28H56: 2,2,4,10,12,12-hexamethyl-7-(3,4,5-trimethylhexyl)tridecane. vi: 6,10-dimethyl-4,5,9-undecatrien-2-one. xix: 3,7,11,15-tetramethyl-1-hexadecen-3-ol (isophytol). xiv: 6,10-dimethyl-2-undecanone (hexahydropseudoionone). e Nikitin et al. (1994).
where Tc is in K, Pc is in bar, and Qw is the van der Waals surface area. The Qw values are estimated using the group increments given by Bondi (1968), and they are readily available in all UNIFAC tables (e.g., Fredenslund et al., 1975). The physical Qw values (as originally given by Bondi) should be employed for all groups. Caution should be exercised for the hydroxyl group for which the physical value ()0.584) should be used (the value available in the UNIFAC tables is based on fitting vapor-liquid experimental data). Equation 1 is shown (together with the experimental Tc/Pc ratios) in Figure 1 and is based on an extensive database including medium to high molecular weight and complex compounds for some of which experimental critical properties became only very recently (19911995) available. Details for the database and the derivation of eq 1 are given by Kontogeorgis et al. (1997). The acentric factor is estimated by the recent GC method of Constantinou et al. (1995), which includes both first- and second-order contributions. Alternatively, if a reliable value for the acentric factor of the compound in question is available from experimental measurements, it could be used as well. An important note should be made here regarding eq 1, which implies that the two critical properties are intercorrelated. The intercorrelation of the critical properties, in general, has been recognized by several investigators, and some (Constantinou and Gani, 1994; Somayajulu, 1991) have stated that the methods used for predicting the various critical properties should be self-consistent in their limiting behavior, a requirement often overlooked in several group-contribution and other
estimation methods. Several authors have proposed equations similar to eq 1, (e.g., Dohrn, 1992; Soave et al., 1994). In particular, Elhassan et al. (1992) proposed correlations of Tc2/Pc and Tc/Pc for n-alkanes and 1-alcohols (separately) with the molecular weight and the number of carbon atoms. Sanchez (1985) proposed GC methods for TcVc2/3 and PcVc5/3 based on the apparent linearity of these quantities with the molecular weight. A theoretical explanation of Elhassan/Sanchez relations is given by Kontogeorgis et al. (1995). Finally, Vetere in a series of publications (1987a,b, 1989, 1992, 1995) has demonstrated how the classical and modified forms of the Rackett equation (which represents a relationship of all three critical properties and the saturated liquid volume) can be used for estimating some of the critical properties based on liquid volume data. Kontogeorgis et al. (1995) used the Rackett-Vetere approach for investigating the chain length dependence of the critical densities of organic homologous series and comment on the experimental discrepancy between recent investigations. Kontogeorgis et al. (1997) showed that the ratio Tc/Pc is a reasonable choice for interrelating these two critical properties. The following procedure is used to evaluate the Tc and Pc values once their ratio is determined from equation (1) and the ω value from the method of Contantinou et al., (1995): 1. Assume a value for Pc. 2. Determine the corresponding Tc value. 3. Calculate the vapor-pressure value (with the cubic EoS) at the given datum.
4010 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 2. Average Absolute Deviation between Experimental and Calculated Vapor Pressures Using the t-mPR and the Proposed Method ref
T-range (K)
n-docosane
Morgan and Kobayashi, 1994
453-573
0.24-20.7
n-tetracosane
Morgan and Kobayashi, 1994
453-588
0.09-17.5
n-octacosane
Morgan and Kobayashi, 1994
483-588
0.0821-6.01
n-octacosane
Chirico et al., 1989
453-575
0.0131-3.885
1-chlorotetradecane
Kemme and Kreps, 1969
414-570
0.693-101.9
1-chlorohexadecane
Kemme and Kreps, 1969
260-600
0.680-102.2
n-dodecylbenzene 2,2,24,4,6,8,8-heptamethylnonane
Daubert and Danner, 1989 Ambrose and Ghiassee, 1988
325-746 423-545
1.98 × 10-4-1120 7.1-172
4-methylpentanone-2b
Ambrose et al., 1988
309-416
4.9-416
2-(1,2-dimethylpropyl)-5,6-dimethylheptenal 2-pentylnonenal
Mills et al., 1987 Mills et al., 1987
367-535 384-553
0.281-101.1 0.276-110.1
C28H56c n-decylcyclohexane
Morecroft, 1964 Mokbel et al., 1995
368-393 313-467
3.09 × 10-5-7.41 × 10-3 8.28 × 10-4-6.377
vic
Baglay et al., 1988
349-422
0.12-4.16
xixc
Baglay et al., 1988
439-469
0.73-2.2
xivc
Baglay et al., 1988
379-473
0.57-24.3
compound
P-range (kPa)
exp. point (K)
∆Pa (%)
453.1 573.1 641.8 453.1 588.1 483.1 588.1 Tb ()704.8) 453.1 575.3 Tb ()704.8) 414.9 570.2 438.8 599.9 395.9 423.6 545.4 Tb ()519.5) 309.7 415.8 Tb ()389.4) 481.3 432.9 553.3 393.15 333.5 382.1 349.1 421.8 439.4 582.6 379.2 472.9 520.8
0.5 0.3 0.6 2.2 0.8 0.4 0.4 1.1 4.9 1.3 2.1 2.5 5.9 1.9 5.2 3.9 5.9 9.9 7.1 0.6 2.0 1.6 10.2 8.8 8.5 15.4 10.6 7.9 19.0 4.7 13.9 9.5 2.9 4.5 15.4
a ∆P (%): mean absolute percentange deviation in vapor pressures. b The error is 2.5% using experimental critical properties. c See footnote d from Table 1.
4. Repeat steps 1-3 until the calculated vaporpressure value agrees with the experimental one. We decided to use the Peng-Robinson EoS as modified by Magoulas and Tassios (1990), although any EoS is expected to give similar results for n-alkanes up to n-octacosane. We have verified this point by performing a few calculations with the SRK EoS. Since eq 1 and the Contantinou et al. method for the acentric factor are generally valid regardless of the EoS used and since most generalized cubic EoS are of the same quality in the prediction of vapor pressures, it is not surprising that the methodology presented in this work is valid for any generally successful cubic EoS. Tc and Pc are adjusted so that the equation of state reproduces the experimental vapor-pressure point used. In principle, any experimental vapor-pressure point can be used, e.g., the normal boiling point or a point at low reduced temperatures. The method is not restricted by the point chosen, but there might be some (often small) effect in the calculations, as will be illustrated in the next section. However, the possibility of using any experimental point gives an additional flexibility to the suggested method, as will become evident in the next section. Very satisfactory vapor pressures are expected over extended temperature ranges, but especially about (within 50-80 K) the experimental point employed the results are excellent. 3. Results and Discussion Results for several compounds are given in Tables 1 and 2, while typical vapor-pressure predictions are
Figure 2. Predicted vs experimental vapor pressures of 1-chlorohexadecane with the proposed method using P ) 102.21 kPa as the experimental vapor-pressure datum.
shown in Figures 2 and 3. The overall performance of the method is very satisfactory for all compounds investigated. The following conclusions summarize our observations: (i) The good performance of the method is essentially independent of the experimental vapor-pressure point
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 4011
in this work. The method requires knowledge of the structure of the compound and one vapor-pressure point, either in the low- or high-vapor-pressure region. The method is tested for a number of heavy, nonpolar, and slightly polar compounds with very satisfactory results; the error in vapor pressure is typically less than 10% over extended temperature ranges (and down to low vapor pressures), and no failure of the method has been detected for the compounds investigated. Furthermore, the values for the critical properties, as estimated by the proposed method, agree very well with the experimental data, whenever available. This indicates that the proposed method may also be used as a property estimation method for the compounds for which the three-parameter cubic EoS are valid and whenever an experimental vapor-pressure point is available. Acknowledgment The authors thank IVC-SEP Research & Engineering Center (Denmark) where part of this work has been performed. Figure 3. Predicted vs experimental vapor pressures of 6,10dimethyl-2-undecanone with the proposed method using P ) 24.3 kPa as the experimental vapor-pressure datum.
employed. It is actually remarkable that using so different experimental points (e.g., one in the low reduced temperature region and one at the normal boiling point) results in so similar (and good) performance with the suggested approach. In order to verify the aforementioned conclusion, an analysis of the effect of experimental datum in the vapor-pressure predictions has been performed for n-octacosane. Whatever the chosen experimental datum is, the average deviation in vapor pressures of n-octacosane is lower than 0.8%. (ii) Table 1 presents experimental critical properties for two heavy alkanes (n-docosane and n-tetracosane) and some other compounds as well as the values obtained from the suggested approach. The agreement is very satisfactory. The critical pressure obtained from the proposed method is essentially independent of the experimental point used, while the critical temperature dependence is, in most cases, rather small. (iii) The quality of the vapor-pressure calculations obtained with the proposed method is essentially the same regardless of the compound involved (alkanes, chloroalkanes, ketones, and compounds used in the synthesis of vitamins A and E). The successful results for the compounds involving polar groups, using a method based on the corresponding states principle, suggest that the impact of these groups on the behavior of a high molecular weight compound is rather small. This is not, of course, expected to be applicable to smaller polar compounds, especially those involving hydrogen bonding such as alcohols and acids. In the latter cases, the proposed methodsof estimating vapor pressures via generalized cubic equations of statesis not applicable due to the limitations of the three-parameter corresponding states principle rather than the use of eq 1 and the Constantinou method for the acentric factor which are valid for heavy compounds. 4. Conclusions A novel approach for estimating the critical properties of medium and high molecular weight compounds particularly suitable for vapor-pressure calculations with generalized cubic equations of state is developed
Abbreviations EoS ) equation of state GC ) group contribution SRK ) Soave-Redlich-Kwong t-mPR ) translated-modified Peng-Robinson MW ) molecular weight
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Resubmitted for review June 6, 1997 Accepted June 7, 1997X IE960497E X Abstract published in Advance ACS Abstracts, August 1, 1997.
