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Ind. Eng. Chem. Res. 2006, 45, 476-480
CORRELATIONS Approach Suitable for Screening Estimation Methods for Critical Properties of Heavy Compounds Ana Z ˇ bogar, Filipe Vidal Da Silva Lopes, and Georgios M. Kontogeorgis* IVC-SEP Engineering Research Center, Institut for Kemiteknik, Technical UniVersity of Denmark, Building 229, DK-2800 Lyngby, Denmark
A simple theoretical-based correlation of the ratio of the critical temperature to the critical pressure (Tc/pc) as a function of the van der Waals surface area (Qw) has been previously developed based on an extensive database of critical data published prior to 1996. The final equation was the following: Tc/pc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) where Tc is the critical temperature in kelvin and pc is the critical pressure in bar. This correlation is further validated here based on recent experimental data for various families of organic compounds, including some heavy ones (mono- and diacids, alkenes, cyclo/phenylalkanes, and squalane). This and previous validations verify that this correlation has a much broader application range, up to (Tc/pc) ratios of 200, than the data used in its development (compounds with ratios up to 100). It seems that most organic compounds, including the very heavy and complex ones, follow the trend suggested by this equation. This equation can be used for testing existing group-contribution estimation methods. It is shown here that direct comparison of the Joback and Constantinou-Gani methods for two families of compounds (alkenes and carboxylic acids) is in agreement with their validation via the proposed equation. Similar results have been obtained for other compounds. Both group-contribution methods are of equal accuracy for heavy alkenes and acids, provided that experimental boiling point temperatures are available for the Joback method. If such data are not available, e.g., for heavy compounds, the Constantinou-Gani method should be preferred. The proposed correlation does not offer an alternative to group-contribution methods as it only provides a single relationship of the two critical properties. Its universal character, though, and validation for many heavy compounds offer a way to test and compare existing group-contribution methods and, finally, to select the one that best conforms with the proposed correlation. It is recommended that the proposed correlation is indeed used for high molecular weight compounds for which experimental critical properties are typically not available. 1. Introduction Critical properties (temperatures and pressures) are important input parameters in many thermodynamic models such as cubic equations of state (EoS) and generalized corresponding-statesbased correlations, extensively used, e.g., for phase equilibrium calculations. While these parameters can be rather easily experimentally obtained for low molecular weight, and thermally stable, compounds, they are often unavailable for heavy and complex molecules such as high molecular weight hydrocarbons or small biomolecules (cholesterol and vitamins). In these cases, values of Tc and pc can be estimated by methods often based on group contributions, e.g., those by Joback-Reid and Constantinou and Gani, reviewed in ref 1. However, low to medium molecular weight compounds were used for the determination of the group parameters of these methods and, thus, their extrapolation capabilities may be difficult to assess. In many cases, it has been reported2,3 that critical properties estimated by such literature methods are in error when compared with measured values or lead to substantial differences in the design of separation processes, e.g., distillations, based on cubic EoS.4 A correlation method for screening estimation methods for the critical properties has been developed previously.2,5 As * To whom correspondence should be addressed. Tel.: +45 45252859. Fax: +45 45882258. E-mail:
[email protected].
shown there, the method is of semitheoretical origin, based on the lattice theory by Kurata and Isida for n-paraffins, which is mathematically equivalent to Flory’s theory of polymer solutions (suitable for medium to high molecular weight compounds). The correlation provides a universal relation for the Tc/pc ratio as a function of the van der Waals surface area (Qw); the latter is calculated by the group increments of Bondi, as reported in UNIFAC tables.6 The final equation is the following:
Tc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) pc
(1)
where the critical temperature Tc is expressed in kelvin and the critical pressure pc is in bar. The dimensionless parameter Qw is a measure of the van der Waals molecular surface area and is calculated as the sum of the group area parameters, Qk:
Qw )
∑k νkQk
(2)
where νk is the number of times group k appears in the molecule. The group area parameters Qk are available in UNIFAC tables.6 These dimensionless values of Qw as calculated from the UNIFAC tables were used in eq 1. Some calculation examples are given in the Appendix.
10.1021/ie050685h CCC: $33.50 © 2006 American Chemical Society Published on Web 11/30/2005
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 477 Table 1. Comparison of Experimental Tc/pc Ratios and Tc/pc Ratios Calculated from Eq 1, Based on Recently Available Experimental Data compound
Tc/pc (exper)
Tc/pc (eq 1)
ref
phenyldecane phenyltridecane phenylundecane dicarboxylic acid, n ) 10 dicarboxylic acid, n ) 12 dicarboxylic acid, n ) 8 dicarboxylic acid, n ) 5 squalane 2,2,4,4,6,8,8-heptamethylnonane
43.72 51.29 46.5 39.95 45.3 33.8 25.6 134.9 44.1 50 (based on vapor pressure data) 21.82 25.4
39.3 51.6 43.2 39.4 47.5 32.3 23.3 132.1 52.0
9 9 9 10 10 10 10 12 13
n-propylcyclohexane n-butylcyclohexane Figure 1. Comparison of eq 1 with experimental data for alkanoic acids.7 For lower acids, the data by Ambrose et al.16 and Rosenthal et al.17 are also shown.
Figure 2. Comparison of eq 1 with experimental data for 1-alkenes.8
Equation 1 was developed2 using an extensive database of critical properties for medium-high molecular weight compounds, based on critical data available until 1996 (covering Tc/pc ratios up to 100), and was subsequently further validated5 on the basis of new experimental data (1996-1999), which spread over Tc/pc ratios up to 185. Those recent data included critical properties of alkanes up to n-C36 and alkanols up to docosanol, both measured by the group of Prof. Nikitin in Russia. The purpose of this work is twofold: first to assess the applicability of the method (eq 1) against experimental data available over the last 5 years (after the publication of ref 5) and then to present some examples of how the method can be used for screening literature estimation methods. 2. Further Validation of the Method Based on Recent Experimental Data The applicability of the generalized correlation (eq 1) is tested against recent (after 1999) experimental critical temperature and pressure data measured by Nikitin and co-workers7-11 and others.12,13 These data include organic acids (up to 22 carbon atoms), alkenes (up to 20 carbon atoms), dicarboxylic acids (up to C14), alkylbenzenes, and alkylcyclohexanes as well as squalane. None of these systems were included in the development of eq 1, and neither have they been employed for validating the method. Thus, they can be used to further assess the predictive capability of the approach. The results are summarized for two families (acids and alkenes) graphically in Figures 1 and 2 as plots of the ratio Tc/pc calculated from eq 1 and from the experimental data
21.88 24.5
11 11
against the carbon number of the acids or the alkenes. Additional comparisons are provided in Table 1. The results are satisfactory, and no systematic deviations appear, which indicates that the correlation is equally wellapplicable to low and high molecular weight compounds. Some additional observations can be made: (1) Squalane (2,6,10,15,19,23-hexamethyltetracosane) [C30H62] represents a very interesting case, discussed by Cummings and co-workers.3,14 Using the DIPPR values for the critical temperatures and pressures (863 K and 8.3 bar) results in a Tc/pc ratio of 104, which is far from that based on the recent experimental values (795.9/5.9 ) 134.9). Estimation methods would also be far off; for example, the Joback method would result in a ratio of 106.3 or 102.6, depending on the choice of the boiling point temperature. On the other hand, recent molecular simulation data14 result in a much more accurate value for the critical temperature (800 K; thus, the ratio is 800/5.9 ) 135.6). The ratios of the critical properties from the experimental data, the simulation, and the proposed method are close to each other but are quite different from the value estimated by the groupcontribution method of Joback. (2) Wakeham et al.4 discuss the influence of property estimation methods on the design of distillation columns. They compare various methods for estimating critical properties for a few heavy hydrocarbons for which reliable experimental data is not available. They offer also a so-called “best estimate” value. For n-eicosylbenzene, the best estimates of the critical temperature and pressure correspond to a ratio of 90.98, which is very close to what we would get from eq 1, 88.10. It is worth mentioning that the various groupcontribution and other estimation methods discussed by Wakeham et al.4 result in a Tc/pc ratio higher or much lower than the best estimate values. 3. Testing of Literature Group-Contribution Methods Two of the most widely used estimation approaches, those of Joback-Reid and Constantinou and Gani, as described in ref 1, were tested for organic acids and alkenes. All group parameters are obtained from ref 1, while the boiling temperature required in Joback’s method is either determined from the groupcontribution method provided by Joback or taken directly from experiments (DIPPR database). The Constantinou-Gani method does not require the boiling temperatures, and for the two families of monofunctional compounds considered here, it does not contain any second-order groups either. At first, the two methods are compared in their purely predictive scheme, which means that the boiling temperature in Joback’s method is also taken from the method. The results obtained using the two estimation methods are compared against
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Figure 3. Results obtained for the critical temperatures of alkanoic acids using the Joback and Constantinou-Gani methods compared against the experimental data of Nikitin et al.7
Figure 4. Results obtained for the critical pressures of alkanoic acids using the Joback and Constantinou-Gani methods compared against the experimental data of Nikitin.7
the experimental data in Figures 3 and 5 for the critical temperature and in Figures 4 and 6 for the critical pressure. It can be concluded that, for both families of compounds, both estimation methods yield similar results for the critical pressure, while the Constatinou-Gani method is clearly more accurate than Joback’s for the critical temperature. In particular, the predictions of the critical temperature with Joback’s methods are even qualitatively wrong, as seen in Figures 3 and 5. Results are satisfactory only for n lower than 6 in the case of alkanoic acids. It seems that the poor results are largely due to the groupcontribution method (again from Joback) used for estimating the boiling temperature. As shown in Figures 7 and 8, very good agreement is obtained with Joback’s method for both acids and alkenes when the experimental boiling point temperatures are used. Experimental Tb data were taken from the DIPPR compilation.15 4. Use of the Generalized Approach (Eq 1) for Screening of Estimation Methods Figures 3-8 offer a direct comparison of estimation methods, which is only possible when experimental data is available.
Figure 5. Results obtained for the critical temperatures of 1-alkenes using the Joback and Constantinou-Gani methods compared against the experimental data of Nikitin and Popov.8
Figure 6. Results obtained for the critical pressures of 1-alkenes using the Joback and Constantinou-Gani methods compared against the experimental data of Nikitin and Popov.8
Figures 9 and 10 present the Tc/pc ratio for alkenes and organic acids, as estimated from the group-contribution methods and as calculated from the generalized correlation, eq 1. Results with Joback’s method are provided both based on predicted and experimental boiling points. The conclusions are in agreement with those from the direct comparison. The Constantinou-Gani method gives more accurate results (closer to those of the generalized correlation, eq 1) than Joback’s method, when the predicted boiling temperature is used in the latter. However, both methods are very close to the generalized correlation when the experimental boiling temperature is used in Joback’s method. Appendix: Examples of Using the Generalized Method (Eq 1) for Heavy Compounds Example 1: Propylcyclohexane. Nikitin et al.11 have reported values for the critical temperature and pressure equal to 624 K and 28.6 bar, thus giving a Tc/pc ratio equal to 21.82. Step 1. The structure of the compound is (CH2)5-CH(CH2)2-CH3.
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Figure 7. Results obtained for the critical temperatures of alkanoic acids using the Joback method with Tb calculated from the Joback correlation and with Tb taken from experimental data.15 The experimental critical temperatures are from ref 7.
Figure 9. Results obtained for Tc/pc of alkanoic acids using the Joback and Constantinou-Gani methods as well as eq 1.
Figure 10. Results obtained for Tc/pc of 1-alkenes using the Joback and Constantinou-Gani methods as well as eq 1. Figure 8. Results obtained for the critical temperatures of 1-alkenes using the Joback method with Tb calculated from the Joback correlation and with Tb taken from experimental data.15 The experimental critical temperatures are from ref 8.
Step 2. We calculate the Qw value of the compound from the corresponding Qk values we read from the UNIFAC tables (QCH3 ) 0.848, QCH2 ) 0.540, and QCH ) 0.228). Thus, Qw ) 1 × 0.848 + 7 × 0.540 + 0.228 ) 4.856. Step 3. Finally, the ratio is calculated from eq 1:
Tc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) ) pc 9.0673 + 0.43309(4.8561.3 + 4.8561.95) ) 21.88 Example 2: Dicarboxylic Acid, n ) 10 (Dodecanedioic Acid). Nikitin et al.10 have reported values for the critical temperature and pressure equal to 859 K and 21.5 bar, thus giving a Tc/pc ratio equal to 39.95. Step 1. The structure of the diacid with 10 methylene groups is COOH-(CH2)10-COOH. Step 2. We calculate the Qw value of the compound from the corresponding Qk values we read from the UNIFAC tables (QCH2
) 0.540 and QCOOH ) 1.224). Thus, Qw ) 10 × 0.540 + 2 × 1.224 ) 7.848. Step 3. Finally, the ratio is calculated from eq 1:
Tc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) ) pc 9.0673 + 0.43309(7.8481.3 + 7.8481.95) ) 39.4 Example 3: Phenyltridecane. Nikitin et al.9 have reported values for the critical temperature and pressure equal to 790 K and 15.4 bar, thus giving a Tc/pc ratio equal to 51.29. Step 1. The structure of this heavy aromatic hydrocarbon is (ACH)5-AC-(CH2)12-CH3, where, in accordance to the terminology of the UNIFAC tables, we have indicated an aromatic CH group as ACH and an aromatic carbon group as AC. Step 2. We calculate the Qw value of the compound from the corresponding Qk values we read from the UNIFAC tables (QCH3 ) 0.848, QCH2 ) 0.540, QACH ) 0.4, and QAC ) 0.120). Thus, Qw ) 5 × 0.4 + 1 × 0.120 + 12 × 0.540 + 1 × 0.848 ) 9.448.
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Step 3. Finally, the ratio is calculated from eq 1:
Tc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) ) pc 9.0673 + 0.43309(9.4481.3 + 9.4481.95) ) 51.6 Example 4: Squalane. The last example is the very heavy hydrocarbon squalane (C30H62, molecular weight equal to 422), for which critical property data have been recently published.12 The reported values for the critical temperature and pressure are equal to 795.9 K and 5.9 bar, thus giving a Tc/pc ratio equal to 134.9. Step 1. The structure of squalane is [simply written in terms of groups this compound consists of] (CH3)8-(CH2)16-(CH)6. Step 2. We calculate the Qw value of the compound from the corresponding Qk values we read from the UNIFAC tables (QCH3 ) 0.848, QCH2 ) 0.540, and QCH ) 0.228). Thus, Qw ) 8 × 0.848 + 16 × 0.540 + 6 × 0.228 ) 16.792. Step 3. Finally, the ratio is calculated from eq 1:
Tc ) 9.0673 + 0.43309(Qw1.3 + Qw1.95) ) pc 9.0673 + 0.43309(16.7921.3 + 16.7921.95) ) 132.1 Literature Cited (1) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The properties of gases and liquids, 5th ed.; McGraw-Hill: New York, 2001. (2) Kontogeorgis, G. M.; Yakoumis, I. V.; Coutsikos, P.; Tassios, D. P. A generalized expression for the ratio of the critical pressure with the van der Waals surface area. Fluid Phase Equilib. 1997, 140, 145-156. (3) Rainwater; et al. Report on Forum 2000: Fluid Properties for new technologiessconnecting Virtual design with physical reality; NIST special publication 975; NIST: 2001 (http://Forum2000.Boulder.NIST.Gov/ NISTSP975.pdf). (4) Wakeham, W. A.; Cholakov, G. St.; Stateva, R. P. Consequences of property errors on the design of distillation columns. Fluid Phase Equilib. 2001, 185, 1-12.
(5) Yakoumis, I. V.; Nikitin, E.; Kontogeorgis, G. M. Validation of a recent generalized expression of Tc/Pc vs the van der Waals surface area according to recent measurements. Fluid Phase Equilib. 1998, 153, 2327. (6) Fredenslund, Aa.; Gmehling, J.; Rasmussen, P. Vapour-liquid equilibria using UNIFAC; Elsevier: New York, 1977. (7) Nikitin, E. D.; Pavlov, P. A.; Popov, A. P. Critical temperatures and pressures of some alkanoic acids (C2 to C22) using the pulse-heating method. Fluid Phase Equilib. 2001, 189, 151-161. (8) Nikitin, E. D.; Popov, A. P. Critical temperatures and pressures of linear alk-1-enes with 13 to 20 carbon atoms using the pulse-heating technique. Fluid Phase Equilib. 1999, 166, 237-243. (9) Nikitin, E. D.; Popov, A. P.; Bogatishcheva, N. S.; Yatluk, Y. G. Vapor-liquid Critical properties of n-alkylbenzenes from toluene to 1-phenyltridecane. J. Chem. Eng. Data 2002, 47, 1012-1016. (10) Nikitin, E. D.; Popov, A. P.; Bogatishcheva, N. S.; Yatluk, Y. G. Critical temperatures and pressures of straight-chain saturated dicarboxylic acids (C4 to C14). J. Chem. Eng. Data 2004, 49, 1515-1520. (11) Nikitin, E. D.; Popov, A. P.; Bogatishcheva, N. S. Critical point measurements for five n-alkylcyclohexanes (C6 to C10) by the pulse-heating method. J. Chem. Eng. Data 2003, 48, 1137-1140. (12) von Niederhausern, D. M.; Wilson, G. M.; Giles, N. F. Critical point and vapor pressure measurements at high temperatures by means of a new apparatus with ultraflow residence times. J. Chem. Eng. Data 2000, 45, 157-160. (13) Kontogeorgis, G. M.; Smirlis, I.; Yakoumis, I. V.; Harismiadis, V.; Tassios, D. P. Method for estimating critical properties of heavy compounds suitable for cubic equations of state and its application to the prediction of vapor pressures. Ind. Eng. Chem. Res. 1997, 36, 4008-4012. (14) Cui, S. T.; Cummings, P. T.; Cochran, H. D. Configurational bias Gibbs ensemble Monte Carlo simulation of vapor-liquid equilibria of branched and short-branched alkanes. Fluid Phase Equilib. 1997, 141, 4561. (15) Daubert, T. E.; Danner, R. P. Physical and thermodynamic properties of pure compounds: data compilation; Hemisphere: New York, 1989. (16) Ambrose, D.; Ghiassee, N. B. Vapor-pressures and criticaltemperatures and critical pressures of some alkanoic acidssC1 to C10. J. Chem. Thermodyn. 1987, 19 (5), 505-519. (17) Rosenthal, D. J.; Gude, M. T.; Teja, A. S.; Mendez-Santiago, J. The critical properties of alkanoic acids using a low residence time flow method. Fluid Phase Equilib. 1997, 135 (1), 89-95.
ReceiVed for reView June 12, 2005 ReVised manuscript receiVed October 25, 2005 Accepted November 2, 2005 IE050685H
