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Method for the Prediction of Pure-Component Vapor Pressures in the Range 1 kPa to the Critical Pressure Bert Wlllman and Amyn S. Teja” School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0 100
A model based on the effective carbon number (ECN) or the boiling point ( T b )has been developed for predicting the pure-component vapor pressures of paraffinic, naphthenic-, aromatic-, and sulfur-containing compounds. Good representation of the vapor pressures is obtained in the range 1 kPa to the critical pressure, with an overall average absolute deviation of 3.97% for the 92 substances studied. The method requires a knowledge only of the normal boiling point of the substance. The method is compared with the modified UNIFAC vapor-pressure method of Jensen, Fredenslund, and Rasmussen.
Many expressions have been proposed for the representation of the vapor pressure of a pure liquid between the triple point and the critical point. Waring (1954) has pointed out the complexity of the vapor-pressure curve and has noted that an adequate representation of the changes in curvature of the function over the whole liquid range would require at least a four-constant vapor-pressure equation. Ambrose (1978) also compared a number of vapor-pressure equations and concluded that the Wagner (1973) equation can represent the vapor-pressure behavior of pure liquids over a wide range with good accuracy by using a relatively small number of adjustable constants (four plus the critical temperature T, and critical pressure P,) and that it may be used to extrapolate low pressure data (1-200 kPa) to the critical point. This was also the conclusion reached recently by McGarry (1983) who reported Wagner equation constants for 251 substances. Finally, Wilsak and Thodos (1984) compared four vaporpressure functions containing five adjustable constanta plus the critical temperature T , and could find no significant difference between the models. It therefore appears that the Wagner equation is as good as other equations containing the same number of adjustable constants for the representation of vapor pressures over a wide range. Experimental vapor-pressure data are, however, required to obtain the Wagner constants. Frequently, only the structure and the normal boiling point of the pure substance are known. In this work, therefore, an attempt has been made to develop a method for the prediction of vapor pressures when only the normal boiling point is available. The constanta of the Wagner equation were first correlated with the normal boiling points of the n-alkanes from ethane through n-eicosane. The effective carbon number (ECN) concept of Ambrose (1976) and Chase (1984) was then employed to extend the vapor pressure correlation to the isomeric alkanes, alkenes, alkynes, cycloalkanes, alkadienes, aromatics, and sulfur compounds. Ambrose (1976) proposed the concept of a nonintegral ECN and showed that this could be used to predict the properties of branched chain alcohols from the straight-line correlation of the normal boiling points of the straight-chain alcohols. As shown below, the ECN can be used successfully to predict the properties of a wide variety of fluids from a knowledge of the properties of the n-alkanes. In addition, it can distinguish between various isomers. Therefore, it can be used as a single characterization variable in studies involving “continuous thermodynamics” techniques. Current studies (e.g., Cotterman and Praus0196-4305/85/1124-1033$01.50/0
nitz, 1985) using these techniques employ the molecular weight as the characterizationvariable and must, therefore, “lump” all isomers together. Development of the Method The functional form of the Wagner vapor-pressure relationship used in this work is given by hl PR = [A(1 - TR)
+ B(1 - T R ) ~+’c(1 ~ - TR)3+ D(1 - T R ) ~ ] / T(1) R
where PR = PIPc is the reduced (vapor) pressure and TR = T I T , is the reduced temperature. This form of the Wagner equation was recommended by Ambrose (1978) and McGarry (1983) although a more complex form is superior than eq 1 for water and ammonia. The Wagner equation can be used to fit experimental vapor-pressure data over the entire liquid range. For prediction purposes, however, the constants of the equation as well as the critical temperature and pressure (which are not necessarily available for all substances) must be correlated with some readily available property such as the normal boiling point or, in the case of a homologous series, the number of carbon atoms in the molecule. Corbin et al. (1947), Kudchadker and Zwolinski (1966), and Dutt (1982) have demonstrated that the constants of the Anbine vapor-pressure equation for a homologous series can be related to the number of carbon atoms and/or the normal boiling point. Their methods, however, were restricted to vapor pressures below 200 kPa, and in addition, different correlations of the constants were necessary for different homologous series. In the work described below, a single correlation obtained for the n-alkanes is shown to be applicable to other substances via the concept of the effective carbon number (ECN). Ambrose (1976) and, more recently, Chase (1984) have shown that a simple relationship (often a straight line) exists between the normal boiling point and the ECN for the straight-chain members of a homologous series; this relationship can then be used to determine the ECN for the branched chain and unsaturated members from their normal boiling points. It should be noted that nonintegral values of the ECN will normally be obtained for the branched chain and unsaturated members of the homologous series. However, straight-chain, branched, and unsaturated members of the homologous series can now be handled by the same relationships. This establishes both the normal boiling point and the ECN as logical parameters for characterizing, for example, hydrocarbon mixtures 0 1985 American Chemical Society
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Process Des. Dev., Vol.
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Table 11. Correlation of the Vapor Pressures of t h e n -Alkanes n-alkane press range, kPa AAD 70 max dev % ethane 1-4865 0.67 1.88 1-4287 1.21 propane 1.56 n-butane 1-3776 0.81 1.48 n-pentane 1.1 1.32 1-3354 n-hexane 1.0 1-3009 1.54 n-heptane 0.24 1-2724 0.56 n-octane 1-2487 0.75 0.32 n-nonane 0.48 1-2286 0.95 n-decane 1-2115 0.7 1.41 n-undecane 0.13 1-200 0.36 n-dodecane 0.18 0.33 1-200 n-tridecane 0.35 1-200 1.03 n-tetradecane 0.29 1-200 0.74 n-pentadecane 0.88 0.31 1-200 n-hexadecane 1.76 0.79 1-200 1.11 2.15 n-heptadecane 1-200 4.75 2.24 n-octadecane 1-200 2.52 4.88 n-nonadecane 1-200 3.86 n-eicosane 1-200 11.02 0.97 overall AAD %
1100
J
400: 300
200
Table I. A, A, A2 A3
I
i
Constants of Ea 3 = 95,504 418 920 07 = 3.742 203 001499 = 2295.530315 13 = -1042.572 560808
~~
= -22.662 298 239 25 = -1660.893 846 582 As = 439.132269615 A4
A5
(the molecular weight, which is widely used as a characterization parameter in many continuous thermodynamics studies, cannot, of course, distinguish between straightchain and branched-chain isomers). Figure 1 shows a plot of the normal boiling point vs. ECN for the n-alkanes between ethane and nClW using data from API Project 44. The analytical form of the relationship is
Tb(K) = A,
+ Al(ECN) + Az(ECN)0.667+A3(ECN)0'5+ A4 In (ECN) + A,(ECN)O.* + A,(ECN)o.9 (2)
where the constants A1-A6 are given in Table I. The correlation can now be used to obtain the ECN for any hydrocarbon from a knowledge of its boiling point. Using data from Ambrose (1980) for the critical temperatures and pressures of the n-alkanes from ethane through tetradecame and from API Project 44 (1971) for the vapor pressures of the alkanes from ethane through n-eicosane, the following correlations were obtained for the critical properties and the Wagner constants:
T,(K) = Tbll.0
+ (1.25157 + 0.137242ECN)-l]
(3)
P,(MPa) = (2.337 61 8.164 48ECN)/(0.873 159 + 0.542 85ECN)' (4)
+
A = -6.902 37 - 0.041 529ECN - 0.006 503ECN2 (5)
B = 3.551 30 - 0.534943ECN + 0.021 867ECN' (6) C = -4.26807 + 0.460 198ECN - 0.029 179ECN2 (7) D = 5.541 03 - 1.931 88ECN
+ 0.029081ECN'
(8)
The results of the correlation of the API Project 44 vapor-pressure data between 1kPa and the critical pressure of the n-alkanes from ethane through n-eicosane are given in Table 11. It should be noted that the errors increase as the molecular weight of the n-alkane increases. This is mainly because the critical temperatures of the higher alkanes obtained from eq 3 are greater than those reported in the API tables and the difference increases as the mo-
lecular weight of the n-alkane increases. Agreement between the API vapor pressures and the values calculated by using eq 1-8 could have been "improved" if the API values of the critical properties had been used throughout. However, we believe that the Ambrose (1980) values of the critical properties are the best available at the present time. In any case, the overd average absolute deviation of 0.97% for the 19 n-alkanes is very satisfactory. Equations 1-8 can now be used to predict the vapor pressure of pure substances in the range 1 kPa to the critical pressure from a knowledge solely of the normal boiling point (or the ECN). It should be added that the form of eq 2 was chosen so that the correlation can be extrapolated to high ECN values (and still yield physically reasonable Tb) so that the method can be used in "continuous thermodynamics" applications.
Predictions Using the Method Vapor pressures were predicted for 92 substances including alkane isomers, cycloalkanes, alkenes, alkynes, alkadienes, aromatics, and sulfur compounds. Experimental vapor-pressure data and normal boiling points were obtained from API Project 44 as follows. The API Project 44 data are reported in the form of two correlations-an Antoine equation valid from 1 to 200 kPa and a fourconstant equation valid from 200 kPa to P,. Sixteen data points were obtained at evenly distributed points from each of these correlations for each substance. These data were then compared with the calculated data by using eq 1-8 and the normal boiling point of the substance. The results are shown in Table I11 and summarized for each class of compound in Table IV. Considering the wide pressure range over which these predictions have been made, the agreement between the calculated and experimental vapor pressures is excellent. The highest errors (AAD = 8.05%) were obtained for the 3 alkynes, and the lowest errors (AAD = 1.55%) were obtained for the 12 alkenes. What is surprising is that the vapor pressures of these unsaturated compounds can be obtained from a correlation for the n-alkanes with quantitative agreement between prediction and experiment over a wide range of temperatures. Comparisons with Other Methods Most other methods for the prediction of the vapor pressure using a minimum of experimental data such as those of Corbin et al. (1947), Kudchadker and Zwolinski (1966), and Dutt (1982) utilize different correlations for
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Table 111. Predictions of t h e Vapor Pressures of P u r e Substances substance AAD % eq 3-9 3.23 isobutane 2.24 isopentane 3.22 neopentane 1.72 isohexane 2.55 3-methylpentane 5.34 2,2-dimethylbutane 4.17 2,3-dimethylbutane 1.57 2-methylhexane 1.89 3-methylhexane 2.5 3-ethylpentane 4.52 2,2-dimethylpentane 2,3-dimethylpentane 3.26 3.12 2,4-dimethylpentane 1.25 2-methylheptane 1.54 3-methylheptane 1.54 4-methylheptane 2.15 3-ethylhexane 3.75 2,2-dimethylhexane 3.35 2,3-dimethylhexane 2,4-dimethylhexane 3.33 2.65 2,5-dimethylhexane 3,3-dimethylhexane 5.45 3,4-dimethylhexane 3.80 4.35 2-methyl-3-ethylpentane 3-methyl-3-ethylpentane 6.99 6.91 2,2,3-trimethylpentane 2,2,4-trimethylpentane 6.61 7.86 2,3,3-trimethylpentane 4.91 2,3,4-trimethylpentane 2,2,3,3-tetramethylbutane 1.55 2-methyloctane 0.58 2,2-dimethylheptane 2.36 2,2,5-trimethylhexane 4.02 2,2,3,3-tetramethylpentane 8.99 2,2,3,4-tetramethylpentane 8.17 2,2,4&tetramethylpentane 8.30 2,3,3,4-tetramethylpentane 7.36 3,3,5-trimethylheptane 6.30 2,2,3,3-tetramethylhexane 9.51 2,2,5,5-tetramethylhexane 2.64 3,3-dimethylpentane 6.2 2,2,3-trimethylbutane 7.46 cyclobutane 1.37 cyclopentane 1.17 cyclohexane 3.11 cycloheptane 6.15
AAD % UNIFAC 5.34 9.45 2.63 3.17 17.71 10.49" 1.52 1.07 4.76" 9.14" 2.30" 3.25" 2.75 4.14" 2.93O 1.09 2.20" 6.16" 9.33" 4.74" 2.29 5.14 3.09" 10.06" 11.49" 3.08" 3.23 2.43" 3.11" 16.22" 3.74" 7.85" 10.71" 5.92" 17.51a 3.73" 2.52" 8.12" 3.89" 28.07" 6.14" 5.46" 22.92" 1.68 4.54 12.23"
substance cyclooctane methylcyclopentane ethylcyclopentane methylcyclohexane trans-1,4-dimethylcyclohexane 1-butene cis-2-butene trans-2-butene 2-methylpropene 1-pentene cis-2-pentene trans-2-pentene 2-methyl-1-butene 2-methyl-2-butene hexene heptene octene 1,3-butadiene 1-butyne 2-butyne pent-1-yne benzene toluene ethylbenzene o-xylene m-xylene p-xylene propylbenzene isopropylbenzene 1,2,3-trimethylbenzene 1,2,4-trimethylbenzene 1,3,5-trimethylbenzene butylbenzene isobutylbenzene 4-isopropyl-1-methylbenzene 1,4-diethylbenzene 1,2,3,5-tetramethylbenzene 1,2,4,5-tetramethylbenzene methylnaphthalene methanethiol ethanethiol tetrahydrothiophene thiophene dimethyl sulfide ethyl methyl sulfide diethyl sulfide
AAD % eq 3-9 8.08 2.51 4.28 6.58 8.42 0.9 3.09 1.72 0.82 4.71 1.30 1.08 1.10 1.51 0.73 0.67 0.94 1.86 11.01 5.26 7.87 0.75 1.86 2.90 2.76 2.03 2.78 3.70 3.73 2.77 2.29 0.96 4.47 6.00 6.38 4.23 3.41 2.55 10.17 8.23 4.83 2.52 2.24 4.66 2.59 10.89
AAD % UNIFAC 11.93" 23.78" 2.50" 4.80" 12.47" 8.20 b b b 10.56 b b b b 8.84 8.66 8.42 18.64" b b b 2.19 11.04 18.25 2.10 17.16 19.67 10.58" 25.93 5.75 13.38 24.09 16.85" 5.68" 0.90 13.22" 11.63" 14.80" 20.77" b b b b b b b
"Vapor-phase fugacity coefficient set equal to 1 due to lack of pure component data. Fugacity coefficients of all other substances calculated by using the method of Hayden and O'ConnelI (1975). bUNIFAC group parameters not available for this substance. Table IV. Summary of Vapor-Pressure Predictions this work UNIFAC no. of no. of class AAD 5% comDds AAD % comDds n-alkane isomers 4.27 42 6.38 42 cycloalkanes 4.63 9 10.76 9 12 alkenes 1.55 8.94 5 1 1 18.64 alkadiene5 1.86 3 a a alkynes 8.05 18 13.00 18 aromatics 3.54 7 sulfur 5.14 a a 92 8.83 75 overall 3.97 (I
UNIFAC group parameters not available for these substances.
different homologous series. The only general method valid for all classes of compounds studied here is the modified UNIFAC method of Jensen et al. (1981) and Yair and Fredenslund (1983). This method is however valid only for vapor pressures in the range 1-270 kPa and requires that the appropriate parameters for the groups be available. Moreover, if the fugacity coefficient required in the calculations is to be evaluated, this requires critical temperature, critical pressure, and/or dipole moment and
Table V. Predictions of t h e Vapor-Pressures of t h e High Molecular Weight n -Alkanes substance AAD % this work AAD % UNIFAC 0.11 1.3 0.39 1.18 0.38 2.94 1.59 2.93 2.51 2.43 5.81 1.06 6.23 1.49 7.66 2.91 8.41 1.97 12.28 4.79 18.64 8.42 5.82 2.86
radius of gyration data (the fugacity coefficient may, however, be set equal to one without a great loss in accuracy). UNIFAC predictions for 75 substances studied by us are shown in Table I11 (predictions for the remaining 17 substances could not be made because the appropriate group parameters were not available). As can be seen from the table, the UNIFAC method is less accurate than the method proposed here. On the other hand, the UNIFAC
Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 1036-1042
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method appears to extrapolate better to very high molecular weight hydrocarbons. This is demonstrated in Table V and is, in part, due to the fact that our critical property correlations (eq 4 and 5) show increasing divergence from the API Project 44 values (noting that the vapor-pressure data for the substances studied were obtained from API Project 44). Conclusion A method has been developed for the prediction of the vapor pressures of alkane isomers, cycloalkanes, alkenes, alkynes, alkadienes, aromatics, and sulfur compounds for pressures between 1 kPa and the approximate critical pressure of the substance. The method requires the normal boiling point of the substance and employs the concept of the effective carbon number (ECN). The correlations were developed by using the known properties of the n-alkanes, and the results presented here indicate that the same relationship may also be valid for different homologous series. Only the normal boiling point of a pure substance need be known to predict its vapor pressure between 1 kPa and its critical point.
Literature Cited Ambrose. D. National Physical Laboratory Report Chemistry 19, Nov 1972 (corrected Feb 1974). Ambrose, D. National Physical Laboratory Report Chemistry 57, Dec 1976. Ambrose, D. J. Chem. Thermodyn. 1978, 10, 765. Ambrose, D. National Physical Laboratory Report Chemistry 107, Feb 1980. American Petroleum Institute Project 44, Texas A&M University, College Station, TX, 1971. Chase, J. D. Chem. Eng. Prog. 1984, 80 (4), 63. Corbin, N.; Alexander, M.; Egloff, G. J . Phys. Chem. 1947, 51, 528. Conerman, R. L.;Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 194.
Dun. N. V. K. Can. J . Chem. Eng. 1982, 6 0 , 707. Hayden, J. G.; O’Connell, J. P. Ind. Eng. Chem. Process Des. Dev. 1975, 14, 209.
Jensen, T.; Fredenslund, A.; Rasmussen, P. Ind. Eng. Chem, Fundam. 1981, 20, 239.
Kudchadker, A. P.; Zwolinski, B. J. J. Chem. Eng. Data 1966, 1 1 , 253. McGarry, J. Ind. Eng. Chem. Process Des. Dev. 1983, 2 2 , 313. Wagner, W. Cryogenics 1973, 13, 470. Waring, W. Ind. Eng. Chem. 1954, 46, 762. Wilsak, R. A.; Thodos, G. Ind. Eng. Chem. Fundam. 1983, 2 3 , 75. Yair, 0 . B.; Fredenslund, A. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 433.
Received for review June 13, 1984 Revised manuscript received November 5, 1984 Accepted January 10, 1985
Generalized Correlation for Solvent Activities in Polymer Solutions Mlchael J. Mlsovich and Erlc A. Grulke” Department of Chemical Engineering, Michigan State University, East Lansing, Michigan 48824
Robert F. Blanks Amoco Chemicals Corporation, Naperville, Illinois 60566
A correlation for solvent activities is developed by applying an athermal form of the ASOG (Analytical Solution of Groups) group-contributiinmodel to polymer solutions. The new model provides a correction which accounts for differences in the free volume between the solvent and polymer and does not require groupcontribution calculatiis. Instead, a closed-form solution containing only one adjustable parameter is derived for the weight fraction solvent activity coefficient as a function of weight fraction composition. The adjustable parameter represents an experimentally measurable infinite dilution weight fraction solvent activity coefficient. The new model is compared with the Flory-Huggins and UNIFAGFV models for 29 isothermal binary systems, including systems with sizable enthalpy of mixing effects. Calculated activity coefficients of the new model are found to agree within the 10% to experimentally observed values for 120 of 130 data points. This agreement represented a better performance than either the Flory-Huggins or the UNIFAC-FV model.
An understanding of the thermodynamics of polymersolvent systems is important in many practical applications; processing steps such as polymerization, devolatilization, plasticization, and addition of other additives all require a knowledge of polymer solution thermodyamics. Diffusion phenomena in polymer melts and solutions are often strongly affected by nonideal solution behavior. Proper design of many polymer processes depends greatly upon accurate modeling of thermodynamic parameters such as solvent activities. This work presents a thermodynamic correlation method for solvent activities in polymer solutions as a function of concentration. The method is developed theoretically from consideration of athermal solutions; however, it shows good 0196-430518511124-1036$01.50/0
agreement with experimental data available for some polymer-solvent systems which have enthalpic interactions. The model is based upon an athermal form of the ASOG (Analytical Solution of Groups) group-contribution model for calculation of activity coefficients in solution and uses weight fractions to describe concentrations. A correction is made to account for the difference in the free volume between the solvent and polymer, as evidenced by their different densities. Since only athermal terms are considered in the model development, group-interaction parameters used in calculating enthalpy effects are not included and the final model reduces to a single equation. The model shows good agreement with experiment over the entire range of concentrations reported in the literature 0 1985 American Chemical Society
