Ind. Eng. Chem. Res. 1988,27, 1058-1065
1058 rtd =
characteristic time for thermal diffusion Registry No. ZnSO,, 7733-02-0; ZnS04.7H20, 7446-20-0.
Literature Cited Antal, M. J., Jr. “Thermogravimetric Signatures of Complex Solid Phase Pyrolysis Mechanisms and Kinetics”. In Thermal Analysis; Miller, B., Ed.; Wiley: New York, 1983; Vol. 11, p 1490. Antal, M. J., Jr. “Biomass Pyrolysis: A Review of the Literature. Part 11. Lignocellulose Pyrolysis”. In Advances in Solar Energy; Boer, K. W., Duffie, J. A., Eds.; American Solar Energy Society: New York, 1985; Vol. 2. Antal, M. J., Jr.; Friedman, H. L.; Rogers, F. E. Combust. Sci. Technol. 1980,21, 141-152. Antal, M. J., Jr.; Mok, W. S.-L.; Roy, J. C.; T-Raissi, A. J . Anal. Appl. Pyrol. 1985, 8, 291. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969; Chapter 11, p 242. Bowman, M. G. “Solar Thermal Hydrogen”. Presented at the Proceedings of the International Symposium on Hydrogen Production from Renewable Energy, Honolulu, 1984. Brown, M. E.; Dollimore, D.; Galwey, A. K. Comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. H. F., Eds.; Elsevier: New York, 1983; Vol. 22, p 41. Come, G. M. comprehensive Chemical Kinetics; Bamford, C. H., Tipper, C. H. F., Eds.; Elsevier: New York, 1983; Vol. 24, p 249. Cvetanovic, R. J.; Singleton, D. L.; Paraskevpoulos, E. J. Phys. Chem. 1979, 83(1),50-60. Ducarroir, M.;Romero-Paredes, H.; Steinmetz, D.; Sibieude, F.; Tmar, M. ”On the Kinetics of the Thermal Decomposition of Sulfates Related with Hydrogen Water Splitting Cycles”. Proc. 4th World Hydrogen Energy Conf. 1982,2, 451-463. Flynn, J. Thermochim. Acta 1980,37, 225-238. Friedman, H. L. J. Polym. Sci. 1964, C6, 183-195. Garn, P. D. J. Thermal Anal. 1975, 7, 475. Hildenbrand, D. L. ”High Temperature Chemistry of Hydrogen Production Cycles”. Technical Status Report, September 1978; SRI International, PYU-6788. Hildenbrand, D. L. ”High Temperature Chemistry of Hydrogen Production Cycles”. Technical Status Report, October 1979; SRI International, PYU-6788.
Hildenbrand, D. L. ”High Temperature Chemistry of Hydrogen Production Cycles”. Technical Status Report, October 1980; SRI International, PYU-6788. Hosmer, P. K.; Krikorian, 0. H. “Solar Furnace Decomposition Studies of Zinc Sulfate” Lawrence Livermore Laboratory Report, Preprint UCRL 83634, 1979. Hosmer, P. K.; Krikorian, 0. H. High Temp.-High Pressures 1980, 12, 281. Ibanez, J. G.; Wentworth, W. E.; Batten, C. F.; Chen, E. C. M. “Kinetics of the Thermal Decomposition of Zinc Sulfate”. Rev. Int. Hautes Temp. Refract. 1984, 21, 113-124. Ingraham, T. R.; Kellogg,H. H. Trans. Metall. SOC.M M E 1963,227, 1419-1426. Kirk-Othmer, Encyclopedia of Chemical Technology, 3rd ed.; Wiley-Interscience; New York, 1984; Vol. 24, p 807. Kolta, G. A.; Askar, M. H. Thermochim. Acta 1975, 11, 65-72. Krikorian, 0. H.; Shell, P. K. Int. J. Hydrogen Energy 1982, 7(6), 463-469. Mok, W. S.-L.; Antal, M. J., Jr. Thermochim. Acta 1983, 68, 165. More, J. J. “The Levenberg-Marquardt Algorithm. Implementation and Theory”. In Numerical Analysis; Lecture Notes in Mathematics 630; Springer-Verlag; New York, 1977; pp 105-116. Mu, J.; Perlmutter, D. D. Znd. Eng. Chem. Process. Des. Dev. 1981, 20, 640. Nowak, U.; Deuflhard, P. “Towards Parameter Identification for Large Chemical Reaction Systems”. International Workshop on Numerical Treatment of Inverse Problems for Differential and Integral Equations, Heidelberg, FRG, 1982. Ostroff, A. G.; Sanderson, R. T. J . Inorg Nucl. Chem. 1959,9,45-50. Sestak, J.; Satava, V.; Wendlandt, W. W. Thermochim. Acta 1973, 7, 333. Shell, P. K.; Ruiz, R.; Yu, C. M. “Solar Thermal Decomposition of Zinc Sulfate”. Lawrence Livermore National laboratory Report UCRL 53370, 1983. Tagawa, H. Thermochim. Acta 1984, 80, 23-33. Urban, D. L.; Antal, M. J., Jr. Fuel 1982, 61, 799-806. Wendlandt, W. W. Thermal Methods of Analysis; Interscience: New York, 1964. Zsako, J. J . Thermal Anal. 1976,9, 101. Received for review April 28, 1987 Accepted January 12, 1988
Parameters for the Perturbed-Hard-ChainTheory from Characterization Data for Heavy Fossil Fuel Fluids S h a o - H w a Wangt and Wallace
B. Whiting*
Department of Chemical Engineering, West Virginia University, Morgantown, West Virginia 26506-6101
Correlations for pure-component parameters have been developed for the Perturbed-Hard-Chain equation of state in terms of conventional characterization data (molecular weight, normal boiling point, and specific gravity). The correlations are comparable to and often better than correlations using molecular-structure information (which require elaborate measurements) in predicting saturated liquid densities and vapor pressures for heavy fossil fuel fractions. A universal temperature dependence of the hard-core volume is introduced. The Perturbed-Hard-Chain equation of state with the proposed parameter correlations and hard-core-volume temperature dependence gives better VLE calculation results than typical cubic equations of state for heavy hydrocarbons. This thermodynamic model is extended for continuous mixtures. The economic value of recovering more liquid from heavy fossil oil is increasing. It is, therefore, important to be able to calculate the properties of these heavy hydrocarbons which have not been studied as extensively as have lighter hydrocarbons.
* Author t o whom correspondence should be addressed. Presently at Morgantown Energy Technology Center, Morgantown, WV 26505.
0888-5885/88/2627-lO58$01.50/0
Conventionally, the Redlich-Kwong equation of state and its modifications have been used to describe the thermodynamic properties of low molecular weight hydrocarbon mixtures. However, theoretically this type of equation of state is limited to simple spherical molecules. As indicated by Sim and Daubert (1980) and Alexander et al. (1985), such equations of state are inadequate in predicting the phase equilibria of ill-defined heavy hydrocarbons. The Perturbed-Hard-Chain (PHC) equation 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1059 of state (Beret and Prausnitz, 1975; Donohue and Prausnitz, 1978; Gmehling e t al., 1979) is suitable for small as well as large molecules whose shape is closer to that of a flexible chain than to that of a sphere. Using the PHC equation of state, Wilhelm and Prausnitz (1985) have developed a correlation to predict the thermodynamic properties of narrow-boiling petroleum fractions by using characterization data such as molecular weight (M,), fractional aromaticity, fractional naphthenicity, and number of methyl groups per molecule. These characterization data provide more information concerning molecular properties than the traditional characterization factors, Le., specific gravity (S,), normal boiling point (Tb), and molecular weight. However, for detailed structural analyses, the following measurements have to be performed for each hydrocarbon fraction: lH NMR, elemental analysis, PNA analysis, and IR spectroscopy. Furthermore, Wilhelm and Prausnitz (1985) found that the correlation for one of the equation of state parameters (the flexibility parameter c) fails to extrapolate to high molecular weights and has to be tuned using a measured vapor pressure datum. We have developed new correlations in accordance with perturbation theory for the PHC equation of state parameters using conventional, readily accessible properties of fossil fuel fractions, Le., M,, S,, and Tb. The PHC parameters of the n-alkanes are correlated with M,. Parameters for the other hydrocarbons are expressed as perturbations to those of the n-alkanes, and the perturbations are correlated with Tband S,. This new correlation is compared with the correlation of Wilhelm and Prausnitz (1985). To increase the predictive capability for high-temperature systems, a universal temperature dependence is introduced for the hard-core-volume parameter of hydrocarbons. Comparisons of the PHC equation of state (with this new correlation) and typical cubic equations of state are made using the vapor-liquid equilibrium data of two coal liquids (Lin et al., 1985a,b). Since the new correlations of parameters are continuous functions of characterization data and most continuous hydrocarbon mixtures are of high normal boiling point, the PHC equation of state with these new correlations is a particularly suitable thermodynamic model for the phase-equilibrium calculations of continuous mixtures.
Thermodynamic Framework We use a semitheoretical equation of state (Gmehling et al., 1979) to describe thermodynamic properties of simple and complex hydrocarbons and their mixtures a t liquid- and gaslike densities. The equation of state is essentially a modification of the Perturbed-Hard-Chain (PHC) equation of state of Donohue and Prausnitz (1978). At low densities, it meets the ideal-gas boundary condition. For complex (polymer) molecules a t liquidlike densities, it reduces to the theories of Prigogine (1957) and Flory (1970). For small, spherical molecules, it becomes the perturbed-hard-sphere theory of Alder (1972) and Barker and Henderson (1972). Appendix A gives the Perturbed-Hard-Chain equation of state used in this work. The fundamental molecular parameters for the truncated version of the PHC equation of state are u*, T * ,and c. These parameters were obtained by simultaneously fitting the equation of state to experimental vapor pressures and liquid densities. Table I gives characterization data and PHC parameters for 61 hydrocarbons. Since the critical properties (which are used in conventional equations of state) can be neither measured for very large molecules nor estimated with satisfactory
accuracy, an equation of state in terms of adjustable (molecular) parameters is more suitable for heavy hydrocarbons. To apply the PHC equation of state to fossil fuel fractions, it is necessary to estimate the molecular parameters. Appendix B shows the correlation of PHC parameters with molecular structure developed by Wilhelm and Prausnitz (1985). This correlation requires data from lH NMR, elemental analysis, PNA analysis, and IR spectroscopy. These analyses are not generally available because of their high cost. A new approach which utilizes only readily available characterization data of fossil fuel fractions (i.e., M,, Tb,and S,) is developed as follows. Since experimental data are most plentiful for n-alkanes, a straight-chain molecule of a given molecular weight is chosen in the following correlations as a reference. For n-alkanes, physical properties (S, and Tb)and PHC parameters can be accurately expressed as a function of molecular weight by
eA= el + e , ~ , + e,Mw2 + e 4 ~ , 3+ B,/M,
(1)
where OA represents either S,, Tb, or PHC parameters of an n-alkane. The correlated S, and Tb values will be required to calculate the PHC parameters of other hydrocarbons. Table I1 gives the coefficients 01-05 and the regression results. It is noted that parameter c is obtained from the division of q t l k by T*. In the correlation, Tb is in kelvins. The specific gravity at 60 OF160 OF is used because API degrees [=141.5/Sg(60 OF160 O F ) - 131.51 are widely used in the petroleum industry, and many data in these units are reported, for example, in the API Technical Data Book (1983). For the data base including n-C4 to n-C30 and n-C40, the average absolute deviations (AAD’s) of the correlations are 0.085% for S, and 0.102% for Tb. The AAD’s of the correlations for the 15 n-alkanes listed in Table I1 are 0.630% for u*, 0.697% for T * , and 0.557% for q t l k . Parameters for hydrocarbons other than n-alkanes are correlated by the second-order perturbation equation: 8 = 4- culAs, + a2ATb+ a3AS; 4- a4AsgATb+ a5ATb2(2) with AS, = S, - SgA
(3)
AT, = Tb - TbA
(4)
and where quantities subscripted with A stand for the parameter or properties of the hypothetical n-alkane with the same M , as that of the hydrocarbon of interest. The coefficients aiin eq 2 are defined as ai = ai + biMw (5) where aiand bi are given in Table 111. The AAD’s of the correlation for the 46 hydrocarbons listed in Table I1 are 0.789% for u*, 1.497% for T * , and 2.074% for q t l k . The following universal temperature dependence for the u* of hydrocarbons is proposed: u* = uo* exp(0.057393 - 0.088538T) (6) where the hard-core volume uo* is determined from eq 2 and T (=TIT*)is the reduced temperature. The coefficients in eq 6 were obtained by fitting the saturated liquid densities of hydrocarbons at different temperatures. The AAD of u* from correlating seven typical compounds (toluene, n-decane, n-dodecane, m-xylene, methylcyclohexane, n-eicosane, and n-triacontane) is 3.310%. In re-
1060 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 Table I. Physical Properties and Molecular Parametere for 61 Hydrocarbon Fluids compound M,,d m o l u * , cm3/mol qclk, K S. at 60 OF/60 OF n-Alkanes methane 16.043 20.03 151.1 0.300 ethane 30.070 27.98 282.0 0.3564 44.097 37.49 374.2 propane 0.5077 58.124 47.06 464.1 butane 0.5844 56.59 556.1 72.151 pentane 0.6310 65.48 638.7 86.178 hexane 0.6640 100.205 74.56 726.6 heptane 0.6882 114.232 85.42 807.1 octane 0.7068 128.259 94.47 899.8 nonane 0.7217 102.3 142.286 985.8 decane 0.7342 1119 123.7 170.340 dodecane 0.7526 1447 164.1 226.448 hexadecane 0.7773 199.7 1749 282.556 eicosane 0.7924 2445 299.9 422.826 triacontane 0.8133 3064 396.8 563.096 tetracontane 0.8252
Th,K
C
111.7 184.5 231.1 272.7 309.2 341.9 371.6 398.8 424.0 447.3 489.5 560.0 617.0 722.7 798.1
1000 1.253 1.436 1.606 1.802 1.952 2.138 2.302 2.519 2.697 2.900 3.510 4.194 5.577 6.752
isobutane isopentane 2-methylhexane 3-methylhexane 2-methylheptane 2,2,4-trimethylpentane 2,2,5-trimethylhexane squalane
58.124 72.151 100.205 100.205 114.232 114.232 128.259 422.826
Isoalkanes 47.15 55.97 75.18 74.68 84.83 85.85 94.39 299.0
435.6 521.0 693.2 694.0 776.1 688.9 783.4 2146
0.5631 0.6247 0.6579 0.6689 0.7021 0.6962 0.7182 0.8129
261.3 301.0 363.2 365.0 390.8 372.4 397.3 697.8
1.553 1.683 2.063 2.044 2.233 1.990 2.242 4.896
cyclopentane cyclohexane bicyclohexyl
70.135 84.162 166.308
Naphthenes 48.20 57.52 103.3
561.8 620.9 1080
0.7504 0.7834 0.9176
322.4 353.9 511.2
1.632 1.680 2.468
methylcyclohexane ethylcyclohexane n-butylcyclohexane n-hexylcyclohexane n-decylcyclohexane n-hexadecylcyclohexane
98.189 112.216 140.270 168.324 224.432 308.594
Alkylnaphthenes 66.27 76.32 95.67 115.1 154.1 210.4
661.8 765.2 921.7 1072 1382 1857
0.7740 0.7922 0.8031 0.8115 0.8224 0.8316
374.1 404.9 454.1 498.1 570.9 652.0
1.735 1.993 2.308 2.593 3.217 4.230
indane cis-decalin trans-deralin tetralin
118.179 138.254 138.254 132.206
Naphthene Aromatics 68.67 863.5 84.81 887.7 87.21 852.9 74.44 940.0
0.9685 0.9011 0.8744 0.9747
451.2 468.9 460.4 480.8
1.960 1.982 1.920 2.045
benzene naphthalene diphenyl phenanthrene anthracene
78.114 128.174 154.212 178.234 178.234
Aromatics 47.67 71.07 83.72 93.95 93.95
630.1 943.0 1083 1207 1206
0.8844 1.1780 1.0801 1.1602 1.1617
353.2 491.1 528.4 611.6 613.1
1.670 1.954 2.223 2.065 2.050
toluene ethylbenzene n-butylbenzene n-hexylbenzene n-decylbenzene n-hexadecylbenzene o-xylene m-xylene p-xylene tert-butylbenzene
92.141 106.168 134.222 162.276 218.384 302.546 106.168 106.168 106.168 134.222
Alkylbenzenes 57.29 65.75 85.16 105.3 143.3 198.8 65.19 65.75 66.35 84.89
716.9 781.9 930.9 1080 1390 1928 799.3 791.6 784.4 879.6
0.8718 0.8718 0.8646 0.8617 0.8596 0.8586 0.8848 0.8687 0.8657 0.8710
383.8 409.3 456.5 498.9 570.9 651.3 417.6 412.3 411.5 442.3
1.869 1.972 2.280 2.580 3.214 4.447 1.969 1.989 1.965 2.192
1-methylnaphthalene 1-ethylnaphthalene 1,2-dimethylnaphthalene 1-n-propylnaphthalene 1-n-butylnaphthalene
142.201 156.228 156.228 170.255 184.282
Alkylnaphthalenes 80.28 89.00 86.75 98.35 108.7
1031 1108 1199 1150 1182
1.0244 1.0122 1.0219 0.9940 0.9808
517.9 531.8 539.5 545.8 562.4
2.115 2.329 2.588 2.404 2.464
diphenylmethane isopropyldiphenyl ditoluenemethane dicumenemethane tetraisopropylphenylmethane
168.239 196.293 196.293 252.401 336.563
Phenylmethanes 95.77 107.3 114.9 150.9 204.5
1183 1232 1236 1489 1624
1.0104 1.0025 0.9793 0.9719 0.9403
538.2 554.3 567.1 610.2 618.9
2.584 2.623 2.593 3.219 3.753
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1061 Table 11. Coefficients and Regression Results of Equation 1 61
62
93
64
0.791593+00* 0.227253+03 0.706913+01 0.264883+03 0.138683+03
0.33908343 0.213493+01 0.667253+00 0.119453+01 0.614653+01
-0.801093-06 -0.30 1463-02 0.965603-04 -0.28228342 -0.141723-02
0.671103-09 0.18595345 -0.92633347 0.23280345 -0.46847346
@A
s* Tb
U*
T* Q'lk
"Average absolute relative deviation. *The notation E*n stands for
Table 111. Coefficients and Regression Results for Equations 2-5
e T*
46lk
0.672293+03 -0.104OOE+01 -0.266593+04 0.193523+02 -0.47354341 -0.163363+01 0.375733-02 0.748043+01 -0.463133-01 0.174743-03 1.497
-0.393043+02 0.333313+01 -0.275233+03 -0.113613+02 0.831403+01 -0.793803+01 0.184173-01 0.275823+02 -0.913983-01 -0.300673-03 2.074
U*
%
-0.297503+02 0.11105E+00 0.225563+03 -0.171553+01 0.29560342 -0.684393+00 -0.394973-03 -0.444103-01 0.63179342 -0.107053-04 0.789
1.1
0.6 280
320
360
400
440
480
520
580
6M)
Temperature, K A +
985). pr0pos.d cornlaflon. ......corrdotlon of (1Wllhdm and Pmusnltl (198s)
0 Q data of 4l.xand.r
1.ol
X
6s -0.13044E+02 -0.406083+04 0.342563+02 4.214073+04 -0.135823+04
AAD," % 0.085 0.102 0.630 0.697 0.557
lo*" here and in other tables in this paper.
Table IV. Molecular Weight, Normal Boiling Point, and Specific Gravity for Fossil Fuel Fractions (Data of Alexander (1985)) Mw, S, at 60 T,.K OF/60 O F sample fraction d m o l Belridge crude oil 1 148 450.0 0.803 52 Belridge crude oil 2 189 506.5 0.848 83 Belridge crude oil 3 245 575.0 0.893 30 4 Belridge crude oil 340 661.0 0.949 66 5 Belridge crude oil 464 758.0 0.957 37 1 Hendrick Station 154 450.15 0.79450 pipeline mixture 2 Hendrick Station 189 505.56 0.824 30 pipeline mixture Hendrick Station 3 247 575.15 0.86840 pipeline mixture 4 Hendrick Station 344 665.28 0.902 90 pipeline mixture 5 Hendrick Station 510 762.50 0.933 60 pipeline mixture Exxon donor solvent 1 490.33 0.935 50 133 process product" 2 Exxon donor solvent 146 498.70 0.913 20 process product" 3 Exxon donor solvent 154 517.40 0.93466 process producta 4 Exxon donor solvent 165 537.45 0.94802 process producta 5 Exxon donor solvent 174 558.65 0.970 54 process producta Exxon donor solvent 6 184 582.20 0.99346 process product" Exxon donor solvent 7 189 589.90 0.994 46 process product" Exxon donor solvent 8 232 692.17 1.060 13 process product" ~~
a Tis are estimated by fitting a vapor pressure datum using the PHC equation of state for the Exxon donor solvent process product. 0.6
280
320
360
400
440
480
520
580
800
Temperature, K A +
00
data of A1eXand.r (IOSS), p l o p o r M ownlotton. ...... oorr.latlon of Wllhmlm and Pmusnm (loas)
Figure 1. Comparison of saturated liquid density calculations for Belridge crude oil fractions using PHC EOS with (a, top) untuned parameters and (b, bottom) tuned parameters. (A,+, *, 0,0)Data correlation of of Alexander (1985); (-) proposed correlation; Wilhelm and Prausnitz (1985). (.-e)
ality, molecules have more energy a t higher temperatures and are, therefore, capable of interpenetrating one another to a greater extent. Therefore, proposed eq 6 compensates in some way for the inadequacies of the PHC equation of state, by making the hard-core volume temperature dependent. (For an alternate approach to the temperature dependence of hard-core volume, see Vilmachand and Donohue (1985).)
Results and Discussion Physical properties required in the characterization of fossil fuel fractions are reported for Belridge crude oil, a Hendrick station pipeline mixture, and Exxon donor solvent process product by Alexander et al. (1985). These
data are transformed and rearranged and are listed in Table IV. The structure-characterization data are given for Belridge crude oil by Wilhelm and Prausnitz (1985). Table V and Figure 1show the comparison of saturated liquid density calculations for Belridge crude oil using the Perturbed-Hard-Chain equation of state with different parameter correlations. Without tuning to a vapor pressure datum, the correlation of Wilhelm and Prausnitz (1985) gives very poor results because of the inaccuracy of the c parameter estimations. After fitting to a vapor pressure datum, the parameter c is corrected and predicted results are significantly improved. Our proposed correlation gives much better results than the original correlation of Wilhelm and Prausnitz (1985), and it even gives better results for the first two fractions than the Wilhelm and Prausnitz correlation with tuned parameters. By adjustment of the parameters to a vapor pressure datum and/or by introduction of the temperature dependence of the hard-core volume, the predicted results are improved and are better than the results yielded by the correlation of Wilhelm and Prausnitz (1985). Table V also lists the average deviations of saturated liquid density calculations for the Hendrick station pipeline mixture and for the Exxon donor solvent process
1062 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 Table V. Comuarison of Saturated Liauid Density Calculations Using the PHC Equation of State for Fossil Fuel Fractions AAD (70)in saturated liquid density calculation fraction
no. of data
(1)"
(2)O (3)" Belridge Crude Oil 1.889 1.015 0.615 2.704 4.175 1.117 1.423 3.707 2.681 3.907
1 2 3 4 5
9 19 15 14 11
1.696 2.635 4.484 4.948 6.315
1 2 3 4 5
7 7 12 20 23
Hendrick Station Pipeline Mixtureb 2.556 0.835 4.697 3.545 3.602 0.659 4.205 5.819 7.554 3.164
7 9 10 9 12 12 23 23
Exxon Donor Solvent Process Productb 1.093 2.027 1.793 1.409 1.776 2.074 1.154 2.316 1.432 2.204 1.335 2.129 1.759 2.065 2.894 4.932
(4)"
(5Ia
(6)"
0.809 0.611 0.918 0.659 0.708
12.283 17.555 18.299 19.219 27.944
3.704 3.893 1.929 2.079 0.995
"PHC equation of state parameters are estimated by (1) eq 1-5 with M,, S,, and Tb;(2) same as (l), except c fit to a vapor pressure datum; (3) same as (l), except u* is temperature dependent; (4) combination of (2) and (3); (5) eq Bl-B3 with structural data; (6) same as (5), except c fit to a vapor pressure datum. bStructural-characterization data for PHC parameters are not available for these fluids.
product obtained by using the PHC equation of state with the proposed correlation. The predictions are very good but become less accurate for heavier fractions. This is expected because the characterization data of these heavy fractions are out of the range of the PHC parameter correlation. Similar results are obtained for vapor pressure calculations, as shown in Table VI and Figure 2. The proposed correlation is better than the correlation using structural information and a vapor pressure datum. Table VI also compares the average absolute deviations of vapor pressure predictions for fossil fuel fractions obtained by using the PHC equation of state and the Soave-Redlich-Kwong (SRK) equation of state with various parameter correlations. By use of conventional characterization data, the PHC equation of state obtains better results for vapor pressure predictions than the SRK equation of state. The SRK equation of state with parameters estimated from structural characterization data (Alexander et al., 1985) calculates vapor pressure for lighter fractions reasonably well, sometimes even better than does the PHC equation of state. However, for heavier fractions, the PHC equation of state with the proposed parameter correlations gives better results than either the PHC equation of state or the SRK equation of state with structure-characterization correlations. The above discussions are for the properties of hydrocarbon fractions. Vapor-liquid equilibrium examples are shown in Tables VI1 and VI11 for a coal liquid from a Kentucky No. 9 coal (Lin et al., 1985a) and for a liquid from a Wyoming coal (Lin et al., 1985b). The PHC equation of state with the proposed correlation obtains better results than typical cubic equations of state for the isothermal flash. This also suggests that the PHC equation of state is more useful for high-boiling fossil fuel fluids than typical cubic equations of state. The extension of the PHC equation of state for semicontinuous or continuous mixtures is straightforward (as given in Appendix C). Since the characterization data (Mw, Tb,and Sg)are readily available and the molar distribution function is a continuous function of these variables, the proposed correlation of molecular parameters should be
8.8
1.6
2.8
2.4
3.2
1 /Temperature X 10 , K b
+ . a data of AIorondor (I 085).
-
propond Eomh2tlon
......comlatlm of Wlholm and Pmumlt. (1085)
2.0
1 6
2 4
1 /TemRerature X 10 b
+ *m
2 8
3.2
,K
-
data of AIwander (I 985) propond oomlation oolrdaflon 01 Wlholk and Pmuanlt.(IOIS)
. .. .
Figure 2. Comparison of vapor pressure calculations for Belridge crude oil fractions using PHC EOS with (a, top) untuned parameters and (b, bottom) tuned parameters. (A,+, *, 0,0 )Data of Alexancorrelation of Wilhelm and der (1985);(-) proposed correlation; Prausnitz (1985). (ma.)
useful in the application of continuous thermodynamics.
Conclusion Correlations based on perturbation theory are developed for the Perturbed-Hard-Chain equation of state parameTb, ters using readily accessible characterization data (Mw, and Sg). These correlations give better predictions for saturated liquid density and vapor pressure than those proposed by Wilhelm and Prausnitz (1985),which require structure-characterization information. This suggests that
Ind. Eng. Chem.Res., Vol. 27, No. 6,1988 1063 Table VI. Comparison of Vapor Pressure Calculations Using PHC Equation of State and SRK Equation of State for Fossil Fuel Fractions AAD (%) in vapor pressure calculation fraction no. of data (1)' (2)" (3)" (4)" (5Y (6)b Belridge Crude Oil 30.000 76.035 24.249 1 17 25.349 23.101 11.000 15.415 16.000 12.353 84.257 2 12 15.466 5.000 36.971 49.000 35.059 33.477 81.183 39.000 3 5 52.195 58.000 44.237 42.121 90.687 48.000 4 14 1 2 3 4
21 22 21 15
Hendrick Station Pipeline Mixturec 44.960 43.250 18.820 19.790 17.910 18.940 48.370 45.940
51.000 22.000 23.000 67.000
34.000 19.000 16.000 77.000
"PHC equation of state parameters are estimated by (1)eq 1-5 with M,, S,, and Tb;(2) same as (l), except u* is temperature dependent; (3) eq Bi-B3 with structural data; (4) same as (3), except c fit to a vapor pressure datum. *Results of Alexander et al. (1985). SRK with m fit to a vapor pressure datum; (6) structural-characterization data, with m fit to a vapor parameters are estimated by (5) S, and Tb, pressure datum. Structural-characterization data for PHC parameters are not available for these fluids.
Table VII. Vapor-Liquid Equilibria for a Coal Liquid (from a Kentucky No. 9 Coal) exptl data" PHC CCOR" P, bar T,K wt % vaporized T, K T,K % dev % dev 0.76 0.56 661.9 656.9 660.6 0.342 37.5 662.2 0.82 656.8 659.2 0.37 0.414 21.4 666.6 -0.92 73.3 672.8 667.9 -0.73 0.252 -0.67 62.0 672.7 -0.58 668.2 668.8 0.289 670.9 -0.32 673.1 670.5 -0.39 0.348 48.3 -0.28 673.1 671.2 0.407 34.0 669.2 -0.58 -0.12 29.3 673.1 -0.49 672.3 0.435 669.8 0.357 82.6 698.6 694.2 -0.63 692.3 -0.90 699.6 0.10 689.9 699.4 0.07 0.506 53.7 0.13 699.8 52.3 698.9 699.5 0.09 0.514 701.7 0.45 699.7 0.17 698.5 0.590 39.1 701.9 0.48 698.5 0.680 22.8 697.4 -0.16 0.480 75.1 710.8 709.3 -0.21 707.6 -0.45 0.541 63.1 711.0 710.3 -0.10 709.5 -0.21 0.617 51.2 710.8 711.8 0.14 712.3 0.21 0.733 29.6 710.2 708.9 -0.18 712.5 0.33 0.34 0.45 AAD (%)
SRK"
T,K 663.0 662.0 669.1 670.4 672.6 671.9 672.6 695.2 701.6 701.8 702.7 701.4 710.6 712.1 714.3 712.8
% dev
0.94 0.72 -0.56 -0.34 -0.07 -0.18 -0.07 -0.49 0.39 0.42 0.60 0.41 -0.03 0.16 0.49 0.36 0.39
"Data and results of Lin et al. (1985a). PHC: Perturbed-Hard-Chain equation of state. CCOR Cubic Chain-of-Rotators equation of state. SRK: Soave-Redlich-Kwong equation of state. Dev: relative deviation.
Table VIII. Vapor-Liquid Equilibria for a Coal Liquid (from a Wyoming Coal) exptl data" PHC P, bar wt % vaporized T,K T,K % dev 1.52 4.2 545.7 561.1 2.82 1.27 13.6 545.5 556.5 2.01 3.59 34.6 622.3 627.7 0.87 3.47 38.0 622.3 627.4 0.82 3.05 50.2 621.6 626.7 0.82 2.39 67.2 621.5 623.7 0.35 1.95 81.3 621.9 622.3 0.06 6.62 13.3 652.3 659.5 1.10 5.81 36.0 652.8 661.1 1.27 5.14 44.2 652.8 656.9 0.63 4.45 57.0 652.3 654.7 0.37 3.71 73.6 650.9 653.6 0.41 8.97 22.7 681.7 687.7 0.88 8.10 35.3 681.6 685.5 0.57 6.95 60.3 681.6 687.2 0.82 6.51 67.1 681.4 686.4 0.73 6.08 73.5 680.5 685.4 0.72 10.58 33.6 702.4 706.2 0.54 9.19 58.5 702.6 707.1 0.64 AAD (%) 0.87
CCOR"
T,K
% dev
557.2 552.9 629.4 629.2 628.7 625.9 624.7 663.4 664.4 659.8 657.3 655.9 693.0 690.0 690.2 689.0 687.7 711.6 710.3
2.12 1.38 1.16 1.11 1.14 0.70 0.47 1.71 1.79 1.09 0.78 0.79 1.67 1.23 1.27 1.12 1.07 1.31 1.10 1.20
"Data and results of Lin et al. (1985b). PHC: Perturbed-Hard-Chain equation of state. CCOR: Cubic Chain-of-Rotators equation of state. Dev: relative deviation.
conventional characterization data Can Still characterize ill-defined hydrocarbons well for phase-equilibrium calculations if suitable formulations are used. The cost, time, and effort for measuring structural information may be unnecessary.
Table IX. Universal Constants A,, in Equation of State n
1
1 -7.04677 2 -3.56999
2 -7.226 26 11.35209
m 3 -3.165 39 -10.853 75
4 5 14.34352 -1.262 27 -3.613 10 7.343 34
1064 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
The PHC equation of state with the proposed parameter correlations should be especially useful for continuous thermodynamic calculations where molecular parameters are expressed as continuous functions of conventional characterization data. We are presently studying this extension of the PHC equation of state.
Acknowledgment This work was supported, in part, by the U.S.Department of Energy and the Energy Research Center of the State of West Virginia.
Nomenclature A,, = constants used in eq A3 and A8 a, b = coefficients in eq 5 c = Prigogine's flexibility parameter FA = fractional aromaticity FN = fractional naphthenicity F M = methyl groups per molecule I, I+ = identity variable for continuous mixtures k = Boltzmann's constant k . . ku, kII+ = binary interaction parameter $,,= molecular weight q = external surface area per molecule S, = specific gravity at 60 OF/60 O F T = temperature, K T = reduced temperature T * = characteristic temperature, K Tb = normal boiling point, K u = molar volume, cm3/mol = reduced volume u*, uo* = molar hard-core volume, cm3/mol z = mole fraction 2 = compressibility factor
where u is the volume per mole and u* is the molar hardcore volume. Based on molecular-dynamic results of Alder et al. (1972), the 10 coefficients (Anm)(listed in Table IX) in eq A3 are estimated by Gmehling et al. (1979). The reduced volume (8) and reduced temperature (T)are given as ij = u / u * (A51 = T/T* = T / [ ( q ~ / k ) / c ]
(A6)
The characteristic temperature, T *, is the ratio of qE/k and c. The effective potential energy ( q t l k ) is defined in terms of q , the external surface area per molecule, and E, the interaction energy per segment; k is Boltzmann's constant. The parameter E / k is determined from the slope of q e l k vs carbon number for hydrocarbons and has the value 105 K. This value is arbitrary because Elk and q always appear as a product. Mixtures. Fluid-mixture properties can be calculated using the purecomponent parameters and only one binary interaction parameter. For a mixture, eq A2 and A3 can be rewritten:
2
mA,,(cT*)(T*)(")
6
,?,(attractive) = C C
T ,ijm
n=l m=l
tA8)
where
Greek Symbols
The brackets ( ) denote a mixture property. By use of the averaging procedures discussed by Donohue and Prausnitz (1978), the mixture properties are
a = coefficient in eq 2 = reduced density 6 = interaction energy per segment 8 = physical property or molecular parameter 8 = coefficient in eq 1
(c) =
Subscripts A = n-alkane C = continuous homologue D = discrete component
CZiCi i
( u * ) = CZiUi* i
(All) (A121
Appendix A. The Perturbed-Hard-Chain Equation of State Pure Substances. Like other van der Waals type equations, the PHC equation of state also takes into account repulsive and attractive interactions: Z = Z(repu1sive) + Z(attractive) (AI) where Z(repu1sive) = 1 + c
[:i 1
(A21
where the subscripts represent components i and j, respectively, and zi is the mole fraction. In eq A13 and A15, the characteristic potential energy between a segment of molecule i and a segment of molecule j is
and where k , is the adjustable interaction parameter. The Prigogine's flexibility parameter ( c ) is to account for the effect of density on rotational and vibrational degrees of freedom. The reduced density is defined by
Appendix B. Correlation of Perturbed-Hard-Chain Equation of State Parameters with Molecular Structure (Wilhelm and Prausnitz, 1985) An n-a.lkane molecule of a given molecular weight is used in the following correlations as a reference. The equa-
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1065
tion-of-state parameters are correlated with the following structural characterization factors: the number of methyl groups per molecule, FM; the fractional aromaticity, FA; and the fractional naphthenicity, FN. By use of the molecular weight as a measure of the molecule’s size, the parameters are estimated f r o m u* (cm3/mol) = u*(n-alkane)
the identity variable could be Mw or Tb,but all three physical properties (Mw, Tb,and S ) are required to evaluate parameters, the integrals in tke mixing rules are essentially line integrals. The interactive terms in the above equations are similar t o eq A16 and are given as
+ Au*(aromatic) + Au*(naphthene)
v*(n-alkane) = 6.209
+ 0.692Mw
- 0.2256Mw)FA = (-2.97 - 0.0770Mw)FN
Au*(aromatic) = (4.961 Au*(naphthene)
(Bl)
q e / k (K)=
+ Aqe/k(aromatic) + Aqc/k(CHJ qe/k(n-alkane) = -5.01 + 16.17Mw0~82w
qe/k(n-alkane)
- 0.1427Mw)FA - FA - F N ) - F M ]
Aqc/k(aromatic) = (51.89
= 45.65[2(1
Aqe/k(CHJ
(B2)
+ Aching) + Ac(CH,) c(n-alkane) = 0.6520 + 0.0355Mw08138 Ac(aromatic) = (0.5169 - 0.00847Mw)(FA + F N ) c = c(n-alkane)
Ac(CH3) = 0.108[2(1
- FA - F N ) - FM]
(B3)
Appendix C. Perturbed-Hard-Chain Equation of State for Semicontinuous Mixtures For a semicontinuous mixture with ND discrete components and one continuous homologue, the composition averages in eq All-A15 are rewritten as
+ ~ ~ J F u )c
( c ) = cZD,CD,
c ~U )
(CI)
1
( u * > = CZD,uD,*
+ z C l F ( I ) uC*(I) dl
(c2)
I
IC1 ~1z D , z D j ~ D , [ ~ ( i , j ) / k l ~ D+j *CZDFCqD, JFU) [ E ( i ~ ) / kuC*U) ] dl + CZD,ZD~D,* x I
(cT*) =
JFU) [ e ( j J ) / k l q c ( I ) dl + zc2J JFU) q c ( I ) [E(IJ+)/kI %*(I+)
m+)x
dl d l + ) / ( U * )
(C3)
(T*)(’) =1
L
(T*)(’) =
(C4)
CzD,zD,sD,uD,*qD,[E(i,j)/kl/cD, + I
CzDFC(qD,2/CD,)$F(I)
[ e ( i J ) / k ]uC*(I)
+
1
c
i
~
D
~
CJFU) ~ D ~
[ e ( j ~ ) / k /l c c ( ~ )dl +
z c 2 1I F ( I ) FU+)q c ( I ) 2 u c * ( I + ) [ e U J + ) / k l /
where Z is an identity variable used to characterize chemical species in the continuous homologue and F ( I ) is the molar d i s t r i b u t i o n f u n c t i o n of the continuous part. Parameters of continuous species, c ( I ) , u*(I),and T * ( I ) ,are required in the above equations. These parameters can be evaluated b y the proposed correlation (eq 1-5). Since
where k ( i , I ) and k ( I J + ) are continuous functions of the identity variable. The continuous interaction parameters could be set equal to zero for some cases but could be significant for others. Registry No. Methane, 74-82-8; ethane, 74-84-0; propane, 74-98-6; butane, 106-97-8; pentane, 109-66-0; hexane, 110-54-3; heptane, 142-82-5; octane, 111-65-9; nonane, 111-84-2; decane, 124-18-5; dodecane, 112-40-3; hexadecane, 544-76-3; eicosane, 112-95-8; triacontane, 638-68-6; tetracontane, 4181-95-7; isobutane, 75-28-5; isopentane, 78-78-4; 2-methylhexane, 591-76-4; 3methylhexane, 589-34-4; 2-methylheptane, 592-27-8; 2,2,4-trimethylpentane, 109-66-0;2,2,5-trimethylhexane, 110-54-3;squalane, 111-01-3; cyclopentane, 287-92-3; cyclohexane, 110-82-7; bicyclohexyl, 92-51-3; methylcyclohexane, 108-87-2; ethylcyclohexane, 1678-91-7; butylcyclohexane, 167893-9;hexylcyclohexane, 4292-75-5; decylcyclohexane, 1795-16-0; hexadecylcyclohexane, 6812-38-0; indane, 496-11-7; cis-decalin, 493-01-6; tram-decalin, 493-02-7; tetralin, 119-64-2; benzene, 71-43-2; naphthalene, 91-20-3; diphenyl, 92-52-4; phenanthrene, 85-01-8; anthracene, 120-12-7; toluene, 10&88-3; ethylbenzene, 100-41-4;butylbenzene, 104-51-8; hexylbenzene, 1077-16-3; decylbenzene, 104-72-3; hexadecylbenzene, 1459-09-2; a-xylene, 95-47-6; m-xylene, 108-38-3; p-xylene, 106-42-3; tert-butylbenzene, 98-06-6; 1-methylnaphthalene, 9012-0; 1-ethylnaphthalene, 1127-76-0; 1,2-dimethylnaphthalene, 573-98-8; 1-propylnaphthalene, 2765-18-6; 1-butylnaphthalene, 1634-09-9; diphenylmethane, 101-81-5; isopropyldiphenyl, 25640-78-2; ditoluenemethane, 1335-47-3; dicumenemethane, 25566-92-1; tetraisopropylphenylmethane, 108-88-3.
Literature Cited Alder, B. J.; Young, D. A,; Mark, M. A. J. Chem. Phys. 1972,56, 3013-3329. Alexander, G. L. Ph.D. Dissertation, University of California, Berkeley, 1985. Alexander, G. L.; Schwarz, B.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1985,24,311-315. API Technical Data Book American Petroleum Institute: Washington, D.C., 1983; “Petroleum Refining”. Vol. 1. Barker, J. A.; Henderson, D. Ann. Rev. Phys. Chem. 1972, 23, 439-484. Beret, S.;Prausnitz, J. M. MChE J. 1975,21,1123-1132. Donohue, M. D.; Prausnitz, J. M. AZChE J. 1978,24,849-860. 1970,49,7-29. Flory, P. J. Discuss. Faraday SOC. Gmehling, J.; Liu, D. D.; Prausnitz, J. M. Chem. Eng. Sci. 1979,34, 951-958. Lin, H. M.; Kim, H.; Guo, T.; Chao, K. C. Ind. Eng. Chem. Process Des. Dev. 1985a,24, 1049-1055. Lin, H. M.; Leet, W. A,; Kim, H.; Chao, K. C. Ind. Eng. Chem. Process Des. Deu. 1985b,24,1225-1230. Prigogine, I. The Molecular Theory of Solutions; North-Holland: Amsterdam, 1957; p 327. Sim, W. J.; Daubert, T. E. Ind. Eng. Chem. Process Des. Dev. 1980, 19,386-393. Vilmachand, P.; Donohue, M. D. Ind. Eng. Chem. Fundam. 1985,24, 246-257. Wilhelm, A,; Prausnitz, J. M.Fuel 1985,64,501-508. Received for review April 20, 1987 Revised manuscript received December 1, 1987 Accepted February 16, 1988
