Characterization of Heavy Oils and Bitumens 2. Improving the

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Characterization of Heavy Oils and Bitumens 2. Improving the Prediction of Vapor Pressures for Heavy Hydrocarbons at Low Reduced Temperatures Using the Peng-Robinson Equation of State G. N. Nji, W. Y. Svrcek, H. Yarranton, and M. A. Satyro* Department of Chemical and Petroleum Engineering, UniVersity of Calgary, Alberta T2N 1N4, Canada ReceiVed August 21, 2008. ReVised Manuscript ReceiVed October 15, 2008

An enhanced form of the temperature-dependent attractive pressure term of the Peng-Robinson equation of state (PR EOS) was developed. The function was developed using experimental vapor pressures from the NIST Standard Reference Data Base #103 as well as vapor pressures estimated based on the Nji et al. (2008) vapor pressure prediction correlation. 6000 vapor pressure data points from 237 diverse hydrocarbons were used in this study to develop the new attractive term. The quality of predicted vapor pressures using the modified PR EOS for heavy hydrocarbons showed a significant improvement when compared against the standard equation of state. Between reduced temperatures of 0.3 and 0.8, the average and maximum absolute percentage deviations for the predicted vapor pressures for the modified PR EOS are 4.9% and 78.0%, respectively, as compared to 10.4% and 230% for the standard PR EOS. For hydrocarbons from methane to tetralin corresponding to C1-C10, the average and maximum absolute percentage deviations in the vapor pressures using the standard Peng-Robinson are 5.7% and 70.7%, respectively. Using the modified PR EOS, the average and maximum percentage deviations in the calculated vapor pressures are 4.6% and 35.6%, respectively. This enhanced attractive pressure term combined with the critical property estimation method presented by Nji et al. (2008) provides a simple and self-consistent method for the prediction of thermodynamic properties of heavy hydrocarbons.

1. Introduction Cubic equations of state (CEOS’s) provide a simple, accurate, and flexible platform for the calculation of thermodynamic equilibrium for hydrocarbon systems. The accuracy of these models for phase equilibrium calculations depends on the availability and quality of experimental vapor pressures that are used to tune the model parameters. Typically, the attractive term of the CEOS is modified with an expression that is a function of the acentric factor. This modification has been based largely on data from relatively light hydrocarbons. To extend these models to heavier hydrocarbons, the relatively small amount of experimental vapor pressure data for heavy hydrocarbons must be carefully screened and used as a consistent basis for estimation of vapor pressure values for high boiling point materials (Nji et al., 2008). These data, either from experiment or from correlation, must then be incorporated into a CEOS framework to allow easy use in existing process or reservoir simulation programs. CEOS predictive capability for compounds with hypothetical normal boiling points up to approximately 1200 °C is required, and this corresponds to conditions far removed from the data used for the original tuning and generalization of traditional models such as the Peng-Robinson1 and Soave-Redlich-Kwong.2 Note that although actual industrial operating temperatures for production and processing will not be that high, reasonably accurate vapor boiling point predictions for vacuum operations are required, and the hypothetical components used to represent heavy oils or * To whom correspondence should be addressed. E-mail: [email protected]. (1) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59–64. (2) Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197–1203.

Figure 1. Comparison of predicted vapor pressures for 2-methylpyrene, chrysene, and 1-phenylheptadecane with experimental data from NIST.3

bitumens will have estimated normal boiling points that may very well approach temperatures above 1000 °C. From another point of view, when processing heavy hydrocarbon materials, it is important to be able to estimate vapor pressures of oil fractions at temperatures significantly lower than the boiling point. The original Peng-Robinson and Soave parametrizations were performed using vapor pressure data at or above the normal boiling point, and extrapolations are usually poor. For instance, Soave2 initially used only the critical point and the calculated vapor pressure at a reduced temperature of 0.7. Peng and Robinson1 followed a similar procedure using vapor pressure data from the normal boiling point to the critical point. In Figure 1, the predicted vapor pressures for some compounds for the PR EOS are compared to experimental data from the NIST Standard Reference Database #103.3 The original PR EOS overpredicts vapor pressures, and the magnitude of this overprediction increases with decreasing reduced temperatures.

10.1021/ef8006855 CCC: $40.75  2009 American Chemical Society Published on Web 12/16/2008

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Similar results have been observed for other heavier hydrocarbons, demonstrating the limit of the original PR EOS when applied to heavy hydrocarbon VLE predictions. The objective of this study is to use a broader vapor pressure database to develop a more accurate correlation for the prediction of heavy hydrocarbon vapor pressures using a CEOS. The NIST database3 provides the necessary database, and all experimental data are from this source and are referred to as NIST data. 2. CEOS Model Framework For modeling purposes, the PR EOS structure was selected because it is commonly used for simulating hydrocarbon phase behavior. Other CEOS’s formulations such as the Soave-Redlich-Kwong will also provide good platforms for equilibrium calculations as long as the details related to vapor pressure and density are addressed; that is, many other CEOS’s could be proposed with similar final results.4 The PR EOS is defined by eqs 1-4. RT a V - b V2 + 2bV - b2

(1)

a(T) ) a(Tc, Pc) · R(Tr, ω)

(2)

a(Tc, Pc) )

0.457235R2T2c Pc

(3)

b(Tc, Pc) )

0.0777969RTc Pc

(4)

P)

where R(Tr,ω) is the alpha function, an empirical function used to fit vapor pressure data that is discussed below. All CEOS’s must be tuned to match vapor pressure data if quantitative results are desired. Formally, the fugacity of the vapor and liquid phases must be equal for two phases in equilibrium at the saturation temperature and pressure. In practice, this equality is usually obtained by adjusting the alpha function in the attractive term of the CEOS. Peng and Robinson1 and Soave2 found that, for nonpolar or slightly polar species, the alpha function can be written as: R(T) ) [1 + fω(1 - √Tr)]2

(5)

The PR EOS parametric function fω is then defined as follows: fω ) 0.37464 + 1.54226ω - 0.26992ω2

(6)

where

()

ω ) -log

Pv Pc

T )0.7 Tc

- 1.000

(7)

Equations 5 and 6 are routinely used for vapor-liquid equilibrium calculations in the hydrocarbon processing industries because they provide accurate vapor pressures and equilibrium ratios and somewhat improved liquid density values when compared to the Soave-Redlich-Kwong formulation. Different functional forms of eq 5 have been proposed, and in some cases eq 6 has been abandoned in exchange for empirical parameters that are characteristic to each pure compound.5 (3) NIST ThermoData Engine. Standard Reference Database #103, version 2.0; National Institute of Standards and Technology, Gaithersburg, MD, 2005; http://www.nist.gov. (4) Soave, G. Improvement of the van der Waals Equation of State. Chem. Eng. Sci. 1984, 39, 357–369.

Soave6 proposed a new temperature dependence of the attractive parameter of the SRK EOS to provide a more reliable extrapolation for heavy hydrocarbon vapor pressures below their normal boiling point. He developed a correlation based on “fictitious vapor pressures” generated by the Lee-Kesler equation.7 However, heavy hydrocarbon vapor pressures predicted by the Lee-Kesler equation7 are known to be unreliable.8 Peng and Robinson9 also proposed a modification to their alpha function to improve the performance of their equation for heavier compounds with acentric factors greater than 0.49. Changing the vapor pressure expression changes other derived properties as well such as the specific heat (enthalpy). The binary interaction parameters fit to equilibrium data would also have to be recorrelated/estimated. Although these two functional forms would work relatively well in their respective regions, problems or discontinuity can always be encountered between these two regions. Therefore, the need for a continuous function, applicable over the whole range, is desirable. Because the generalized correlations are intrinsically dependent on the original database used in their development, the functional forms of Soave2,6 and Peng and Robinson1,9 alpha functions resulted in large vapor pressure deviations when used to model heavy oil and bitumen systems. In this study, the alpha function is retuned on the basis of experimental vapor pressures from NIST3 as well as vapor pressures generated from the Nji et al.8 vapor pressure prediction correlation when experimental data are not available. The form of eq 5 has been maintained, and it was assumed that the modeling of heavy hydrocarbons can be improved by modifying the correlating function defined by eq 6. 3. The Data Set A total of 237 diverse hydrocarbons were used in this study. The selected data include 26 paraffins, 62 iso-paraffins, 44 naphthenes, 26 alkenes, and 79 aromatics. Of these 237 hydrocarbons, 82 components either had no experimental vapor pressures or critical data, or experimental vapor pressure data but no experimental critical data. The physical properties of these components were predicted from Nji et al.8 correlation. This method predicts the boiling point temperatures at different pressures as well as critical constants by applying a perturbation theory using n-paraffins as a reference system and modeled using a function of only the molecular weight and specific gravity at 15.6 °C. To determine the vapor pressure at any temperature, the constants of the Riedel10 vapor pressure correlation are then determined by regressing the predicted boiling points and critical constants. The rest of the data were obtained from the NIST Standard Reference database. It was also necessary to calculate acentric factors for compounds with incomplete data. Predictive methods for the (5) Sandler, S. I.; Orbey, H.; Lee, B.-I. In Equations of State In Models for Thermodynamic and Phase Equilibria Calculations, Sandler, S. I., Ed.; Marcel Dekker, Inc.: New York, 1994; pp 87-186. (6) Soave, G. Improving the Treatment of Heavy Hydrocarbons by the SRK EOS. Fluid Phase Equilib. 1993, 84, 339. (7) Lee, B. I.; Kesler, M. G. A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States. AIChE J. 1975, 22, 510– 527. (8) Nji, G. N.; Svrcek, W. Y.; Yarranton, H. W.; Satyro, M. A. Characterization of Heavy Oils and Bitumens 1. Vapor Pressure and Critical Constant Prediction Method for Heavy Hydrocarbons. Energy Fuels 2008, 22, 455–462. (9) Robinson, D. B.; Peng, D. Y. The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs, Research Report 28, Gas Producers Association (GPA), Tulsa, 1978. (10) Riedel, L. Chem.-Ing.-Tech. 1954, 26, 679.

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Figure 2. Relationship between the acentric factor at Tr ) 0.78 and the molecular weight.

Figure 3. Dispersion plot of the predicted acentric factors (Nji et al. and Riazi-Sahhaf) as compared to measured acentric factors (NIST). Table 2. Numeric Constants for Equation 10

Table 1. Numeric Constants for Equation 8 numerical constant

value

numerical constant

value

a b c d e

-2.867 5.822 0.01418 -15.47 4.166

a b c d e

0.8467 0.3069 0.2557 0.0015 0.3128

acentric factor are summarized by Riazi.12 The accuracy of the acentric factors by these methods depends on the accuracy of the critical constants and vapor pressure values. Also, previous attempts to correlate the acentric factor with parameters such as the normal boiling point and specific gravity have failed.12 The standard method for estimating acentric factors for undefined hydrocarbons or petroleum fractions for refining is the Lee-Kesler equation based on their proposed correlation for vapor pressure.7 Also, for hydrocarbons with no vapor pressure data, the Riazi-Sahhaf equation (eq 13) is used. Riazi and Sahhaf developed the correlations for different homologous series of hydrocarbons as follows: n-alkanes, n-alkylcyclopentanes, n-alkylcyclohexanes, and n-alkylbenzenes, and for components from C5 to C20. This equation depends only on molecular weight, implying that the estimated acentric factors are the same for components with the same molecular weight. We have previously shown that the molecular weight alone is not sufficient to properly characterize an undefined petroleum fraction.8 Here, the same perturbation procedure will be used as in ref 8. The acentric factors of the n-paraffins were calculated from their vapor pressures provided in ref 3 using eq 7 and can be correlated as a function of their molecular weight: a

ω)

+e (8) 1 [1 + exp(b - cMW)] ⁄d The constants of eq 8 are provided in Table 1. Equation 8 and Figure 2 show that the limiting value of the acentric factor for normal paraffins as the molecular weight goes toward infinity is 1.3. The following perturbation equation is proposed to predict the acentric factor of any other hydrocarbon:

[ (1(1 -+ 2f)2f) ]

ω ) ω◦

2

(9)

where ω° is the acentric factor of the n-paraffins, and ω is the acentric factor of a hydrocarbon of interest. The form of the (11) Soave, G. Direct Calculation of Pure-compound Vapor Pressures through Cubic Equations of State. Fluid Phase Equilib. 1986, 31, 203–207.

perturbation function, f, is given by: f ) a∆SG2 + b∆SG + c∆MW2 + d∆MW + e∆SG∆MW (10) where ∆SG ) ln

SG◦ SG

(11)

MW◦ (12) MW The constants for eq 10 are provided in Table 2. In Figure 3, the acentric factors predicted by methods of Riazi-Sahhaf and Nji et al. are compared. The equations for the acentric factor developed in this study are within a % AAD of 11.58 and a % MAD of 57.26, as compared to 18.6 and 59.47 for the Riazi-Sahhaf correlation, respectively. Because the method here is generalized and consistent with the vapor pressure prediction method by Nji et al., it was used to generate acentric factors. ∆MW ) ln

4. Development of a Generalized Alpha Function The following two parametric forms of fω were developed, and the predicted vapor pressures were compared to heavy hydrocarbon vapor pressure data: (1) an alpha function based on the acentric factor calculated using Pitzer’s definition at Tr ) 0.7; and (2) an alpha function based on a modified acentric factor calculated at Tr ) 0.5, which is closer to the melting point of heavy hydrocarbons and the corresponding low vapor pressures of interest in the processing of heavy oils and bitumens. The first step in developing the alpha functions defined in 1 and 2 is to determine a value of fω from the data or quasiexperimental data. Equation 5 can be linearized as follows:

√R - 1 ) fwτ

(13)

τ ) (1 - √Tr)

(14)

where

Characterization of HeaVy Oils and Bitumens 2

Figure 4. A plot of
R - 1 versus (1 -
Tr) to obtain fω is equal to 1.394 for 2-butyl-3-hexylnaphthalene.

Figure 5. Relationship between characteristic constants and acentric factor at Tr ) 0.7. Closed symbols are NIST data; open symbols were obtained from Nji et al. method.

Energy & Fuels, Vol. 23, 2009 369

Figure 6. Relationships between characteristic constants and acentric factor at Tr ) 0.5. Closed symbols are NIST data; open symbols were obtained from Nji et al. method.

Figure 7. A comparison of the characteristic constants of both the original PR and the new PR function based on ω at Tr ) 0.7. Table 3. Comparison of Vapor Pressure Predictions characteristic constant

Peng and Robinson1,9 iteratively determined a characteristic constant, fω, for each compound, while Soave11 developed a noniterative procedure to determine this constant from the vapor pressure. Here, using Soave’s approach,11 R at each vapor pressure point was calculated for each compound as a function of τ. Equation 13 was then used to calculate fω as shown for 2-butyl-3-hexylnaphthalene in Figure 4. The data used to determine fω for each hydrocarbon included the vapor pressures from NIST as well as vapor pressures computed using the Nji et al.8 correlation as previously mentioned. The characteristic constants were then correlated against the acentric factors and the modified acentric factors. 4.1. Alpha Function Based on the Acentric Factor Calculated at Tr ) 0.7. Acentric factors defined at Tr ) 0.7 are normally used in the parameter correlation for CEOS’s. The acentric factor at Tr ) 0.7 was determined using eq 7 with the NIST vapor pressures and critical constants, and, for some compounds, the Nji et al.8 correlations were used. The fω values calculated for all components were correlated to the acentric factor at Tr ) 0.7, and the following cubic polynomial was used to fit the data:

% AAD

% MAD

% BIAS

10.38 4.88 4.38

230.13 77.51 29.51

10.79 -2.02 -3.02

5.70 4.60

70.67 35.62

7.34 -2.44

Entire Database fω0 fω1 fω2 C1-C10 Hydrocarbons fω0 fω1

4.2. Alpha Function Based on the Acentric Factor Calculated at Tr ) 0.5. The reduced temperature used to define the acentric factor was changed from 0.7 to 0.5 in an attempt to investigate the possibility of a better correlation using a parameter defined at reduced temperatures closer to the actual processing temperatures to be encountered when processing heavy fractions. The modified acentric factor defined at a reduced temperature of 0.5 is calculated as follows:

()


) -log

Pv Pc

T )0.5 Tc

- 1.000

(16)

fω1 ) 0.0617513ω3 - 0.29014ω2 + 1.56132ω + 0.388187 (15)

The same database and procedure described in section 4.1 were used, and the fω was recalculated as shown in eq 17:

Figure 5 shows the functional form of the characteristic constants correlated against the acentric factor at Tr ) 0.7.

fω2 ) 0.0018935
3 - 0.041109
2 + 0.62674
- 0.36423 (17)

(12) Riazi, M. R. Characterization and Properties of Petroleum Fractions, 1st ed.; ASTM Manual Series; American Society of Testing and Materials: Philadelphia, PA, 2005; eq 2-42, p 50.

Figure 6 shows the functional form of the characteristic constants correlated against the acentric factor at Tr ) 0.5.

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Table 4. Values of the Characteristic Constants “fω” and the Acentric Factors used in the Correlations compound

fω (NIST)

ω




compound

fω (NIST)

ω




methane ethane ethene propane propene n-butane 2-methylpropane 1-butene n-pentane 2-methylbutane 2,2-dimethylpropane 1-pentene cis-2-pentene trans-2-pentene 2-methyl-1-butene 2-methyl-2-butene cyclopentane n-hexane 2-methylpentane 3-methylpentane 2,3-dimethylbutane 2,2-dimethylbutane 1-hexene cis-3-hexene cyclohexene benzene methylcyclopentane cyclohexane n-heptane 2-methylhexane 3-methylhexane 2,2,3-trimethylbutane 2,2-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 3-ethylpentane 2,3-dimethylpentane 1-heptene toluene ethylcyclopentane 1,1-dimethylcyclopentane trans-1,2-dimethylcyclopentane trans-1,3-dimethylcyclopentane methylcyclohexane cycloheptane n-octane 2,2,3,3-tetramethylbutane 2,2,4-trimethylpentane 2,3,3-trimethylpentane 3,3-dimethylhexane 2,2,3-trimethylpentane 2,3,4-trimethylpentane 3,4-dimethylhexane 2,3-dimethylhexane 2,4-dimethylhexane 4-methylheptane 3-methylheptane 2,2-dimethylhexane 2,5-dimethylhexane 2-methylheptane 2-methyl-3-ethylpentane 3-ethylhexane 3-methyl-3-ethylpentane 1-octene trans-2-octene o-xylene p-xylene m-xylene ethylcyclohexane n-propylcyclopentane cyclooctane ethylbenzene n-nonane 3,3-diethylpentane 4-methyloctane 2,3,4-trimethylhexane 3,4-dimethylheptane 1-nonene 1,3,5-trimethylcylcohexane

0.3953 0.5391 0.5266 0.6241 0.6169 0.6838 0.6715 0.6833 0.7639 0.7330 0.6715 0.7475 0.7711 0.7581 0.7382 0.7955 0.6840 0.8290 0.8098 0.8014 0.7719 0.7337 0.8183 0.8128 0.7235 0.7100 0.7494 0.6988 0.8942 0.8799 0.8703 0.7492 0.8221 0.8444 0.7928 0.8504 0.8310 0.8883 0.7797 0.8011 0.7408 0.7596 0.7790 0.7571 0.7418 0.9622 0.5746 0.8348 0.8208 0.8666 0.8329 0.8564 0.8808 0.8949 0.8953 0.9377 0.9375 0.8949 0.9116 0.9487 0.8800 0.9145 0.8283 0.9588 0.9652 0.8378 0.8448 0.8571 0.8166 0.8900 0.7742 0.8426 1.0252 0.8885 0.9102 0.9255 0.9234 1.0029 0.8485

0.0109 0.0992 0.0866 0.1525 0.1425 0.1997 0.1851 0.1919 0.2495 0.2237 0.1968 0.2365 0.2595 0.2487 0.2333 0.2816 0.1954 0.3004 0.2774 0.2722 0.2463 0.2331 0.2930 0.2829 0.2224 0.2103 0.2313 0.2099 0.3495 0.3303 0.3232 0.2476 0.2865 0.3020 0.2658 0.3119 0.2942 0.3466 0.2594 0.2711 0.2381 0.2435 0.2571 0.2366 0.2399 0.3975 0.1297 0.3041 0.2905 0.3214 0.2984 0.3162 0.3391 0.3470 0.3443 0.3717 0.3710 0.3390 0.3568 0.3809 0.3310 0.3618 0.2988 0.4089 0.3966 0.3123 0.3232 0.3276 0.2907 0.3388 0.2645 0.3025 0.4401 0.3460 0.3523 0.3686 0.3672 0.4229 0.2802

1.3521 1.6032 1.5857 1.7631 1.7408 1.8959 1.8204 1.8798 2.0427 1.9718 1.9039 1.9992 2.0558 2.0401 1.9970 2.1139 1.8934 2.1901 2.1309 2.1119 2.0445 1.9963 2.1538 2.1410 1.9595 1.9561 2.0002 1.9547 2.3374 2.2787 2.2573 2.0456 2.1569 2.2031 2.0922 2.2268 2.1754 2.2986 2.0762 2.1101 2.0066 2.0233 2.0639 2.0150 2.0283 2.4810 1.6841 2.1979 2.1573 2.2516 2.1886 2.2350 2.3050 2.3294 2.3304 2.4040 2.4024 2.3120 2.3605 2.4263 2.2818 2.3739 2.1867 2.4543 2.4758 2.2151 2.2384 2.2541 2.1541 2.3049 2.0963 2.2020 2.6107 2.3268 2.3435 2.3933 2.3882 2.5544 2.1222

isopropylcyclohexane n-propylcyclohexane 1,2,4-trimethylbenzene 1,3,5-trimethylbenzene 1,2,3-trimethylbenzene n-propylbenzene isopropylbenzene 1-ethyl-2-methylbenzene 1-ethyl-3-methylbenzene 1-methyl-4-ethylbenzene indane isopropenylbenzene indene n-decane 2,7-dimethyloctane 3,3,5-trimethylheptane 2,2,3,3-tetramethylhexane 2,2,5,5-tetramethylhexane 1-decene butylcyclohexane 1-methyl-4-isopropylcyclohexane isobutylcyclohexane sec-butylcyclohexane tert-butylcyclohexane naphthalene trans-decalin 1-methyl-2-isopropylbenzene 1-methyl-3-isopropylbenzene 1,4-dimethyl-2-ethylbenzene 1,3-dimethyl-2-ethylbenzene 1,4-diethylbenzene 1,2,3,5-tetramethylbenzene 1,2,4,5-tetramethylbenzene butylbenzene isobutylbenzene tert-butylbenzene sec-butylbenzene tetralin n-undecane 3-methyldecane 3-ethylnonane 2,6-dimethylnonane 1-undecene n-pentylbenzene 1-methylnaphthalene 2-methylnaphthalene pentamethylbenzene n-dodecane 5-methylundecane 4-methylundecane 3-methylundecane 2-methylundecane 5-ethyldecane 1-butyl-3-ethylcyclohexane 1-dodecene 1,2-dimethylnaphthalene acenaphthene biphenyl phenylcyclohexane acenaphthylene n-tridecane 5-methyldodecane 4-methyldodecane 3-methyldodecane 2-methyldodecane heptylbenzene 1-isopropylnaphthalene diphenylmethane fluorene 1-tridecene 1,1-diethyl-2-propylcyclohexane n-tetradecane 2,11-dimethyldodecane 1-tetradecene anthracene phenanthrene n-pentadecane nonylbenzene 1-pentadecene

0.8537 0.8759 0.9184 0.9409 0.9102 0.9024 0.8717 0.8895 0.8746 0.8977 0.8423 0.8852 0.8080 1.0810 1.0490 0.9637 0.9075 0.9223 1.0833 0.9387 0.9910 0.9189 0.9770 0.8646 0.8182 0.8214 0.8660 0.7921 0.8557 0.8834 0.9636 0.9712 0.9908 0.9585 0.8932 0.9059 0.9094 0.9249 1.1470 1.1088 1.4522 0.9723 1.1187 1.0371 0.8987 0.8966 1.0329 1.1961 1.1440 1.1658 1.1745 1.1794 1.0948 1.1308 1.1814 1.0689 1.0309 0.9633 0.9340 1.1066 1.2454 1.2169 1.1397 1.2368 1.2427 1.1190 1.0391 1.0138 1.0314 1.2393 1.1392 1.2991 1.2897 1.2801 0.9660 0.9934 1.3316 1.2796 1.3458

0.3206 0.3251 0.3729 0.3916 0.3667 0.3454 0.3229 0.3394 0.3278 0.3563 0.3101 0.3411 0.2918 0.4847 0.4681 0.3964 0.3583 0.3721 0.4797 0.3685 0.4594 0.3591 0.4087 0.3236 0.3139 0.3019 0.3048 0.3168 0.3036 0.3338 0.4049 0.3939 0.4259 0.3932 0.3577 0.3476 0.3554 0.3363 0.5442 0.5027 0.7722 0.3994 0.5110 0.4433 0.3548 0.3704 0.4609 0.5721 0.5283 0.5394 0.5487 0.5548 0.4892 0.5480 0.5577 0.4800 0.4403 0.3981 0.3408 0.5476 0.6131 0.5836 0.5245 0.6010 0.6068 0.4987 0.4476 0.4327 0.4570 0.6090 0.5509 0.6462 0.6460 0.6346 0.4095 0.4190 0.6790 0.6349 0.7105

2.2388 2.2791 2.3816 2.4325 2.3626 2.3170 2.2752 2.3063 2.2719 2.3413 2.2085 2.3129 2.1518 2.7464 2.6656 2.4728 2.3583 2.4052 2.7191 2.4171 2.5312 2.3683 2.5072 2.2648 2.2018 2.1762 2.2076 2.2757 2.2623 2.3236 2.4816 2.5064 2.5499 2.4531 2.3584 2.3437 2.3533 2.2788 2.8900 2.7829 3.6068 2.4844 2.8203 2.6270 2.3390 2.3394 2.6304 3.0109 2.8652 2.9164 2.9369 2.9523 2.7516 2.8138 2.9630 2.7050 2.6529 2.4998 2.4296 2.7470 3.1234 3.0344 2.8573 3.0855 3.1035 2.8101 2.6507 2.5868 2.6384 3.0970 2.8409 3.2492 3.2201 3.1999 2.5144 2.5577 3.3350 3.1896 3.3596

Characterization of HeaVy Oils and Bitumens 2

Energy & Fuels, Vol. 23, 2009 371 Table 4. Continued

compound

fω (NIST)

ω




compound

fω (NIST)

ω




n-hexadecane 1-hexadecene pyrene fluoranthene n-heptadecane 1-heptadecene undecylbenzene 2-methylpyrene 1,2-benzofluorene n-octadecane 1-octadecene chrysene naphthacene triphenylene m-terphenyl 1,6-diphenylhexane 2,2-bis(4-methylphenyl)butane 1-phenyldodecane n-nonadecane 3-methyloctadecane 7-hexyltridecane 2-methyloctadecane 1-nonadecene tetradecylcyclopentane 1-dodecyl-3-methylcyclohexane tridecylcyclohexane 1,1-dicyclohexylheptane tricyclohexylmethane tridecylbenzene icosane 2-butyl-3-hexyldecalin 7-butyl-1-hexyldecalin tetradecylbenzene 2-butyl-3-hexylnaphthalene 1,4-dimethyl-5-octylnaphthalene triphenylethylene n-heneicosane 8-hexylpentadecane

1.3865 1.3833 1.0685 1.0967 1.4466 1.4221 1.3843 2.2009 1.9247 1.4989 1.4235 2.2301 2.1383 2.1849 1.2475 1.4658 1.4550 1.4983 1.5752 1.4817 1.4057 1.4778 1.5810 1.2374 1.6890 1.5647 1.2606 1.0699 1.4332 1.5882 1.3251 1.3251 1.6216 1.3941 1.4951 1.3893 1.6418 1.4520

0.7055 0.6666 0.5074 0.5262 0.7558 0.7304 0.7606 1.4228 1.1752 0.8043 0.7522 1.4511 1.3587 1.4045 0.6185 0.7852 0.7734 0.8649 0.8455 0.7957 0.6914 0.7921 0.8769 0.6096 0.9703 0.8604 0.6324 0.4244 0.7562 0.8756 0.6770 0.6770 0.8759 0.7199 0.8333 0.7562 0.9273 0.7430

3.4650 3.4715 2.7024 2.7634 3.5956 3.5673 3.4366 5.6295 4.8519 3.7352 3.5566 5.7136 5.4556 5.5746 3.1303 3.6354 3.6064 3.7064 3.9558 3.6750 3.5122 3.6684 3.9263 3.0911 4.2150 3.8866 3.1952 2.7431 3.5541 3.9573 3.2927 3.2927 4.0334 3.4919 3.7051 3.4421 4.0848 3.6139

3-methylicosane 1-heneicosene pentadecylcyclohexane hexadecylcyclopentane n-docosane 3-methylheneicosane hexadecylcyclohexane 1,5-dicyclopentyl-3-(2-cyclopentylethyl)pentane 1,1-bis(decahydro-1-naphthyl)ethane 1,5-dicyclopentyl-3-(2-cyclopentylethyl)-2-pentene 1,2-bis(decahydro-1-naphthyl)ethane 1-phenylhexadecane 1,10-diphenyldecane n-tricosane 9-hexylheptadecane 9-cyclohexylheptadecane 1-phenylheptadecane n-tetracosane 2-methyltricosane 1-cyclohexyloctadecane hexapropylbenzene octadecylbenzene 1,3,5-triphenylbenzene n-pentacosane 9-octylheptadecane 9-(3-cyclopentylpropyl)heptadecane 9-(2-cyclohexylethyl)heptadecane decahydro-1-pentadecylnaphthalene 1-cyclohexyl-3-(2-cyclohexylethyl)undecane 1,7-dicyclopentyl-4-(3-cyclopentylpropyl)heptane 1-cyclohexyl-3-(2-cyclohexylethyl)-6-cyclopentylhexane (3-octylundecyl)benzene 1-pentadecylnaphthalene n-hexacosane 9-dodecylanthracene squalane (1-hexadecylheptadecyl)cyclohexane (1-hexadecylheptadecyl)benzene

1.5777 2.0173 1.9942 1.4697 1.6979 1.6225 1.6622 1.3218 1.1950 1.2587 1.2331 1.6735 1.7443 1.7724

0.8627 1.2544 1.2312 0.7997 0.9680 0.9143 0.9572 0.6800 0.5679 0.6201 0.5886 0.9838 1.0153 1.0409 0.6979 0.9560 1.3955 1.0574 0.8813 1.0682 0.7075 1.0279 0.9430 0.9020 0.9413 1.1116 1.0542 0.7611 0.7723 0.5610 0.7176 1.0066 1.0261 1.1394 1.1078 1.2765 1.6234 1.2891

3.9381 5.1161 5.0288 3.6500 4.2197 4.0355 4.1778 3.3549 2.9716 3.2419 3.0657 4.1580 4.3480 4.4001 3.6972 4.0878 5.5273 4.4408 4.1159 4.5234 3.4092 4.3289 4.2594 4.1624 4.1246 4.3950 4.3787 3.5850 3.6058 3.3136 3.4600 4.2804 4.4091 4.6593 4.3950 4.9854 6.2706 5.5863

5. Comparisons The vapor pressures of each hydrocarbon in the entire database were calculated using each of the alpha forms to determine which provides the best results. Table 3 shows the percentage errors obtained for the different alpha functional forms. Subscript “0” represents the original alpha function of the PR EOS. The percentage average absolute deviation (% AAD), percentage maximum absolute deviation (% MAD), and the percentage bias (% BIAS) for each compound were calculated using eqs 18, 19, and 20, provided in the Appendix. The alpha function form fω2 provided the best vapor pressure predictions for the entire database; however, fω1 does provide a more balanced predictive capability when the % BIAS is taken into consideration and is therefore recommended. A comparison of the predicted vapor pressures between the new and standard PR EOS for light hydrocarbons corresponding to C1-C10 is shown in Table 5 in the Appendix. The percentage average and maximum absolute deviations in the vapor pressures using the standard Peng-Robinson are 5.7% and 70.7%, respectively. Using the enhanced EOS, the percentage average and maximum absolute deviations in the calculated vapor pressures are 4.6% and 35.6%, respectively. Figure 7 compares the characteristic constants from the standard PR EOS, fω0, with values provided in fω1 representing the polynomial function in the acentric factor defined at a reduced temperature of 0.7. For acentric factors up to 0.5, the calculated characteristic constants of both functional forms are similar, as discussed in the previous paragraph.

1.6502 2.1644 1.7772 1.6343 1.8082 1.3720 1.7494 1.6701 1.6692 1.6537 1.7771 1.7576 1.4443 1.4532 1.3150 1.3901 1.7215 1.7586 1.8569 1.7695 2.0005 2.4247 2.1316

The standard PR alpha function was developed from hydrocarbons with acentric factors up to 0.49. However, with an increase in the acentric factor, corresponding to heavier hydrocarbons and especially aromatic containing hydrocarbons, the new functional form begins to deviate significantly to higher values. This implies that the standard PR EOS would always predict higher vapor pressures, as shown in Figure 1. In fact, for hydrocarbons with MW above 200 g/mol, the standard PR EOS vapor pressure predictions result in large errors, sometimes much greater than 150%. The new correlation performs better as compared to the original PR EOS function for light hydrocarbon vapor pressures. Therefore, it is recommended to completely replace the temperature-dependent attractive pressure term of the standard PR EOS with the one proposed in this study. 6. Conclusions A single set of “recommended” acentric factors and a new alpha function based on the acentric factor definition of Pitzer have been developed and are included in the Appendix (Tables 4 and 5). The new alpha function was used to accurately predict the vapor pressures of heavy hydrocarbons. This extension of the PR EOS is based on a large vapor pressure database for 237 hydrocarbons that contains data for highly aromatic heavy hydrocarbons, naphthenes, as well as branched paraffins that are characteristic of heavy oils and bitumens. The predicted vapor pressures between reduced temperatures of 0.3 and 0.8 for all of the data showed an average and maximum absolute percentage deviation of

372 Energy & Fuels, Vol. 23, 2009

Nji et al.

Table 5. Comparison of Predicted Vapor Pressures for C1-C10 Using the Original and Modified Peng-Robinson Equation of State standard Peng-Robinson EOS

enhanced Peng-Robinson EOS

compound

% AAD

% MAD

% BIAS

% AAD

% MAD

% BIAS

methane ethane ethane propane propene n-butane 2-methylpropane 1-butene n-pentane 2-methylbutane 2,2-dimethylpropane 1-pentene cis-2-pentene trans-2-pentene 2-methyl-1-butene 2-methyl-2-butene cyclopentane n-hexane 2-methylpentane 3-methylpentane 2,3-dimethylbutane 2,2-dimethylbutane 1-hexene cis-3-hexene cyclohexene benzene methylcyclopentane cyclohexane n-heptane 2-methylhexane 3-methylhexane 2,2,3-trimethylbutane 2,2-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 3-ethylpentane 2,3-dimethylpentane 1-heptene toluene ethylcyclopentane 1,1-dimethylcyclopentane trans-1,2-dimethylcyclopentane trans-1,3-dimethylcyclopentane methylcyclohexane cycloheptane n-octane 2,2,3,3-tetramethylbutane 2,2,4-trimethylpentane 2,3,3-trimethylpentane 3,3-dimethylhexane 2,2,3-trimethylpentane 2,3,4-trimethylpentane 3,4-dimethylhexane 2,3-dimethylhexane 2,4-dimethylhexane 4-methylheptane 3-methylheptane 2,2-dimethylhexane 2,5-dimethylhexane 2-methylheptane 2-methyl-3-ethylpentane 3-ethylhexane 3-methyl-3-ethylpentane 1-octene trans-2-octene o-xylene p-xylene m-xylene ethylcyclohexane n-propylcyclopentane cyclooctane ethylbenzene n-nonane 3,3-diethylpentane 4-methyloctane 2,3,4-trimethylhexane 3,4-dimethylheptane

1.1261 6.2445 10.9341 7.3241 7.2062 5.1918 0.8572 6.8975 5.0373 4.7677 6.1641 3.3827 2.6774 4.8868 5.0926 1.6560 8.7084 5.1741 5.8969 6.0443 7.2570 4.1254 2.5278 5.5802 7.4277 11.1610 9.4832 11.9730 5.8911 3.5188 4.6367 5.8513 4.9487 5.2379 5.0059 5.0513 4.8747 1.5298 6.1759 4.9276 5.0128 5.8120 5.0394 8.7187 6.8996 4.8112 1.1253 3.1071 3.4358 3.7820 4.2284 3.7098 4.4159 4.5555 5.3083 4.9517 5.1443 4.4701 4.1867 4.2865 4.3638 4.9051 4.2068 3.4398 5.5856 4.3266 2.5135 3.1070 3.0253 5.3692 6.4198 5.3643 5.1289 4.3612 3.1489 4.1176 4.1639

1.8397 29.8327 34.8086 24.9042 24.0152 17.1605 1.2105 23.7942 18.9268 17.1696 22.4958 14.0666 12.3989 16.8425 18.8057 6.6317 29.7357 21.2363 24.6757 27.3373 32.4396 19.5694 12.5755 22.6545 31.5626 33.1869 36.6874 41.2545 24.6815 15.1221 20.1065 22.4860 19.1486 20.3143 19.5268 19.7325 19.3424 7.3207 22.2010 17.7962 22.4598 25.6275 20.9586 33.4525 23.3119 18.5734 3.1321 11.2994 14.5255 15.2045 16.1601 15.1954 17.6628 19.5734 22.9551 19.3679 20.2721 18.1823 17.7245 18.7420 17.7832 19.4282 15.4786 5.5766 21.4610 19.1689 13.0341 13.8219 12.4097 20.8777 22.5783 24.7829 18.9930 15.8912 12.3097 16.1646 15.4917

1.1261 6.2445 10.9341 7.3241 7.2062 5.1918 -0.8476 6.8975 5.0101 4.7250 6.1201 3.2731 1.8104 4.8769 5.0486 -0.1208 8.7084 5.0949 5.7942 5.8669 7.1163 3.8986 1.9355 5.5049 7.2493 267.8635 9.4802 11.9730 5.8443 3.4009 4.5417 5.8220 4.8803 5.1761 4.9679 4.9693 4.7940 0.2013 6.1757 4.8962 4.8047 5.6279 4.9250 8.7055 6.8996 4.8108 1.1253 3.0273 3.3325 3.6962 4.1371 3.6186 4.3530 4.4872 5.2164 4.9207 5.1154 4.3666 4.1104 4.2648 4.2829 4.8679 4.1449 -3.4174 5.5848 4.2204 2.0591 2.9551 2.8022 5.3262 4.1948 5.2004 5.1289 4.3324 3.0874 4.1029 4.1362

2.9731 3.7615 4.5148 2.9154 2.9485 3.3029 7.7566 3.0558 3.4016 2.9520 2.9467 4.1221 5.9674 3.3098 3.2676 7.7650 3.4249 3.5639 3.3130 3.9217 4.2091 3.4769 5.7589 3.4125 4.4394 3.7462 4.3779 5.6787 3.6818 3.7256 3.6954 3.1803 3.1520 3.1971 3.3504 3.3466 3.2015 7.5879 3.2621 3.1763 3.9896 4.1164 3.6758 4.1018 3.0235 3.5438 4.9369 4.3538 4.1605 3.8019 3.3251 3.9577 3.6435 3.6876 3.4550 3.4489 3.4640 3.2229 3.4962 3.9622 3.6068 3.5216 3.4841 11.3097 3.6116 4.9054 6.4781 6.3766 4.9694 3.4916 5.2403 3.5065 3.3625 3.4286 3.8878 3.6241 3.5607

4.7109 9.7329 18.2661 6.4107 8.6703 4.2113 12.5098 7.1452 4.4130 3.9252 8.4760 5.1525 7.2746 4.2904 4.3166 9.8968 11.2519 4.6229 7.4672 9.4216 14.8361 4.6769 7.1901 4.7193 10.5616 17.9995 16.5781 23.2575 5.9137 4.7793 4.8264 7.2993 4.1234 4.8130 4.2945 4.3876 4.0192 10.1877 6.0350 4.1973 6.0347 8.2751 4.7642 14.5526 8.1721 4.6054 7.1951 4.9197 5.2137 4.8100 4.3374 5.0071 4.7328 4.8453 6.8981 4.5163 4.5373 4.2627 4.5953 5.2637 4.7050 4.6164 4.3464 17.9261 4.6799 6.2804 8.1727 7.6094 5.8474 4.5476 7.6760 7.1759 4.3745 4.2548 5.0692 5.0415 4.5805

-2.9731 -2.0161 2.7639 -1.5085 -0.5752 -3.2372 -7.7566 -1.3468 -2.9126 -2.4847 -0.6584 -4.1221 -5.9674 -3.1742 -2.6930 -7.7650 -0.1104 -3.1706 -1.8759 -2.0393 -0.4500 -2.8016 -5.7589 -2.6925 -2.1972 71.4713 0.5987 3.6083 -2.7135 -3.6923 -3.3340 -1.6199 -2.4847 -2.3694 -2.7533 -3.0042 -2.8238 -7.5879 -2.0361 -2.6903 -3.0174 -2.5582 -3.0101 0.0920 -0.7202 -3.3177 -4.9369 -4.3538 -4.1605 -3.7854 -3.1734 -3.9524 -3.5095 -3.2990 -2.2257 -3.1569 -3.0501 -2.8075 -3.2563 -3.7980 -3.4094 -3.2171 -3.4707 -11.3097 -3.2482 -4.9054 -6.4781 -6.3766 -4.9694 -3.1255 -4.4019 -2.2720 -3.0954 -3.3746 -3.8878 -3.5556 -3.5425

Characterization of HeaVy Oils and Bitumens 2

Energy & Fuels, Vol. 23, 2009 373 Table 5. Continued standard Peng-Robinson EOS

enhanced Peng-Robinson EOS

compound

% AAD

% MAD

% BIAS

% AAD

% MAD

% BIAS

1-nonene 1,3,5-trimethylcylcohexane isopropylcyclohexane n-propylcyclohexane 1,2,4-trimethylbenzene 1,3,5-trimethylbenzene 1,2,3-trimethylbenzene n-propylbenzene isopropylbenzene 1-ethyl-2-methylbenzene 1-ethyl-3-methylbenzene 1-methyl-4-ethylbenzene indane isopropenylbenzene indene n-decane 2,7-dimethyloctane 3,3,5-trimethylheptane 2,2,3,3-tetramethylhexane 2,2,5,5-tetramethylhexane 1-decene butylcyclohexane 1-methyl-4-isopropylcyclohexane isobutylcyclohexane sec-butylcyclohexane tert-butylcyclohexane naphthalene trans-decalin 1-methyl-2-isopropylbenzene 1-methyl-3-isopropylbenzene 1,4-dimethyl-2-ethylbenzene 1,3-dimethyl-2-ethylbenzene 1,4-diethylbenzene 1,2,3,5-tetramethylbenzene 1,2,4,5-tetramethylbenzene butylbenzene isobutylbenzene tert-butylbenzene sec-butylbenzene tetralin

4.2333 3.4139 2.3175 6.8124 1.6563 0.4967 1.5394 3.0415 7.2423 3.0969 2.9417 1.8072 2.2939 5.4022 5.9088 11.3405 2.2338 4.2354 10.4346 5.4132 4.6587 17.4884 11.1039 4.7146 12.6072 7.6439 1.4993 4.6734 5.9207 20.1896 25.6380 12.3170 5.8161 18.0304 2.3458 3.0669 11.5251 9.2047 5.8857 7.3110

15.3927 15.4232 9.5119 22.4801 7.5696 2.7595 8.3704 12.3891 21.5311 12.6381 11.9891 6.3135 9.8151 20.6798 17.1488 48.0861 6.1876 18.6482 47.7421 21.6824 6.3832 70.6707 24.1036 19.2251 56.5992 42.5894 1.8794 24.6937 28.8803 64.3872 61.2802 28.9018 27.4938 59.6738 4.4184 14.1975 53.2890 48.4283 28.3878 38.2659

4.2333 3.3528 1.9594 6.8050 0.6871 0.4839 0.8698 2.9685 7.2423 2.9577 2.7842 1.7392 1.6534 5.3657 2.0168 11.3405 2.2338 4.0706 10.3164 5.3824 -4.6587 17.4835 -10.7018 4.6748 12.6069 7.4632 -1.4993 4.0788 5.7465 20.1768 25.6325 12.3077 5.8161 18.0304 2.3458 3.0580 11.4659 9.0446 5.7855 6.4658

3.7354 3.8361 5.6552 2.9346 7.6661 7.4431 7.0380 4.6327 2.6492 3.7015 3.7428 5.9218 5.9970 3.7888 6.3126 3.7146 5.6768 4.1727 4.2007 3.7404 4.6587 7.7859 17.6193 3.5898 4.5678 4.6930 8.1327 5.9671 3.5179 8.7520 13.3353 4.1593 5.7299 7.7371 4.2372 5.1220 5.3631 5.0883 3.5547 6.1684

4.6812 6.0658 6.4807 6.5016 9.6162 11.4423 8.8839 5.5410 5.7163 4.5222 4.5156 7.6180 6.9428 4.9971 9.5868 17.8090 8.5619 5.6084 17.8463 5.0903 6.3832 35.6211 35.3830 4.7047 15.0276 19.3833 10.6762 8.0670 8.2155 27.9207 30.3218 8.6006 7.6773 27.8784 5.7455 6.4345 20.4943 23.6091 6.3550 8.1465

-3.7354 -3.7141 -5.6552 -1.2792 -7.6661 -7.4431 -7.0380 -4.6327 -0.8536 -3.7015 -3.7428 -5.9218 -5.9970 -3.1407 -6.1425 -0.4230 -5.9098 -4.0240 -1.2427 -3.1594 -4.6587 5.0882 -17.6193 -3.1335 -1.4631 -0.7478 -8.1327 -5.8162 -2.1394 6.9319 12.1196 2.2890 -5.7299 5.7634 -4.2372 -5.1220 -0.5307 0.4530 -2.6111 -5.1161

10.4% and 230%, respectively, for the standard PR EOS. The new and recommended PR EOS showed values of 4.9% and 78%, respectively. For light hydrocarbons corresponding to C1-C10, values of 5.7% and 70.7% for the original PR EOS and 4.6% and 35.6% for the new PR EOS were calculated, respectively. This simple correlation of attractive term using acentric factors from a more relevant database for heavy hydrocarbons combined with the Nji et al.8 method for the predication of critical temperatures and pressures provides a simple framework for the estimation of thermodynamic equilibrium for systems containing heavy oil fractions. Acknowledgment. We wish to thank Shell Canada Energy for financial support. Technical support provided by Virtual Materials Group Inc. is gratefully acknowledged.

( ) ( )

Appendix

n

m

∑

1 % AAD ) m j)1

∑ |e | ji

i)1

n

j

n m % MAD ) 100max(max(|eji|)i)1 )j)1

(18) (19)

n

m

∑

1 % BIAS ) m j)1

∑e

ji

i)1

n

j

(20)

where eji are the relative errors, n is the number of vapor pressure data points per hydrocarbon, and m is the number of hydrocarbons. EF8006855