Estimation of Isomeric Distributions in Petroleum Fractions - Energy

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Estimation of Isomeric Distributions in Petroleum Fractions Zhanyao Ha,† Zbigniew Ring,‡ and Shijie Liu*,† Department of Chemical and Material Engineering, University of Alberta, Edmonton, T6G 2G6, Canada, and National Center for Upgrading Technology, 1 Oil Patch Drive, Devon, AB, T9G 1A8, Canada Received November 8, 2004. Revised Manuscript Received March 11, 2005

This paper proposes a new approach to quantify the compositional distribution of different hydrocarbon isomers in an “isomeric lump” of a crude oil, determined using gas chromatographymass spectrometry (GC-MS) methods. The concentration distribution of isomers can be determined with good accuracy by minimizing the Gibbs free energy of the mixture containing a set of isomers, subject to the stoichiometric constraint and the measured average boiling point of that isomeric lump. The simulated compositions of the hexane and heptane isomers were compared with the reported analytical results for 18 crude oils. The correspondence between predicted and measured distributions was found to be satisfactory. The experimental distributions of hexane and heptane isomers in those crudes are far from the thermodynamic equilibria, but the introduction of additional experimental information, in the form of the average boiling point of the lump, made it possible to model its isomeric distributions. This finding is important for the derivation of molecular representation for distillates in advanced kinetics modeling of refinery conversion processes.

1. Introduction Characterization of petroleum fractions is critically important to advanced kinetics modeling of various conversion processes in petroleum refining. Recent attempts to model the hydrocracking and catalytic cracking processes require molecular representation of the feedstock.1-2 This trend in process modeling is driven by the desire to predict, in a fundamental way, not only the yields of individual product fractions but also their detailed properties. On the other hand, positive identification and quantification of large numbers of isomers is beyond the capabilities of today’s analytical techniques. The isomeric lump frequently sets the limit for molecular characterization.3 Even if it were possible to know the exact composition of the fraction, computational limitations make it impossible to use this amount of information. Therefore, usually there is a practical limit as to how much a process modeler may know about feedstock composition. For example, structural-oriented lumping4 and single-event kinetics5 models rely on characterization of feedstock in terms of several molecular classes (homologous series) distrib* Author to whom correspondence should be addressed. † University of Alberta. ‡ National Center for Upgrading Technology. (1) Souverijns, W.; Martens, J. A.; Froment, G. F.; Jacobs, P. A. J. Catal. 1998, 174, 177-184. (2) Mizan, T. I.; Klein, M. T. Catal. Today 1999, 50, 159-172. (3) Briker, Y.; Ring, Z.; Iacchelli, A.; McLean, N.; Rahimi, P. M.; Fairbridge, C. Energy Fuels 2001, 15, 23-37. (4) Quann, R. J.; Jaffe, S. B. Ind. Eng. Chem. Res. 1992, 31, 24832497. (5) Vynckier, E., Froment, G. F. In Kinetic and thermodynamic lumping of multicomponent Mixtures; Astarita, G., Sandler, S. I., Eds.; Elsevier Science Publishers B. V.: Amsterdam, 1991.

uted by carbon number. However, a strong assumption in lumping isomers (by type and carbon number) is that the physical and chemical properties of those isomers are identical. This assumption is not true for most of hydrocarbons. Many thermophysical properties of various isomers are widely spread. For example, the difference in normal boiling points (NBP) among terphenyls (o-, m-, p-) is 50 °C. The maximum difference of normal freezing points among octane isomers is 227 K, with a maximum of 374 K for 2,2,3,3-tetramethylbutane and a minimum of 147 K for 2,3-dimethylhexane.6 If isomeric lumps are considered instead of individual molecules, it is not possible to reliably estimate bulk properties for an arbitrary stream. However, if the molecular makeup of a refinery stream is known at the molecular level, an efficient property estimation model could be used to estimate its properties. One possible way to achieve this is to use a quantitative-structure-property-relationship (QSPR) model to estimate a particular property of each individual hydrocarbon in the stream7 and then estimate this property for the whole stream using appropriate mixing rules. The chemical activities and reaction paths of hydrocarbons are also dependent on isomer distribution. For example, different isomers produce different carbonium or carbenium ions during catalytic cracking. As a result, they go through different elementary reaction paths. In a catalytic cracking study of three C6 isoparaffins (2-methylpentane, 3-methylpentane, and (6) API Technical Data Book - Petroleum Refining, 5th ed.; American Petroleum Institute: Washington, DC, May, 1992. (7) Ha, Z.; Ring, Z.; Liu, S. Energy Fuels 2005, 19, 152-163.

10.1021/ef049712r CCC: $30.25 © 2005 American Chemical Society Published on Web 04/09/2005

Estimation of Isomeric Distributions in Petroleum Fractions

2,3-dimethylbutane), Wojciechowski8 found that these three C6 isomers followed quite different reaction paths in the initiation, propagation, and β-cracking. As a result, their corresponding products significantly differed in terms of the kinetic chain length (3.38, 3.12, and 27.03, respectively), paraffin/olefin ratio (3.38, 1.21, and 10.75, respectively), and volume expansion (1.30, 1.83, and 1.09, respectively). Therefore, the distribution of isomers is important in the estimation of bulk physical properties, as well as in the detailed kinetic study of complex mixtures. The capabilities on analytical techniques rapidly decrease with boiling range. Composition of refinery streams in the naphtha boiling range can be measured at the molecular level using the DHA (detailed hydrocarbon analysis) method. However, mass spectrometry, probably the most capable method for distillate characterization, is incapable of distinguishing various isomers in middle distillates. Hence, an isomeric lump is the practical limit for the compositional detail available from the analytical laboratory. Although advances in characterization of petroleum fractions benefit from the development of new more advanced analytical techniques, this may not be the only way to deliver the detail necessary for reliable modeling of product quality. Finding isomer distribution within an isomeric lump of petroleum has been considered an intractable problem.9 These distributions reflect the reaction conditions during the crude maturing processes. Consequently, the abundances of individual isomers in the isomeric lump would be expected to depend on the kinetics of the reactions they undergo and their thermodynamic stabilities. So far, the thermodynamic equilibrium among isomers has been assumed in isomeric lumping for kinetics modeling.10 However, the actual distribution of isomers differs from the equilibrium distribution in most cases, especially for saturates.11 Although it is infeasible to quantify each individual isomer in a large isomeric lump (e.g., >C10), only a relatively small fraction of the set of all possible molecules is actually present in various petroleum fractions in quantities that affect their processability and, ultimately, quality.11 This paper proposes a deterministic way of finding isomer distribution in an isomeric lump independent of the limitations of analytical methods. We found that the distribution of isomers in the isomeric lump could be calculated by minimizing the Gibbs free energy of the lump subject to a constraint in addition to the stoichiometric one; the independently measured boiling point distribution within this lump. By default, this boiling point distribution is measured with decreasing degree of accuracy for isomer systems of increasing carbon number. The approach was applied to estimate the hexane and heptane isomer distributions, and the results were compared to the distributions in light petroleum fractions published in the open literature. The validity of this approach and the uniqueness of the solution to the associated mathematical problem were (8) Wojciechowski, B. W. Catal. Rev.-Sci. Eng. 1998, 40, 209-328. (9) Kuo, J. C. W. In Chemical Reactions in Complex Mixtures; Sapre, A. V., Krambeck, F. J., Eds.; Van Nostrand Reinhold, New York, 1991. (10) Krambeck, F. J. In Kinetic and thermodynamic lumping of multicomponent Mixtures; Astarita, G., Sandler, S. I., Eds.; Elsevier Science Publishers B. V.: Amsterdam, 1991. (11) Tissot, B. P., Welte, D. H. Petroleum Formation and Occurrence, 2nd ed.; Springer-Verlag: Berlin, 1984.

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examined. Our approach avoids some pitfalls of the techniques based on Monte Carlo simulations,12 such as creating redundant molecules and neglecting thermodynamic stability aspects of molecular composition. 2. Isomer Distribution within an Isomeric Lump To simplify the problem, the isomeric lump is frequently assumed to be a closed ideal system.13 This approach is also taken here. In such a system, the Gibbs free energy of an ideal solution of N isomers can be expressed as N

∆G on )

N

xi∆G oi + RT ∑ xi ln xi ∑ i)1 i)1

(1)

subject to the stoichiometric constraint ∑xi ) 1. The equilibrium composition of the isomeric lump can be obtained by minimizing ∆G on with respect to the system composition (xi)14 to give

xi ) exp[(∆G on - ∆G oi )/RT] where ∆G on is defined as

[∑ N

∆G on

) -RT ln

i)1

(2)

]

exp( -∆G oi /RT)

(3)

Using this approach and on the basis of limited data available in the open literature, the measured isomeric distribution in virgin crude oils is not consistent with thermodynamic equilibrium. Martin et al.15 measured distributions of small alkane isomers in naphtha from 18 crude oils. The averaged distribution (they found little variation in heptane isomer distributions among these crudes) is compared with the calculated equilibrium distributions in Table 1 for 298 and 400 K, with respect to the gas and liquid phases.11 Clearly, the match between the measured and equilibrium distributions is inadequate. The same was found true for hexane isomer distributions.15 A solution to this problem is proposed below. We assume that all the existing isomers are in a state that can be estimated by considering the classical equilibrium problem, subject to the stoichiometric constraint, with an additional constraint of partial information about the system composition. One way to obtain this partial information would be to measure the boiling point distribution of the mass in the isomeric lump by an appropriate GC technique. Note that detailed (rather than partial) information about boiling point distribution would be equivalent to knowing the system composition in detail. This partial information could be the average boiling point of the lump if the lump consists of a relatively small number of isomers with widely spread (relative to the accuracy of measurement) boiling points. The average boiling point measured with finite accuracy may not provide a (12) Neurock, M. N.; Nigam, A.; Trauth, D.; Klein, M. T. Chem. Eng. Sci. 1994, 49, 4153-4177. (13) Smith, W. R., Missen, R. W. Chemical reaction equilibrium analysis: theory and algorithm; John Wiley and Sons: New York, 1982. (14) Alberty, R. A. In Chemical Reactions in Complex Mixtures; Sapre, A. V., Krambeck, F. J., Eds.; Van Nostrand Reinhold: New York, 1991. (15) Martin, R. L.; Winters, J. C.; Williams, J. 6th World Pet. Congr., Sec. V 1963, 231-260.

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Table 1. Abundances of Heptane Isomers in Virgin Crude Oils11 abundance, wt% isomers/distribution n-heptane 2-methylhexane 3-methylhexane 3-ethylpentane 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 2,2,3-trimethylbutane a

ave. 18

oilsa

55.5 13.8 19.2 2.6 0.6 6.1 1.7 0.4 0.1

isomerization equilibrium, wt% HMb

298 K (g)

298 K (l)

400 K (g)

400 K (l)

52.0 16.0 22.4 2.4 0.4 4.8 2.0

1.25 9 5.1 0.45 32 25 9.9 11.4 5.9

2.25 11.2 6.85 0.6 24.9 30 8.15 11.3 4.75

4.3 15.4 11.3 1.3 16.7 28.8 8.4 10.4 3.4

8.55 15.65 12.15 1.45 13.2 29.5 6.8 9.8 2.9

Reference 15. b Hassi-Messaoud crudes (monophasic sample). Table 2. Properties of Hexane and Heptane Isomers and Average Predicted Results isomers\properties

∆Gog,298K kcal/mol

∆G0l,298K kcal/mol

n-hexane 2-methylpentane 3-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane n-heptane 2-methylhexane 3-methylhexane 3-ethylpentane 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 2,2,3-trimethylbutane

-0.016 -1.275 -0.512 -2.089 -0.7464 1.9515 0.8294 1.2247 2.7199 0.1315 1.3664 0.8142 1.1735 1.1186

-1.03 -1.97 -1.34 -2.90 -1.69 0.3564 -0.5631 -0.2176 1.3215 -1.2331 -0.072 -0.341 -0.0875 -0.0139

a

predicted wt%

BP, °C

F15°C g/mL

298 K (g)

298 K (l)

abundancea wt% in 18 crudes

68.73 60.26 63.27 49.73 57.98 98.43 90.05 91.85 93.47 79.19 89.78 80.49 86.06 80.88

0.6651 0.6577 0.6693 0.6539 0.6662 0.690 0.682 0.692 0.704 0.682 0.699 0.676 0.696 0.695

53.778 19.650 20.729 1.803 4.040 56.157 15.925 16.099 2.379 0.870 5.822 0.449 1.988 0.311

56.821 17.559 17.064 2.838 5.719 56.590 15.510 15.961 2.064 1.204 6.186 0.416 1.796 0.273

52.787 24.916 18.581 0.591 3.125 55.5 13.8 19.2 2.6 0.6 6.1 1.7 0.4 0.1

Averaged abundance in reported 18 crude oils.15

sufficient amount of information for a lump with a relatively narrow boiling point spread or consisting of a large number of isomers. In those cases, as much information as possible about the boiling point distribution should be provided. The concentration of structural isomeric lumps (SILs), defined as hydrocarbon species of the same carbon number within a hydrocarbon homologous series, can be quantified using relatively low-cost advanced analytical techniques such as GC-FIMS. With the help of appropriate GC retention time calibration, each SIL can be assigned a boiling point distribution. The use of n-paraffin standards to link the boiling points with retention time (basis of the ASTM D2887 simulated distillation method) is assumed to be sufficiently accurate in this work, but in principle, other more sophisticated methods could be used for retention calibration for individual hydrocarbon groups (e.g., retention calibration for each individual hydrocarbon type). The boiling point differences among individual isomers are reflected in differences of their retention times. However, it should be noted here that when the number of isomers is large, usually it is not possible to resolve their corresponding individual peaks by standard chromatography. Therefore, the boiling point distribution of a SIL cannot be measured in the detail required for determination of its composition and other sources of information are required to achieve it (e.g., methodology proposed in this paper). If detailed molecular composition of the SIL is known, for example, through simulation, its boiling point distribution or an average boiling point (BPlump) in less complex cases, can be obtained from the boiling points of individual isomers and their concentrations (in case

of BPlump, an appropriate mixing rule6 is used). Therefore, operationally, partial information about the SIL composition can be used as a constraint to estimate the isomer distribution and, as proposed here, the problem becomes one of constrained minimization of ∆G on defined by eq 1

[

Min.[∆G on] ) Min.

Subject to

N

N

xi∆G 0i + RT ∑ xi ln xi ∑ i)1 i)1

{∑∑

xi ) 1 xiBPi ) BPlump

] (4)

The uniqueness of the solution for this problem in a general case is shown in the Appendix. The calculations discussed below, conducted using Powell’s method16 (a modified Newton-Raphson method), yielded the composition vector (X) of dimension N, predicting the hexane and heptane isomer distributions presented in the following section. 3. Simulated Isomer Distributions and Discussions Minimization of eq 4 subject to the stoichiometric and BPlump constraints was applied to simulate the isomeric distribution of the hexane and heptane isomers. Table 2 lists the densities, boiling points, and free energies of formation in the gas and liquid phases for each isomer used in the calculations. The densities, boiling points, (16) Powell, M. J. D. TOLMIN: A fortran package for linearly constrained optimization calculations, DAMTP Report NA2; University of Cambridge: Cambridge, 1989.



Table 3. General Description of 18 Crude Oils15 field

state/country

gravity, °API

era

temp. °C

vol% up to 111 °C

Alida Bever Lodge Darius Eola Mclish Eola oil creek Hendricks Kawkawlin Lee Harrison North Smyer Pembina Ponca city Redwater South Houston Swanson River Teas Uinta Basin Wafra Wilmington

Saskatchewan N. Dakota Iran Oklahoma Oklahoma Texas Michigan Texas Texas Alberta Oklahoma Alberta Texas Alaska Texas Utah Kuwait California

38.1 40.7 29.0 36.6 44.6 33.1 35.0 25.3 43.2 41.6 42.0 35.3 24.0 31.3 40.0 30.6 18.3 19.3

Paleozoic Paleozoic Mesozoic Paleozoic Paleozoic Paleozoic Paleozoic Paleozoic Paleozoic Mesozoic Paleozoic Paleozoic Cenozoic Cenozoic Paleozoic Cenozoic Cenozoic Cenozoic

38 NA 118 78 75 30 32 NA 66 52 57 49 58 66 64 NA NA 54

19.21 18.43 9.18 15.69 22.83 14.07 10.47 14.30 27.33 21.13 17.09 17.54 3.54 12.11 23.46 4.50 4.58 4.44

Figure 1. Prediction of heptane isomer distribution in Alida crude oil.

Figure 2. Prediction of heptane isomer distribution in Bever Lodge crude oil.

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Figure 3. Prediction of heptane isomer distribution in Darius crude oil.

Figure 4. Prediction of heptane isomer distribution in Eola Mclish crude oil.

and free energies of formation in the gas phase are reported in the API Technical Data Book.6 The free energies of formation in the liquid phase were calculated from the standard free energies of formation in the gas phase, reported heat of vaporization, heat capacities of gas and liquid phases, and the entropies of gas and liquid phases at standard state. Since the original GC data for the reported hexane and heptane isomer distributions were not available, the BPlump was estimated directly from the actual concentration distribution instead. Again, normally, BPlump can be calculated from the GC-MS data. Distributions of hexane and heptane isomers in Table 2 in the gas and liquid phases were calculated using the free energy of formations at standard state in the gas and liquid, respectively.

The normalized simulated distributions of hexane and heptane isomers are compared below to the distributions measured by Martin et al.15 for 18 crude oils. These crude oils spanned a wide range of geological ages and represented compositional extremes with API gravities ranging from 18 to 45. Eleven were found in Paleozoic rocks, two in Mesozoic, and five in Cenozoic. Some important characteristics of the 18 crude oils are given in Table 3. Further details can be found in the original publication by Martin et al.15 The hexane and heptane isomers had been quantified in the naphtha fractions boiling up to 111 °C by GC using a capillary column. The hexane and heptane isomers were quantified in vol% of this naphtha fraction. The total amounts of hexane or heptane isomers were between 2 and 4% in



Figure 5. Prediction of heptane isomer distribution in Eola Oil Creek crude oil.

Figure 6. Prediction of heptane isomer distribution in Hendricks crude oil.

most of the crude oils. Densities at 15 °C were used to convert the vol% to wt% used in this work. The estimated uncertainties in the original results were less than 6% of the amount reported, or one in the last digit, whichever was larger.15 Table 2 also compares the normalized simulated isomeric distributions with the average distribution measured by Martin et al.15 Clearly, the agreement is quite close and much better than the agreement between the equilibrium and measured distributions presented in Table 1. The hexane and heptane isomeric distributions were estimated at temperatures of the individual reservoirs. For those whose reservoir temperatures were not available (4 samples), 60 °C (the averaged temperature for the remaining 14 crude oils) was used. The free energies of formations for individual

hexane and heptane isomers at reservoir temperatures were estimated from the heats of formations and entropies at 300 and 400 K.17 The simulated heptane isomer distributions are compared with the reported distributions of 18 virgin crude oils in Figures 1-18, showing a good agreement. The predictive deviations for both hexane and heptane distributions are tabulated in Table 4. Similarly good agreements between the simulated and measured distributions were also found for five hexane isomers (see Table 4). Since the distribution of possible isomers within a SIL is only related to the relative values of their free energies of formations, the distribution pattern of the hexane and heptane (17) Stull, D. R.; Westrum, E. F., Jr.; Sinke, G. C. The chemical thermodynamics of organic compounds, John Wiley and Sons: New York, 1969.

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Figure 7. Prediction of heptane isomer distribution in Kawkawlin crude oil.

Figure 8. Prediction of heptane isomer distribution in Lee Harrison crude oil.

isomers in the gas phase were expected to be similar to those in the liquid phase. Indeed, since the relative values of free energy of formations among hexane and heptane isomers in the gas phase were found to be very close to those in the liquid phase, similar distributions in gas and liquid phase were obtained for all 18 crudes (see Figures 1-18 for heptanes). Consequently, only the simulated isomeric distributions in the gas phase were used to estimate the prediction errors. As shown in Table 4, the average absolute errors over all hexane and heptane isomers are less than 3% and 1.5% for most of the reported crudes. However, substantial predictive errors are found for the younger crudes, especially for South Houston, Wafra, and Wilmington (see Figure 13, 17, and 18 for heptane isomer

distributions), which were formed during the tertiary Cenozoic Era. The maximum average predictive errors are 11% and 7% for hexane and heptane isomers, respectively, in Wafra crude (see Table 4). The potential significance of the current approach lies on the fact that the approach was able to predict the key isomers (nhexane, 2-methylpentane, 3-methylpentane for hexane isomers; n-heptane, 2-methylhexane, 3-methylhexane, and 2,3-dimethylpentane for heptane isomers) with good confidence. As shown in Table 4, the relative predictive errors were 8.1%, 22.3%, and 16.5% for n-hexane, 2-methylpentane, and 3-methylpentane, respectively. Those for n-heptane, 2-methylhexane, 3-methylhexane, and 2,3-dimethylpentane were 5.1%, 7.6%, 20.4%, and 15.1%, respectively. The three key isomers made up



Figure 9. Prediction of heptane isomer distribution in North Smyer crude oil.

Figure 10. Prediction of heptane isomer distribution in Pembina crude oil.

more than 92% of total hexane isomers except for two younger crudes (South Houston and Wafra), while the four key isomers comprised more than 90% of total heptane isomers, except for those three younger crudes (South Houston, Wafra, and Wilmington). Considering the overall experimental uncertainties and the wide distribution range of the key isomers in those crudes, the predictions for key isomers are satisfactory. According to Martin et al.,15 the high proportion of branched paraffins in the three outlier crudes suggests a possible contribution of catalytic cracking in their early formation period. These authors suggest that thermal cracking was the dominant process that contributed to the isomeric distributions in naphtha of the mature crudes. In addition to this dissimilarity in the maturation

processes, the total amounts of hexane/heptane isomers in these three crudes are relatively small (0.335:0.381%, 0.885:0.847%, and 0.529:0.495% in South Houston, Wafra, and Wilmington crudes, respectively) compared to those in the remaining 15 crudes (>2.0%). This likely resulted in larger measurement errors. Consistent with their relative immaturity (formed in the Cenozoic era), the three crude oils were also the three highest boiling mixtures among the 18. Although our proposed approach resulted in larger prediction errors for the three crudes than for the remaining 15 (e.g., average absolute errors of 4.28%, 7.2% and 4.07% in normalized heptane isomer distributions), our results are much better than the thermodynamic equilibrium distributions calculated from eqs 2 and 3. Those errors are as high as 7.02%,

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Figure 11. Prediction of heptane isomer distribution in Ponca City crude oil.

Figure 12. Prediction of heptane isomer distribution in Redwater crude oil.

11.86% and 13.22% for these three crudes, respectively. The good agreements between the simulated and experimental distributions for the mature crude oils suggest that the proposed method may also be applicable to the predictions of isomer distributions in thermally cracked materials such as coker gas oils. The purpose of this paper was not to devise a method for distributing hexane and heptane isomers but to look for a method that would potentially be generally applicable to any SIL, and particularly for heavier SILs. For heavier materials, isolation and identification of individual molecules are infeasible. However, the boiling point distribution can be obtained experimentally for the individual SILs during GC-MS measurements re-

gardless of their boiling range. Although the number of possible isomeric permutations is very high for larger hydrocarbons, the actual distribution of isomers is not as diverse as might be expected.18 As shown by Martin et al.,15 the distributions of hexane/heptane isomers were dominated by three or four key isomers that accounted for 92/90% of total SIL. The distributions of these key isomers can be estimated with high accuracy using current approach. It is possible to use a limited number of major isomers to represent a SIL in molecular characterizations without losing the intrinsic chemistry (18) Hood, A.; Clere, R. J.; O’Neal, M. J. J. Inst. Petrol. 1959, 45, 168-173.



Figure 13. Prediction of heptane isomer distribution in South Houston crude oil.

Figure 14. Prediction of heptane isomer distribution in Swanson River crude oil.

detail.19 The selection of major isomers can be based on the analyzed structural occurrences and thermodynamic stabilities of individual molecules. The free energies of formation for larger hydrocarbons can be estimated by means of computational chemistry software packages, such as MOPAC, whenever such experimental data are not available. Consequently, at least operationally, it is possible to simulate the isomer distribution of larger hydrocarbons using the proposed approach. The validity of this approach in the generation of molecular composition of higher boiling fractions is explored elsewhere.20 (19) Liguras, D. K.; Allen, D. T. Ind. Eng. Chem. Res. 1989, 28, 674683. (20) Ha, Zhanyao; Liu, Shijie; Ring, Z. Derivation of molecular representations of diesels, in preparation.

4. Conclusions A computational augmentation for GC-MS techniques has been proposed to determine the composition of petroleum samples in detail that is impossible to achieve through GC-MS or any other analytical techniques alone. The proposed methodology uses a partial knowledge of the composition of a structural isomeric lump introduced in the form of a constraint derived from its boiling point distribution in the calculation of the thermodynamic equilibrium among the isomers involved. The resulting molecular distribution closely resembled experimental distributions when this approach was applied to predicting the isomeric distribution among hexane and heptane isomers reported for 18 geologically different crude oils. Excellent agreement

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Figure 15. Prediction of heptane isomer distribution in Teas crude oil.

Figure 16. Prediction of heptane isomer distribution in Uinta Basin crude oil.

was found for 15 crudes, while a worse, but still favorable, agreement was found for the remaining three younger (Cenozoic) crudes. Although, in view of very scarce experimental data, this approach could only be tested using data on isomer distributions in the naphtha boiling range, it has potential applications in predicting the isomer distributions for heavy fractions.20 The boiling point distribution constraint used in the Gibbs free energy minimization captures partial knowledge about the composition of the SIL under consideration, regardless of the boiling point range. It was demonstrated that, for simpler SILs with a boiling point spread that is wide enough (compared with the accuracy of the boiling point distribution measurement), as little compositional information as the average boiling point of the SIL carries sufficient information to help find the isomeric distribution with

excellent accuracy. For SILs that involve more isomers with less boiling point spread, the boiling point distribution (rather than the average boiling point) may be required to achieve the same accuracy. It has been demonstrated here that the composition of petroleum mixtures, going well beyond the capabilities of the analytical methods available today, can be determined with good accuracy through a computer simulation. Appendix. Derivation of Uniqueness for Quasi-Equilibrium Isomer Distribution Smith and Missen13 proved analytically that the chemical equilibrium problem has a unique solution for a single phase of ideal solution. Similarly, the solution uniqueness of the optimization problem in eq 4 can be proved analytically under the same condition. The



Figure 17. Prediction of heptane isomer distribution in Wafra crude oil.

Figure 18. Prediction of heptane isomer distribution in Wilmington crude oil.

constrained optimization problem can be solved by the Lagrange Multiplier method. The Lagrangian function is defined as

F(x,λ) )

∆G 0n

+ λ1(1 -

∑ xi) +

λ2(BPlump -

∑ xi BPi)

(A-1)

Substituting the derivatives of eq 1 into eq A-2 and solving for xi, we obtain

xi ) e(λ1+λ2BPi-∆G i -RT)/RT 0

Therefore, minimizing the Lagrangian function is reduced to solving the following set of nonlinear equations

Minimizing F with respect to x and λ results in a set of nonlinear equations

N

e(λ +λ BP -∆G -RT)/RT - 1 ) 0 ∑ i)1

(A-5)

BPi(e(λ +λ BP -∆G -RT)/RT) - BPlump ) 0 ∑ i)1

(A-6)

1

∂∆G 0n

∂F ) - λ1 - λ2BPi ) 0 i ) 1, 2, ‚‚‚, N (A-2) ∂xi ∂xi

(A-4)

2

0 i

i

and N

and constraints:

1

∂F ) 0 j ) 1, 2 ∂λj

(A-3)

2

i

0 i

Mathematically, eqs A-5 and A-6 can be written as

-5.898 -5.434 -3.624 -7.198 -5.083 -4.326 -2.566 -4.981 -9.587 -5.698 -5.815 -7.538 11.257 -7.122 -6.502 -9.571 1.834 -4.391 0.223 0.337 2.030 -0.883 0.535 3.170 -9.515 4.689 -4.515 -5.119 0.036 1.637 -2.290 3.666 0.526 -0.094 0.763 -25.881 -7.097 0.165

1.683 1.058 1.118 1.843 0.626 4.095 -0.097 4.048 3.838 1.904 1.178 2.636 0.389 2.362 2.002 2.658 7.908 3.347 4.550

1.500 1.381 1.326 2.140 1.288 0.999 -0.187 -0.941 3.366 1.104 1.666 2.440 -11.413 1.168 1.631 2.453 -2.288 1.145 0.547

2.359 2.174 1.803 2.879 2.034 5.537 1.875 4.174 5.882 2.279 2.326 3.931 6.125 2.849 2.638 3.828 11.267 4.595 N/A

ave abs in 2M-Pen. 3M-Pen. 22M-But. 23M-But. 5 isomers 1.707 -0.008 0.768 2.150 -0.999 8.607 -1.081 4.115 5.275 -0.049 0.242 2.557 2.214 -0.484 1.503 0.215 15.557 5.587 0.051

Hep. 0.987 0.858 -1.339 0.354 -0.443 1.862 -0.671 0.803 3.185 -0.081 0.967 2.243 10.378 0.795 0.524 1.018 3.864 7.378 0.076

-5.266 -3.285 -4.043 -5.535 -0.976 13.624 1.061 -5.123 -8.860 -3.485 -2.721 -5.819 0.329 -1.342 -4.702 -1.908 19.855 -5.662 0.204 0.870 1.057 2.057 0.027 1.173 -1.311 0.833 -0.141 -0.670 1.075 0.904 -0.853 -3.732 0.203 0.762 0.404 -3.227 -4.067 0.434

0.453 0.038 0.376 0.570 -0.111 2.730 -0.110 1.251 1.373 0.323 0.123 0.627 4.921 0.737 0.475 0.070 5.664 1.618 1.156

-0.234 0.711 0.949 1.109 1.172 -1.786 0.225 -3.767 -3.263 1.508 -0.227 -0.159 10.270 0.156 0.005 -0.613 -7.639 -7.414 0.151

-0.832 -1.184 -0.866 -0.913 -1.189 -0.902 -0.366 -0.342 -0.623 -2.162 -0.876 -1.104 -4.577 -2.758 -1.080 -0.900 -1.660 -1.179 0.630

1.407 0.995 1.389 1.433 0.826 3.916 0.500 2.083 2.981 1.284 0.850 1.763 4.280 1.019 1.285 0.760 7.196 4.072 N/A

∑ exp(ai + t1 + bit2) - 1 F2 ) ∑ bi exp(ai + t1 + bit2) - c

0.320 0.213 0.339 0.281 0.128 0.951 -0.020 0.520 0.691 0.441 0.188 0.348 -0.679 0.343 0.361 0.176 2.101 0.672 3.683

where t1 ) λ1/RT, t2 ) λ2/RT, ai ) -(1 + ∆G oi /RT), bi ) BPi, and c ) BPlump. The solution to the optimization problem is (λ1, λ2) ) f(T, t1, t2), such that

1.995 1.601 1.761 1.958 1.244 3.473 0.129 2.684 2.892 2.430 1.399 2.160 1.417 2.350 2.152 1.537 5.195 3.068 5.236

ave abs in 2M-Hex. 3M-Hex. 3E-Pen. 22M-Pen. 23M-Pen. 24M-Pen. 33M-Pen. 223M-But. 9 isomers

Ave. Rel. Dev. ) ∑[|predicted-measured|/measured]/15; South Houston, Wafra, and Wilmington are excluded

2.378 0.966 2.063 2.680 -0.001 8.748 -1.839 6.388 7.502 2.654 1.333 4.752 -3.898 3.067 2.963 3.697 18.426 6.996 0.081

Alida Bever Lodge Darius Eola Mclish Eola oil creek Hendricks Kawkawlin Lee Harrison North Smyer Pembina Ponca city Redwater South Houston Swanson River Teas Uinta Basin Wafra Wilmington ARDa in 15 crude oils

a

Hex.

crude/isomers

Table 4. Prediction Deviations (Calculated - Measured) for Normalized Distribution of Hexane/Heptane Isomers (wt%)

1672 Energy & Fuels, Vol. 19, No. 4, 2005 Ha et al.

F (F)(t) ) F1 ) 0 2

F1 )

( )

F(Z1) - F(Z2) ) F′(ξ)(Z1 - Z2) ) 0

m Z1 - Z2 ) m1 2

( )

∑ ea +ξ +b ξ ∑ biea +ξ +b ξ ∑ biea +ξ +b ξ ∑ b2i ea +ξ +b ξ

(

m2

i

EF049712R

1

i

∑ b2i ea +ξ +b ξ i

i 2

1

i

i

1

i

i

i 2

and

∑ m1ea +ξ +b ξ +

∑ m2biea +ξ +b ξ ) 0 (A-13)

∑ m1biea +ξ +b ξ +

∑ m2b2i ea +ξ +b ξ ) 0 (A-14)

1

i 2

i

1

i 2

i 2

m1 ) - m2

- m2(

∑ b2i ea +ξ +b ξ 1

i 2

*(

1

i 2

1

i 2

i

∑ biea +ξ +b ξ i

∑ biea +ξ +b ξ )2 ) 0 i

m1 ) m2 ) 0

1

i

1

1

∑ biea +ξ +b ξ )2 i

i 2

1

i 2

i 2

i 2

1

(A-7)

(A-8)

(A-9)

Assuming Z1 and Z2 are two roots of (F)(t):

(A-10)

Let

(A-11)

If multiple solutions exist, eqs A-10 and A-11 give

)( ) m1 m2 ) 0 (A-12)

Then

N eai+ξ1+biξ2 ) 1, eq With the stoichiometric constraint ∑i)1 A-13 can be rearranged to give

(A-15)

Substituting eq A-15 into A-14 yields

(A-16)

Physically, 0 < exp(ai + ξ1 + biξ2) < 1; and bi ) BPi (K) > 0 for all hydrocarbons. Thus, i 2

Therefore, to satisfy eqs A-15 and A-16, the following equation has to be true

(A-17)

One can conclude, then, that multiple solutions do not exist. The nonlinear eqs A-5 and A-6 have a unique solution of the Lagrange multiplier (λ1 and λ2). Consequently, the composition xi determined by eq A-4 is also unique.

Acknowledgment. Partial funding for NCUT has been provided by the Canadian Program for Energy Research and Development (PERD), the Alberta Research Council (ARC) and the Alberta Energy Research Institute (AERI). The authors thank Professor R. W. Missen (University of Toronto) and Professor W. R. Smith (University of Ontario Institute of Technology) for their valuable comments.

Estimation of Isomeric Distributions in Petroleum Fractions - Energy

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