Statistical Associating Fluid Theory. 1. Application toward Describing

and compiled in the data report from API project 42,3 widely regarded ..... 1 + (1 - 2)Fb. (5) .... employ a two-dimensional graph to represent molecu...
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Ind. Eng. Chem. Res. 2001, 40, 2947-2955

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Statistical Associating Fluid Theory. 1. Application toward Describing Isoparaffins Peter A. Gordon† Corporate Strategic Research, Exxon Mobil Research & Engineering, 1545 Route 22 East, Annandale, New Jersey 08801

We apply recent advances in perturbation theory-based equation of state models to correlate thermodynamic property data for pure compounds. Parameters for the equation of state were regressed for over 240 compounds, including a number of homologous series and branched paraffins. We find that the equation of state is capable of quantitatively correlating all of the pure-component experimental data we tested. Our primary interest, however, is to look for trends in the parameters of the equation of state in order to extrapolate to properties of other types of molecules. For isoparaffins, increased branching for a given carbon number leads to a SAFT model molecule with fewer segments but with larger segment volume and energy parameters. These trends correlate highly with topological descriptors characterizing the degree of branching, suggesting a route toward estimating parameters for molecules for which data are in short supply or do not exist. I. Introduction Although physical properties of paraffin isomers might seem relatively easy to characterize because of their chemical homogeneity, the problem becomes remarkably complex when considering such species in the context of lubricants. Despite chemical similarities, isomers of alkanes will have variations in both thermodynamic (vapor pressure, liquid density, heat of vaporization, etc.) and transport (diffusion, viscosity, thermal conductivity, etc.) properties. The importance of hydrocarbon structure has been well documented in the field of synthetic base stocks. Traction behavior in elastohydrodynamically lubricated contacts, closely related to energy efficiency in machine elements, can be extremely high or low, depending on the molecular structure of the lubricant. In the search for low-viscosity, low-volatility hydrocarbons for use in passenger car engine oils, considerable spread has been observed in the volatility characteristics for similar molecular weight hydrocarbons. In these examples, the principal discriminator among the test candidates is hydrocarbon structure. For fuel-ranged components, the problem of characterizing variations in properties as related to isomer structure may be somewhat more tractable; in this case, the molecular weight distribution of components is mostly confined in a range of carbon numbers between 4 and 10. For example, in this range, there are 147 structurally distinct species of paraffins. While this certainly represents a large number of components, many properties of these species have been measured and tabulated over the years.1-3 As the carbon number increases, however, the number of isomers of a given carbon number increases rapidly. This fact is depicted in Figure 1, where the number of isomers is plotted for paraffins and alcohols as a function of the number of carbon atoms.4 We see that for a carbon number of 30, for example, in the molecular weight range of low† E-mail: [email protected]. Phone: 908730-2546. Fax: 908-730-3031.

Figure 1. Number of structural isomers of paraffins (0) and alcohols (4) as a function of the carbon atom number. Data are taken from Trinajstic.4

viscosity synthetic base stocks (and hence the lower end of the lube-ranged components), there are already 4 billion isomers of the paraffin imaginable. At a carbon number of 50, this number exceeds 1018, or a billion billion isomers. By comparison, approximately 50 species of “lube-ranged” paraffins have been characterized and compiled in the data report from API project 42,3 widely regarded as the most comprehensive source of high molecular weight properties available in the literature. Thus, we can see the enormous gap in knowledge that exists in even pure properties of hydrocarbons at high molecular weight. An example of how the variation in isomer structure impacts physical properties is shown in Figure 2. The normal boiling point of various isoparaffins is plotted as a function of molecular weight. The data come from API project 441 and API project 42; in the latter data set, normal boiling points were extrapolated from available vapor pressure data. A feature that becomes apparent from the low molecular weight isomer data is that the temperature range between the highest and lowest boiling isomer increases with increasing carbon

10.1021/ie001043r CCC: $20.00 © 2001 American Chemical Society Published on Web 05/11/2001

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Figure 2. Distribution of normal boiling points of paraffins as a function of molecular weight. Each point represents a datum for a pure species. The top and bottom curves represent n-alkanes and a possible extrapolation of the lowest boiling point isomers based on data from C6-C10.

number. On the basis of extrapolation of the complete set of isomers between C6 and C10, we have included one possible lower bound on the boiling point. Because the amount of data available for isomers greater than C11 is but a tiny fraction of the total number of isomers conceivable, it is difficult to estimate how large the boiling point range truly is for isomers in the lube range. The plot does illustrate how structural variations may lead to significant differences in physical properties for lube-ranged isomers. Depending on the application, however, differences in isomer distributions may or may not be important. For complicated mixtures as are typically found in crude oil fractions, thousands of components are present, and it is an enormously difficult task to finely resolve product composition. The state-of-the-art techniques available at Mobil through HDHA5 (high detail hydrocarbon analysis) reveal composition interpreted through molecular weight distributions of homologous series for well-defined structural classes but are limited in their ability to resolve isomer distributions within the molecular classes. Despite these restrictions, models based on this structure-oriented-lumping (SOL) classification have been applied with great success in predicting the properties of crudes and crude cuts made through the refining process.6 This leads to the obvious question, if models already exist that can correlate properties and discriminate between potentially good and bad base stocks, then why should we worry about isomer distribution? One possible answer may lie in the difficulties associated with evaluating hydroprocessed base oil quality from compositional indicators. For example, many product quality characteristics of mineral oils, such as oxidative stability, solvency, and volatility, can be correlated to functions of simple compositional descriptors, including aromatics, sulfur, or basic nitrogen content. Hydroprocessed base oils typically contain equivalently negligible amounts of sulfur and nitrogen and a relatively low aromatics content.7 Thus, many of the compositional indicators that can be used to discriminate mineral oil base stock quality cannot be applied to differentiate hydroprocessed stocks derived from different starting materials or processed under different conditions. Yet, marked differences in performance characteristics have been observed in this class of base stock.

Synthetic lubricants offer much greater specificity and purity of molecular composition. The increased ability to control molecular structure during synthesis allows one to program specific performance attributes into the base stock. However, effective design of next-generation synthetic lubricants will require a deeper understanding of the relationship between molecular structure and performance attributes. In mineral oils, where other compositional effects tend to dominate the performance characteristics of the base stock, isomer distribution can be safely ignored. For severely hydroprocessed base stocks, it would appear that the isomer effect rises in importance, and an understanding of how isomer distributions impact product properties appears to be the key in understanding product composition performance relationships. As lubricants move increasingly toward utilizing hydroprocessed and synthetic stocks, an understanding of these more subtle compositional effects will become more important in designing next-generation lubricants. In this paper, we apply recent advances in perturbation theory-based equation of state models to correlate thermodynamic property data for paraffins and a variety of alkyl aromatics. We compare the performance of two different versions of SAFT, benchmarking against the Peng-Robinson (PR) equation of state, a commonly used model in engineering applications. Our primary interest, however, is to assess the potential of these models for extrapolative prediction of lube-ranged hydrocarbon properties. In particular, we are looking for qualitative trends relating the parameters in the equation of state to the degree and type of branching in paraffins. II. Methods: Models Tested and the Parameterization Procedure In the past decade, statistical associating fluid theory (SAFT) has become the basis for a powerful equation of state. The approach is based on thermodynamic perturbation theory, which seeks to describe the thermodynamics of a complex fluid in terms of a closely related but well-defined system. SAFT has resulted from the pioneering work of Wertheim,8-12 who developed a thermodynamic description of associating species. This was extended to describe molecules capable of forming highly directional, short-ranged associative bonds.13,14 In this work, we have chosen two forms of SAFTbased models to compare their performance in describing straight-chain and branched paraffins. The first is due to the work of Huang and Radosz,15,16 denoted as HS-SAFT, and the latter was developed by Kraska and Gubbins,17,18 denoted as LJ-SAFT. The Helmholtz free energy for both SAFT models can be written in terms of separate contributions resulting from ideal gas interactions, monomer segment interactions, and chain forming between segments

ASAFT ) Aideal + Amonomer + Achain

(1)

The free energy of the monomer segments can be written as a perturbation about a hard-sphere reference

Amonomer ) m(Ahs + Adisp)

(2)

Specific expressions for each of these terms can be found in the original papers and references therein. We briefly

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2949 Table 1. Compounds Selected for Parametrization with SAFT Equation of State with Data Obtained from a Variety of Sources series

no. of species selected

C# range

range of vapor pressure data

data source

n-alkanes n-alkylcyclopentanes n-alkylcyclohexanes n-alkylbenzenes C6-C10 isoparaffins API project 42 isoparaffins

15 17 17 17 142 45

4-36 6-21 6-21 6-23 6-10 10-38

0.52 < T/Tc < 0.96 10 < Psat < 1500 mmHg 0.02 < Psat < 2 bar 0.02 < Psat < 2 bar 0.02 < Psat < 2 bar 0.5 < Psat < 10 mmHg

DIPPR TRC TRC TRC TRC API-42

outline the major differences between the equations here. In the HS-SAFT model, the monomer free energy due is expressed as the sum of two terms: the hard-sphere reference is described by the Carnahan-Starling equation of state, and the dispersion term is given by a power series expression in terms of both packing fraction and segment interaction energy.19 This was parametrized to molecular dynamics simulation data of a variety of simple fluids interacting through square-well potentials.20 The free-energy monomer contribution for the LJ-SAFT model employs the Kolafa-Nezbeda equation of state for Lennard-Jones fluids.21 As with the HSSAFT model, the monomer free-energy contribution is split into a hard-sphere reference and dispersion correction. In this case, however, the hard-sphere reference free-energy expression is a somewhat more accurate representation, and the monomer energy contribution has been parametrized to computer simulation data covering a wider range of temperature and density. Both HS- and LJ-SAFT describe the free energy due to covalent bonding between monomer segments as

Achain ) (1 - m)RT ln[g(r)d;F,T)]

(3)

where g(r)d) is the pair correlation function evaluated at d, the contact distance between adjacent segments. The principal difference between the SAFT models employed here is the type of pair correlation function used. The HS-SAFT equation employs the hard-sphere contact value of the pair correlation function given by Carnahan and Starling, while LJ-SAFT utilizes an expression for g(r)σ) of the Lennard-Jones fluid, evaluated at the segment diameter of the Lennard-Jones fluid. The latter expression has been fit to extensive computer simulation data of Lennard-Jones fluids spanning a wide range of density and temperature.22 Several molecular parameters must be defined to apply these formulations of the SAFT model to reproduce experimental data. Interactions of the segments depend on σ, the collision diameter (or segment volume, voo), and uo, a characteristic energy of interaction. The formation of chains requires specification of m, the number of segments in a chain. This gives us three parameters that must be specified in order to describe a molecule, m, voo, and uo. Despite the differences outlined above, both models employ a free-energy description of chain formation derived from first-order thermodynamic perturbation theory (TPT1). At this level of sophistication, there is no accounting for formation of ring structures or specific angle constraints between bonded segments. The accuracy of this approximation has been tested by comparing the theory predictions to computer simulations of a variety of bonded systems.23-27 In general, TPT1 is quite accurate, but deviations from computer simulation predictions are observed for systems where appreciable steric hinderence exists from segments bonding at

adjacent sites. In this case, the assumption that the contact value of the pair correlation function can be approximated as the value of a collection of hard-sphere or Lennard-Jones monomers appears to break down. This situation would arise for highly branched species, particularly at high density. Higher order perturbation theory has also been explored. Second-order perturbation theory (TPT2) can explicitly account for angle constraints between adjacent bonding sites. For example, Muller and Gubbins28 compared TPT1 and TPT2 predictions for bonded hardsphere triatomics with fixed bond angle. TPT2 successfully captures the dependence of the thermodynamic behavior on the bond angle; however, it was concluded that the overall improvement in the model due to the added detail offered by TPT2 is relatively small for angles greater than 120°. In addition, TPT2 requires knowledge of the triplet correlation function, a quantity that has been characterized in relatively few instances.29 In principle, higher ordered perturbation theory can be applied to systems with multiple branch points, but almost no information is known about the fourth- and higher-ordered correlation functions required for their implementation.28 We also note that, although numerous heterosegmented formulations of SAFT have been developed,30-32 both models tested here are homonuclear; there is no differentiation between the energy and volume parameters between different types of chemical subgroups, such as mono-, di-, tri-, and tetravalently bonded carbon segments. These heterosegmented models still employ TPT1 in their description of chain formation, and as such are expected to have the same strengths and weaknesses when applied to highly branched species. Thus, in applying HS- and LJ-SAFT to correlate data for molecules such as isoparaffins, we must recognize that we depart in some respects from the physical basis of the TPT1 model. The thermodynamics of the chain formation contribution does not explicitly include branch structure. Despite these physical limitations, the hope is that we can find a set of SAFT parameters that maps the branch structure to a model that mimics freely jointed chains through an effective volume, segment energy, and segment number characteristic of each molecule. SAFT model parameters for pure compounds are regressed to experimental vapor pressure and liquid density data obtained from a variety of sources.1-3 The compounds selected include a number of homologous series, including n-alkanes, n-alkylcyclohexanes, nalkylcyclopentanes, n-alkylbenzenes, and a number of isoparaffins. Table 1 provides a brief summary of the molecular weight and vapor pressure data range of the compounds. The liquid density data employed generally spanned a smaller range of temperatures. The optimal values of the SAFT parameters (m, voo, and uo) are found through minimization of an objective function describing the average absolute deviation (AAD) with experimental

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Figure 3. AADs of predicted liquid density (a) and saturated vapor pressure (b) of n-alkanes of varying carbon number. Each curve corresponds to results from a specific equation of state model: (0) HS-SAFT; (O) LJ-SAFT; (2) PR.

vapor pressure and liquid density data

(

AAD % ) 100

1

np

∑

calc expt |psat,i - psat,i |

npi)1

expt psat,i

+

1

nd

∑

nd i)1

)

calc expt |Fliq,i - Fliq,i | expt Fliq,i

(4)

where np and nd are the number of data points (at selected temperatures) for which vapor pressure, Psat, and liquid density, Fliq, are computed and compared to experiment. A possible objection to this approach could be that the number of segments per molecule, m, is allowed to fluctuate between whole numbers. This was adopted to allow greater flexibility in the fitting procedure. Huang and Radosz15 adopted a similar regression approach. From a physical standpoint, this implies a rough interpretation of the chain formation process occurring through the overlapping of bonded segments. As a comparison of the quality of fit, the PR equation of state was used to predict the vapor pressure and saturated liquid density for the same data. The PR equation of state is well-known and frequently employed in engineering calculations because of its ease of implementation. The Helmholtz free energy per unit volume, APR, has the form

APR(F,T) ) FkT ln

1 + (1 + x2)Fb Fa F ln 1 - bF 2x2b 1 + (1 - x2)Fb (5)

where a and b are simple functions of the critical temperature, critical pressure, and acentric factor, defined for each pure fluid. III. Results (a) Homologous Series. As a starting point, we compare in Figure 3 the resulting quality of fit for linear alkanes between n-butane and n-hexatriacontane using the HS-SAFT, LJ-SAFT, and PR equations of state. For each species considered, the AADs in liquid density

Figure 4. Trends in LJ-SAFT equation of state parameters m (O), νoo (0), and uo/k (]) for n-alkanes as a function of carbon number. In each case a trend line is shown as a simple function of the carbon number.

(Figure 3a) and saturated vapor pressure (Figure 3b) are computed from data that span a reduced temperature range of approximately 0.52 < T/Tc < 0.96. For both liquid density and saturated liquid density, we find that, as the alkane chain length increases, the PR equation of state becomes noticeably worse relative to the SAFT models. The LJ-SAFT model provides the best overall fit to the experimental data. While both models yield very accurate predictions for the saturated vapor pressure, the HS-SAFT model does not reproduce both the liquid density and vapor pressure simultaneously with the same level of accuracy. This is consistent with the more realistic structural information contained in the LJ-SAFT model; the use of the Lennard-Jones radial distribution function conveys more accurate structural information in this model. Although the error in the vapor pressure and liquid density for the SAFT models rises somewhat with chain length, we note that the uncertainty in the experimental data for the largest n-alkanes is commensurate with the uncertainty in the experimental data used in the regression.2 In Figure 4, we examine the trends in the regressed LJ-SAFT equation of state parameters m, voo, and uo for the n-alkanes. The segment number increases linearly with the carbon number, and the segment volume and energy parameters display asymptotic behavior at higher carbon numbers. Although not plotted, we observe very similar trends in the HS-SAFT parameters; the only qualitative difference is that the segment volume parameter decreases asymptotically to a roughly constant value with increasing carbon number. These trends are consistent with others reported.15,17,31 In the figure, some modest increase in the voo and uo is still observed above C30, but this seems to be due to the greater degree of uncertainty in the experimental data in this range. In addition, we note slight variations in parameters obtained depending on the data source. For example, we have performed parameter regressions on the same n-alkane series with data from TRC1 tables. The best fit obtained for the effective segment number, for example, is m ) 0.7776 + 0.3597C# when derived from TRC data. These variations underscore the need to have a consistent set of data when attempting to develop correlations for SAFT parameters.

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2951 Table 2. Summary of the Overall Agreement with Experimental Data for LJ-SAFT and PR Equations of State for Various Homologous Series AAD %

series

mol wt range (g/mol)

LJ-SAFT HS-SAFT PR Psat Fliq Psat Fliq Psat Fliq

n-alkanes n-alkylcyclohexanes n-alkylcyclopentanes n-alkylbenzenes

58-507 84-309 70-294 78-302

3.2 0.4 0.3 0.3

1.9 0.5 0.4 0.4

3.3 0.6 0.6 0.8

4.8 14.6 11.6 1.6 4.9 19.5 1.4 4.7 16.5 1.4 3.5 17.7

In both cases, however, the trend lines yield good fits to the parameters and provide a means of extrapolation to higher carbon number. For example, using fits to the parameter trends similar to those shown in Figure 4, Huang and Radosz15 estimated the critical constants of long-chain alkanes. These properties are notoriously difficult to measure directly because the molecules thermally decompose below the critical point. Similarly, Kraska and Gubbins17 used a similar extrapolative approach to successfully predict the properties of neicosane based on trends in parameters up to n-decane. It is important to note that the success of this extrapolation is rooted in the sound physical basis of the equation of state. The quality of fit of each equation of state is summarized for a number of different homologous series in Table 2. For both SAFT models, the AAD for predicted vapor pressure and liquid density for the homologous series is quite low. The LJ-SAFT and HS-SAFT equations of state represent a substantial improvement over the PR model. The somewhat higher errors associated with the n-alkane series can be attributed to the range of data employed in the regression. We note that, except for the n-alkanes, the temperature range of the vapor pressure data employed in the parameter regression is typically between 125 and 200 K, and the upper temperature is significantly below the critical temperature. It is well-known that most equations of state have difficulties in describing behavior near the critical point. The restricted temperature range, along with fitting with data away from the critical point, help explain the improved quality of fit relative to n-alkanes. We note that, despite the physical departures implied in applying the SAFT model to alkyl aromatics and n-alkanes, the SAFT models do an excellent job in correlating experimental data. We re-emphasize the importance of the physical underpinnings of the equation of state in being able to accomplish this. (b) Effect of Chain Branching in Paraffins. Thus far, we have seen that the LJ-SAFT equation of state can be quantitative in computing pure-component properties for homologous series, and the smooth variations in the SAFT parameters for chemically similar molecules enable successful extrapolative property prediction, a property seldom seen in correlative work. These features lead us to ask whether similar trends exist with respect to chain branching in paraffins. As mentioned in the Introduction, the amount of data available for paraffins drops off significantly above isomers of decane. The properties of fuel-ranged paraffinic components, however, have been extensively characterized. For this reason, we have used the complete set of C6-C10 paraffin isomers as a data set to investigate the relationship between regressed SAFT parameters and molecular structure. Experimental liquid density and vapor pressure data of the 142 paraffin

Figure 5. Correlations between regressed LJ-SAFT parameters for paraffin isomers between C6 and C10: (a) Segment volume vs segment number; segment energy vs segment number. Symbols correspond to specific paraffin groups, including ([) n-alkanes, (4) C6, (0) C7, (O) C8, (]) C9, (×) C10 isomers. Table 3. Summary of the Overall Agreement with Experimental Data for LJ-SAFT and PR Equations of State for Paraffin Isomers AAD % LJ-SAFT

PR

isomer group

no. of isomers

Psat

Fliq

Psat

Fliq

C6 paraffins C7 paraffins C8 paraffins C9 paraffins C10 paraffins

5 9 18 35 75

0.3 0.3 0.3 0.2 0.2

1.1 1.3 1.7 0.3 0.2

1.6 1.9 1.6 2.3 2.3

4.0 4.0 4.8 2.3 2.1

isomers in this range are obtained from the TRC data compilation.1 To compare with the PR equation of state, it was necessary in some instances to estimate the acentric factor by extrapolating available vapor pressure data. A summary of the overall quality of fit obtained for the LJ-SAFT model, along with the comparison to the PR equation of state, is given in Table 3. As was observed with the homologous series in Table 2, the agreement between experiment and the LJ-SAFT model is excellent. The HS-SAFT equation is also quite successful; for example, the AADs for liquid density and vapor pressure for the C10 isomer data considered are 0.61 and 0.51%, respectively. While the PR model yields reasonably good predictions, the quality of prediction appears to degrade somewhat with increasing carbon number. This degradation would likely continue with higher molecular weight species in a similar manner as was seen for the n-alkane series in Figure 1. Examination of the LJ-SAFT parameters regressed through the fitting procedure for the isoparaffins reveals interesting trends. Figure 5a shows the segment volume parameter, voo, plotted against m, the effective segment number. The lower curve represents the homologous series of n-alkanes. For a given carbon number, the values of voo fall into a line with relatively little scatter, voo increasing with decreasing effective segment number. Figure 5b shows the correlation between the segment energy parameter, uo, and m. In this case, the

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Figure 6. Correlation between regressed LJ-SAFT parameters for C6-C10 paraffins and the Wiener number, a topological index measuring the degree of branching in a hydrocarbon. Symbols are as defined in Figure 5.

same general trend of increasing uo with decreasing m appears; however, the scatter in the data for a given carbon number is somewhat larger. For both the segment energy and volume parameters, the degree of scatter in the data appears to increase somewhat with increasing carbon number. Again, although not shown, the HS-SAFT parameter trends show qualitatively the same trends. These trends can be related to the structure of the molecule through the use of topological indices. A topological index is simply a mathematical construct used to quantify the structure of a molecule. The perfect index will uniquely identify a molecular structure, allowing unambiguous connection between the molecular structure and a specific property of interest. Applied to hydrocarbons, many topological descriptors employ a two-dimensional graph to represent molecular structure, where the carbon skeleton is depicted as a Kekule structure. A commonly used descriptor of branched hydrocarbon structures is the Wiener number,33 defined as the number of bonds between carbon atoms, summed over all carbon-carbon pairs ncarb

W)

1 2

∑ ∑(no. of bonds between i and j) i,j

(6)

This quantity measures the effective size of the molecule; for a given number of carbon atoms, the Wiener number decreases with increasing branching of the structure. In Figure 6, the LJ-SAFT parameters voo and m are plotted against the Wiener index. We see that a strong correlation between the parameters and the molecular structure exists; within a given set of structural isomers, the segment number decreases and the effective segment volume increases with decreasing Wiener number. The effective segment energy parameter, uo (not shown), also increases with decreasing W. As with the trends in Figure 5, the correlation between the segment energy parameter and the Wiener number exhibits more scatter than in the trends with m or voo. The correlations of the parameters with the Wiener index imply that, within the framework of SAFT, increased branching leads to

Figure 7. Regression of LJ-SAFT parameters to API project 42 data for n-dodecane. The lines follow the parameter set {m, νoo, uo/k} over the course of the fit: (]) segment volume νoo; (0) segment energy uo/k. Note that, even though the parameters shift significantly over the course of the fit, the agreement with experimental data is quite good for all sets of parameters.

a molecular model representation characterized by shorter chains with bulkier, more strongly interacting segments. This interpretation is reasonable given the lack of explicit structural connectivity within the SAFT framework. (c) Extension to Larger Paraffins: API-42 Data Set. To extend the SAFT equation of state to larger branched paraffins, we fit parameters with respect to data obtained from API project 42.3 While a large number of species have been characterized through this project, we note that the project focused primarily on rheological properties. Consequently, vapor pressure and liquid density data are available over a relatively small temperature range. This leads to difficulties in the fitting procedure because of the possibility of overfitting the data. An example of such a problem is shown in Figure 7 for n-dodecane. In this example, the experimental data for density cover the temperature range 273 K < T < 372 K, and the vapor pressure data span 377 K < T < 420 K. The danger of fitting to such a narrow temperature range of data is that it becomes too easy to rationalize the data within the SAFT framework and its three adjustable parameters. At the start of the parameter fit, the parameter set {m, voo, uo/k} ) {5.15, 36.97 cm3/mol, 266.5 K} produces an AAD of 3.12% for the vapor pressure and 0.52% for the liquid density. Through the course of the regression, the overall error is reduced to 0.68% and 0.11% AAD for the vapor pressure and density, respectively. The corresponding parameter set at the end of the regression is {m, voo, uo/k} ) {3.952, 49.68 cm3/mol, 303.4 K}. Because the overall fit to the data is good over the entire parameter range, this suggests that we are overfitting the data to some degree. For example, using the final parameters generated by this regression to predict vapor pressure over a larger temperature range (351 K < T < 629 K) yields a degraded AAD of 13.2% with respect to available experimental data. To deal with this problem, we attempt to add a constraint based on the observations of Figure 5, which reveals a strong correlation between the SAFT model parameters for lower molecular weight species. Specifically, we note that the data in Figure 6 for paraffin isomers between C6 and C10 obey a relationship of the form

voo (cm3/mol) ) voo,n-alkane - 11.36(m - mn-alkane) (7)

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2953 Table 4. LJ-SAFT Parameters for Selected Isoparaffins (Parametrizations for the Unconstrained Three-Parameter Fit and Constrained Two-Parameter Fit Using Eq 12 Included for Comparison) LJ-SAFT: two-parameter fit + eq 12

LJ-SAFT: three-parameter fit AAD % νoo, cm3/mol uo/k, K Psat

Fliq

AAD %

species

m

PSU549 PSU581 PSU546 PSU556 PSU582 PSU545 PSU583 PSU500 PSU642 PSU643 PSU511 PSU588 PSU591 PSU163 PSU584 PSU25 PSU1 PSU2 PSU3 PSU4 PSU22 PSU23 PSU27 PSU51 PSU53 PSU55 PSU67 PSU109 PSU5 PSU63 PSU184

4-n-propylheptane 2-methyldecane 5-n-butylnonane 2,2,3,3,5,6,6-heptamethylheptane 2-methylpentadecane 7-n-propyltridecane 2-methylheptadecane 7-n-hexyltridecane 2,6,10,14-tetramethylpentadecane 2,6,11,15-tetramethylhexadecane 5-n-butylhexadecane 3-methyleicosane 10-methyleicosane 9-n-hexylheptadecane 2-methyltricosane 9-n-octylheptadecane 11-n-butyldocosane 9-n-butyldocosane 7-n-butyldocosane 5-n-butyldocosane 6,11-di-n-amylhexadecane 3-ethyl-5-(2-ethylbutyl)octadecane 11-n-amylheneicosane 7-n-hexyleicosane 11-(3-pentyl)heneicosane 5,14-di-n-butyloctadecane 11-neopentylheneicosane 3-ethyltetracosane 7-n-hexyldocosane 9-n-octyleicosane 2,2,4,10,12,12-hexamethyl7-(3,5,5-trimethylhexyl)tridecane 9-n-octyldocosane 2,6,10,15,19,23-hexamethyltetracosane 11-n-decylheneicosane 11-n-decyldocosane 1-n-decyltetracosane 9-n-octylhexacosane 10-n-heptyl-10-n-octyleicosane 13-n-undecylpentacosane

2.967 6.658 3.933 3.815 7.109 5.908 6.609 7.283 6.733 7.131 7.522 7.773 6.704 7.696 9.667 10.149 8.978 8.759 10.836 9.940 8.661 8.549 10.533 12.007 9.901 8.709 9.121 7.463 11.118 13.727 8.515

55.964 25.066 53.293 56.255 34.514 42.047 42.384 39.785 43.515 43.082 42.581 41.427 48.906 45.889 37.475 36.857 44.082 45.279 35.724 39.332 45.380 45.743 36.842 31.855 39.095 45.167 43.229 54.213 37.575 29.743 49.411

311.59 223.97 305.46 311.18 259.42 278.69 284.07 272.65 277.85 277.98 275.97 284.26 301.71 289.92 274.27 267.68 287.22 290.19 265.42 276.59 284.85 285.62 267.42 254.50 272.11 286.48 282.81 318.94 270.92 249.97 277.76

0.5 1.1 1.3 0.4 1.0 1.3 1.7 1.3 1.5 0.7 1.4 0.5 1.0 1.1 1.7 1.5 2.1 2.4 1.9 1.4 1.6 1.7 2.2 2.8 1.5 1.6 2.1 2.0 1.7 1.9 1.5

0.0 3.317 1.2 4.864 0.1 3.844 0.5 5.799 0.7 6.313 0.5 6.258 0.4 6.904 0.6 7.365 0.5 7.308 0.5 7.656 0.5 7.961 0.4 7.914 0.2 7.813 0.4 8.617 0.4 9.108 0.7 9.449 0.4 9.826 0.4 9.819 0.7 9.898 0.5 9.869 0.4 9.874 0.5 9.905 0.6 9.892 0.8 9.927 0.6 9.924 0.4 9.867 0.5 9.843 0.2 9.723 0.6 10.404 0.9 10.457 0.6 10.437

49.175 34.815 54.847 36.835 39.022 39.648 40.565 39.323 39.977 40.026 40.140 40.680 41.828 40.824 39.802 39.731 40.119 40.195 39.299 39.630 39.571 39.211 39.360 38.967 38.997 39.644 39.918 41.285 40.287 39.682 39.916

11.481 9.761 10.385 12.287 12.926 12.044 16.690 21.503

38.985 46.383 45.184 38.758 39.129 42.342 29.945 23.670

274.25 284.83 289.72 273.43 275.11 282.86 245.80 233.85

1.7 1.8 1.3 1.9 2.3 2.3 1.6 1.9

0.6 0.5 0.4 0.6 0.6 0.5 1.0 1.2

41.048 41.316 41.810 41.308 41.609 41.907 40.212 40.710

PSU6 PSU223 PSU8 PSU7 PSU107 PSU164 PSU211 PSU133

The segment volume parameter for an isoparaffin is expressed in terms of the deviation from the n-alkane parameters of the same carbon number. The n-alkane parameters employed in eq 7 correspond to those in Figure 4. With this constraint in place, we have reduced the number of adjustable parameters by one, leaving m and uo to be determined. Detailed results of the two-parameter regression for 45 species of isoparaffins in the API project 42 report are detailed in the appendix. The AAD for the predicted vapor pressure and density averaged over the 45 API project 42 paraffins are 3.3% and 0.9%, respectively. We also performed the parametrization without the constraint of eq 7; the corresponding AADs for this threeparameter fit were 1.6 and 0.5%, respectively. Thus, the additional constraint equation simplifies the parametrization procedure by reducing the number of parameters to be determined, at the expense of only a modest degradation in the quality of fit. When constraint eq 7 was applied to the parametrization procedure, the LJ-SAFT model had difficulties representing several molecular structures. The worst such case gave an AAD of 12.08% for the vapor pressure and of 5.33% for the density. The fits were generally poorest for structures that were very highly branched, indicating a possible limitation of the basic underpinnings of the equation of state or limitations of the constraint equation applied.

m

10.929 10.905 11.190 11.562 12.189 12.163 12.641 12.924

Psat

Fliq

293.78 262.35 309.27 249.83 274.65 270.89 278.20 271.23 267.31 268.95 268.89 281.93 280.86 275.51 281.46 275.97 276.25 276.18 275.68 277.44 269.14 267.94 274.58 275.29 271.83 271.41 273.70 283.58 278.47 278.76 254.56

1.5 6.5 1.3 8.7 2.6 1.9 1.9 1.3 2.1 1.6 1.7 0.6 3.9 2.8 2.0 2.0 2.9 3.1 2.9 1.3 3.6 4.0 2.2 4.7 1.5 3.8 2.8 6.7 2.0 6.7 5.6

0.4 3.83 0.3 5.3 1.2 0.7 0.6 0.5 0.8 0.7 0.6 0.4 1.5 0.9 0.7 0.7 0.7 0.8 0.8 0.5 0.9 1.1 0.7 1.2 0.6 0.9 0.7 2.1 0.7 1.6 1.5

279.82 271.94 280.93 280.15 281.52 281.75 272.76 280.69

1.7 3.3 2.0 2.0 2.2 2.3 7.1 12.1

0.6 0.8 0.6 0.6 0.6 0.5 1.4 2.2

νoo, cm3/mol uo/k, K

PSU no.

Figure 8. Correlation between segment number and segment volume parameters in the LJ-SAFT equation of state. Symbols correspond to specific paraffin groups, including ([) n-alkanes, (4) C6, (O) C8, (×) C10, and (]) paraffins from API project 42 data compilation. Note that parameters regressed for the species in the API project 42 data set have been forced into alignment with the added constraint equation for νoo in terms of m.

Nevertheless, the most important feature demonstrated here is that we have enforced an external constraint on the equation of state parameters through the use of eq 7. Figure 8 shows the resulting segment number and volume parameters for the API project 42 paraffins subjected to the constraint of eq 7. While the additional constraint reduces the overall quality of fit, the approach represents a first step in the development of a rule-based approach toward generating equation

2954

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001

of state parameters for which no data exist. Another possible approach is suggested by Figure 7, where the segment number may be related to a function of one or more topological descriptors of the molecule. Because the descriptors can be generated for any conceivable branched structure, this would create an additional link between a proposed molecular structure and the SAFT equation of state parameters.

comparing predictions of LJ-SAFT and PR to experimental vapor pressures and liquid density over a range of temperatures is also included for a number of nalkanes between n-C10 and n-C30. This material is available free of charge via the Internet at http:// pubs.acs.org.

IV. Conclusions

(1) Selected Values of Properties of Hydrocarbons and Related Compounds. Research Project 44 of the American Petroleum Institute and the Thermodynamic Research Center; Thermodynamics Research Center: College Station, TX, 1985. (2) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Daubert, T. E., Danner, R. P., Eds.; Hemisphere Publishing Corp.: New York, 1989; Vols. 3-5. (3) Dixon, J. A. Properties of Hydrocarbons of High Molecular Weight; Project 42; American Petroleum Institute: Washington, DC, 1966. (4) Trinajstic, N. Chemical Graph Theory; CRC Press: Boca Raton, FL, 1983; Vol. 2. (5) Jacob, S. M.; Quann, R. J.; Sanchez, E.; Wells, M. E. Compositional Modeling Reduces Crude-Analysis Time, Predicts Yields. Oil Gas J. 1998, 96, 51. (6) Quann, R. J.; Jaffe, S. B. Structure-oriented Lumping: Describing the Chemistry of Complex Hydrocarbon Mixtures. Ind. Eng. Chem. Res. 1992, 31, 2483. (7) Galiano-Roth, A. S.; Page, N. M. Effect of Hydroprocessing on Lubricant Base Stock Composition and Product Performance. Lubr. Eng. 1994, 50, 659. (8) Wertheim, M. S. Fluids with highly directional attractive forces. I. Statistical thermodynamics. J. Stat. Phys. 1984, 35, 19. (9) Wertheim, M. S. Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations. J. Stat. Phys. 1984, 35, 35. (10) Wertheim, M. S. Fluids with highly directional attractive forces. III. Multiple attraction sites. J. Stat. Phys. 1986, 42, 459. (11) Wertheim, M. S. Fluids with highly directional attractive forces. IV. Equilibrium polymerization. J. Stat. Phys. 1986, 42, 477. (12) Wertheim, M. S. Thermodynamic Perturbation Theory of Polymerization. J. Chem. Phys. 1987, 87, 7323. (13) Chapman, W. G. Theory and Simulation of Associating Liquid Mixtures; Chapman, W. G., Ed.; Cornell University: Ithaca, NY, 1988; p 172. (14) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Phase Equilibria of Associating Fluids. Chain Molecules with Multiple Bonding Sites. Mol. Phys. 1988, 65, 1057. (15) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (16) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules: Extension to Fluid Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. (17) Kraska, T.; Gubbins, K. E. Phase Equilibria Calculations with a Modified SAFT Equation of State. 1. Pure Alkanes, Alkanols, and Water. Ind. Eng. Chem. Res. 1996, 35, 4727. (18) Kraska, T.; Gubbins, K. E. Phase Equilibria Calculations with a Modified SAFT Equation of State. 2. Binary Mixtures of n-Alkanes, 1-Alkanols, and Water. Ind. Eng. Chem. Res. 1996, 35, 4738. (19) Chen, S. S.; Kreglewski, A. Applications of the Augmented van der Waals Theory of Fluids. I. Pure Fluids. Ber. Bunsen-Ges. 1977, 81, 1048. (20) Alder, B. J.; Young, D. A.; Mark, M. A. Studies in Molecular Dynamics. X. Corrections to the Augmented van der Waals Theory for the Square Well Fluid. J. Chem. Phys. 1972, 56, 3013. (21) Kolafa, J.; Nezbeda, I. The Lennard-Jones Fluid: An Accurate Analytic and Theoretically-Based Equation of State. Fluid Phase Equilib. 1994, 100, 1. (22) Johnson, J. K.; Muller, E. A.; Gubbins, K. E. Equation of State for Lennard-Jones Chains. J. Phys. Chem. 1994, 98, 6413. (23) Amos, M. D.; Jackson, G. Bonded Hard-Sphere (BHS) Theory for the Equation of State of Fused Hard-Sphere Polyatomic Molecules and Their Mixtures. J. Chem. Phys. 1991, 96, 4604.

In this work, we have employed equations of state based on SAFT to correlate pure-component thermodynamic property data over a wide range of species and molecular weight. For nonassociating fluids such as paraffins and alkyl aromatics, the equation of state has three adjustable parameters: the number of segments per molecule, the segment volume, and the segment interaction energy. The parameters in the LJ- and HSSAFT models for a number of species were regressed to vapor pressure and liquid density data, and quantitative agreement was found in all cases. A conceptual advantage of the SAFT equation of state model is that each of the parameters represents a physical feature of the model fluid. For branched alkanes, the fitted parameters in both the LJ-and HSSAFT equations of state were found to follow clear trends with respect to the degree of branching. For a given carbon number, increased branching leads to a SAFT model molecule with fewer segments but with larger segment volume and energy parameters. While quantitative accuracy is certainly desirable, our primary goal is to develop a method by which one can estimate the equation of state properties where experimental data are unavailable. The case of paraffins offers an interesting test for such an approach, because a great deal of experimental data are available for fuel-ranged species but are generally lacking for lube-ranged components. Because there are still a multitude of hydrocarbon structures still to be explored, we hope to be able to link as-yet-unsynthesized structures to thermodynamic properties to aid in the search for next-generation lube components. The possibility of linking molecular structure through the use of topological descriptors looks like a promising route to aiding in this search, and its application to the LJ-SAFT equation of state utilized here will be the subject of an upcoming paper. Given estimates of these parameters, we can envision modeling distributions of molecules in order to link measurable properties, such as the average branch ratio (and distribution), to thermodynamic properties. Acknowledgment The authors thank Exxon Mobil Research & Engineering for support of this work and permission to publish the results. Special thanks go to Roland Saeger and Margaret Wu for helpful discussions. Appendix: Selected SAFT Parameters LJ-SAFT parameters for selected isoparaffins from the API project 42, along with the associated AADs, are listed in Table 4. Supporting Information Available: Complete tabulated SAFT parameters for both the LJ and HS-SAFT models for all species in this work. A representative plot

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Received for review December 1, 2000 Revised manuscript received April 6, 2001 Accepted April 11, 2001 IE001043R