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Ind. Eng. Chem. Res. 2001, 40, 2956-2965
Statistical Associating Fluid Theory. 2. Estimation of Parameters To Predict Lube-Ranged Isoparaffin Properties Peter A. Gordon† Corporate Strategic Research, Exxon Mobil Research & Engineering, 1545 Route 22 East, Annandale, New Jersey 08801
Building on our work in a previous paper,1 we present a method by which statistical associating fluid theory (SAFT) parameters can be generated for any isoparaffin structure. The SAFT parameters that characterize a pure component are linked to functional relationships based on a number of easily computed topological indices. Simple relationships for the effective number of segments, m, and segment volume, voo, are formalized based on observed trends in regressed parameters in our previous work. We also employ a simulated annealing procedure to develop an accurate equation for uo/k, the segment interaction energy. The robustness of the parameter estimation method is demonstrated by comparing the quality of the equation of state predictions to the performance of the original SAFT parameter set determined in part 1. Perhaps most importantly, this approach opens the door for quantitative structure-property relationship studies, with particular emphasis on how branching affects specific properties. This is demonstrated by examining the influence of branching on the normal boiling point for a variety of lube-ranged isoparaffins. I. Introduction
II. Background
In a previous paper1 (referred to hereafter as part 1), we reported that an equation of state based on thermodynamic perturbation theory, statistical associating fluid theory (SAFT), appears to have considerable promise in describing the thermodynamic behavior of branched paraffins. Parameters for the equation of state were regressed to available vapor pressure and liquid density for over 240 pure compounds, comprised primarily of isoparaffins and a number of homologous series. In addition to quantitative reproduction of the data, an important result was that the model parameters vary smoothly with respect to carbon number within a given homologous series and the degree of branching for a given carbon number. This suggests a path toward estimating parameters to predict the thermodynamic behavior of molecular species for which limited or no experimental data are available. In this work, we seek to quantify the trends in the SAFT parameters previously observed in order to provide reasonable estimates of the thermodynamic behavior of branched paraffins relevant to synthetic lubricant components. The process of quantification is accomplished through the use of topological descriptors, which helps link structural features of the molecule to the input parameters of the equation of state. For isoparaffins, we report a refined SAFT parameter set for our original set of test molecules, along with recommendations for determining new parameters depending on whether experimental data are available. The quality of the equation of state predictions is compared to the performance of the original SAFT parameter set in part 1. Finally, we examine trends in normal boiling points of lube-ranged species as a function of chain branching for which no experimental data are available.
As stated in part 1, our main objective in applying the SAFT equation of state to fuel-ranged isoparaffins is not simply to accurately correlate data. We want to estimate model parameters in order to develop reasonable parameter estimates for lube-ranged molecules. If such parameter estimates are sound, a mechanism to study structure-property relationships for a wide variety of thermodynamic properties becomes accessible, including quantities such as boiling point, compressibility, heat capacity, heat of vaporization, and so on. Topological descriptors provide a useful way to quantify the connectivity of hydrocarbons. A large number of descriptors have been designed to transform some aspect of a molecule’s topology into a single quantity. One such example is the Wiener number, defined as the number of bonds between carbon atoms, summed over all of the carbon-carbon pairs in the molecule
† E-mail:
[email protected]. Phone: 908730-2546. Fax: 908-730-3031.
ncarb
W)
1 2
∑ ∑(no. of bonds between i and j) i,j
(1)
Figure 1 shows the Wiener number plotted as a function of the carbon number, C#, for several different types of paraffin structures. The structures in the figure are grouped into four different effective “homologous series” with similar types of branching, including nalkanes, tribranched star molecules, hydrogenated polyisobutylene (PIB), and a highly branched species. The utility of this topological index is evident in its ability to distinguish and catalog a variety of complex molecular architectures with a single measure. Topological descriptors are frequently employed to construct correlations of physical properties for a specific set of molecules. The correlation can then be used to predict the properties of related compounds where data are unavailable. A wide variety of such correlations have been developed, including such examples as boiling
10.1021/ie001044j CCC: $20.00 © 2001 American Chemical Society Published on Web 05/15/2001
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2957
Figure 1. Wiener number as a function of carbon number for four different effective homologous series. Within each series, the nature of branching follows a similar pattern with increasing carbon number: (O) n-alkanes, (4) tribranched stars, (]) hydrogenated PIB, and (0) structures with minimum Wiener number.
point, critical properties, density, solubility, heat of formation, partition coefficients, and various aspects of biological activity, such as toxicity and narcotic effects.2-5 This approach, however, has several inherent disadvantages. For one, a separate correlation must be developed for each physical property of interest. This can become cumbersome for general physical property estimation of a given species. In addition, correlations developed in this manner are not readily generalized to mixtures. Finally, the quality of the correlation developed depends on the availability of data covering the compounds of interest. For example, an abundant amount of liquid density data has been compiled for isoparaffins, but data for a property such as liquid heat capacity are scant in comparison. To avoid some of these difficulties, we propose an alternative approach toward thermophysical property prediction. We attempt to develop topology-based correlations that predict the parameters of the SAFT equation of state, which in turn can be used to compute thermodynamic properties of interest. This approach allows us to simultaneously predict a variety of properties with only a single set of parameter correlations, rather than requiring a separate correlation for each property. Given the pure-component SAFT parameters, mixture properties can be estimated at no additional cost. In addition, the parameter correlations can be used to generate estimates of thermodynamic properties for molecular species where no experimental data are available. We note that this approach is similar in some respects to the equation of state based extrapolation approach commonly employed in the petroleum-based applications. For example, many cubic equations of state such as Peng-Robinson are commonly used in petroleum fluid calculations. Extrapolative predictions with most cubic equations of state require estimates of the critical temperature, pressure, and acentric factor, and much work has been done to develop correlations to predict these input parameters.6 This approach, however, has two basic difficulties. First, the input parameters into the equation of state are based on specific measured properties of the fluid of interest. For heavy petroleum components, these properties cannot be easily measured. This is often because the species decompose before
Figure 2. Schematic representation of a molecule in the SAFT model. The carbon skeleton of the molecule 2,5-dimethyl-3ethylhexane, shown in this example, is modeled as a collection of m bonded segments, each with an interaction energy uo and volume voo. The overlap of segments denotes the possibility of having a fractional number of segments in a molecule.
the critical point is reached. This necessitates approximate methods to determine the required inputs for equation of state applications. Even for cases where reliable estimates for the critical properties are achieved, the equation of state often yields poor property predictions. This is due to the basic theoretical limitations of the model and becomes especially problematic for higher molecular weight species approaching the lube range.7 For SAFT, the critical properties are not required input parameters to the equation of state. In fact, the input parameters can be fit from any available thermodynamic data. In the absence of any data, we can still take advantage of the smooth variation in the parameters to estimate the input parameters. For isoparaffins, we must specify three parameters in order to describe a molecule within the SAFT framework. These parameters each have a physical correspondence to a feature of the molecule, as shown in the example of Figure 2. A molecule is modeled as a collection of m bonded segments (segment number), and each segment is characterized by a volume voo (segment volume) and interaction energy, uo (segment energy). We restrict our attention here to the SAFT description developed by Kraska and Gubbins, which employs a Lennard-Jones potential to describe interactions between segments.8 A brief description of the model, referred to as LJ-SAFT, can be found in part 1. For the compounds in our original data set, these parameters were determined by fitting to available vapor pressure and liquid density data for the pure compound. Parameters found in this manner were able to very accurately correlate the data. The next task is to quantify the link between isoparaffin topology and the SAFT input parameters in order to enable property prediction of luberanged species. III. Methods Reduction of Parameters: Application of Parameter Constraints. In our previous work, we observed a strong correlation between the degree of branching in an isoparaffin and the regressed value of m, the effective segment number, and voo, the segment volume. We can take advantage of these relationships by imposing constraints on the parameters. Referring back to Figure 1, we note that, at fixed carbon number, the n-alkane structure represents the largest possible value of the Wiener number, while the highly branched series represents the structure with the
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lowest value. These two limits represent a boundary that encapsulates the entire range of isoparaffin structures. Using these limits, we can define a type of “road map” for other types of branched paraffins. The tribranched stars and PIB series represent two examples of structures that lie somewhat closer to the n-alkane and highly branched limit, respectively. A ratio can be expressed in terms of the difference between Wn-alkane and Wmin, the upper (n-alkane) and lower (highly branched) bounds on the Wiener number
WΘ )
Wisoparaffin - Wn-alkane Wmin - Wn-alkane
(2)
We term WΘ the fractional Wiener number, and it is evaluated at fixed carbon number. It is defined such that WΘ varies from 0 for a normal alkane to 1 for a highly branched paraffin. This scaling has the advantage that the upper and lower bounds are independent of the carbon number. A correlation designed for predictive purposes should avoid, if possible, excessive extrapolation outside the range of independent variables originally used in building the correlation. Whereas the range of values spanned by W increases with carbon number, WΘ is bounded by the limits (0, 1), independent of molecular weight. With respect to these two variables, one should expect a correlation based on WΘ to exhibit more stable behavior when used to extrapolate to higher molecular weight species. We note here that the minimum value of the Wiener number among a collection of isomers must be determined in order to use eq 2. Defining this limit is not trivial, and details on this specification can be found in Appendix A. Recalling the trends in our original LJ-SAFT parameters, we note that, at fixed carbon number, the segment number decreases with decreasing Wiener number (or increased branching). This can be expressed through the relationship
mΘ )
misoparaffin - mn-alkane ) aWΘb mmin - mn-alkane
(3)
where we have defined a fractional effective segment number, mΘ, in analogy with WΘ in eq 2. Using this equation, the segment number for an isoparaffin obeys proper limiting behavior; in the n-alkane limit, misoparaffin f mn-alkane, and (with a ) 1) misoparaffin f mmin in the opposite extreme. Equation 3 is implicitly a function of carbon number, and we require a means of describing this dependence. The values of mn-alkane from our previous work are well fit to a linear dependence on carbon number
mn-alkane(C#) ) 0.7776 + 0.3597C#
(4)
We have chosen the expression that was fit to n-alkane data from the Thermodynamic Research Center (TRC);17 these data are from the same source as most of the isoparaffin data employed in our data set. The minimum value of m as a function of carbon number was determined only for 4 < C# < 10 and was also assumed to extrapolate linearly outside of this range
mmin(C#) ) 1.6753 + 0.1359C#
(5)
Regression of the parameters in eq 3 yields a ) 1.0145 and b ) 1.8113 and produced a good fit to the original
Table 1. Range and Average of the Regressed Quantity mvoo for Isomers of Carbon Number between 6 and 10a C#
mνoo, cm3/mol
max value, cm3/mol
min value, cm3/mol
6 7 8 9 10
102.66 116.16 129.39 145.06 159.27
103.41 117.41 132.04 147.84 162.27
102.25 114.62 127.24 142.19 155.84
a The deviation around the mean value rarely exceeds 2%, providing justification for the approximation that this quantity is approximately constant at fixed carbon number.
data regressed. For example, for the complete set of 142 isomers at 6 < C# < 10, the average deviation in the predicted values of m from eq 3 is 2.3%. Analysis of the LJ-SAFT parameters reported in part 1 also revealed that the quantity mvoo remains relatively constant among isomers of a given carbon number. Table 1 shows the average value of mvoo as a function of carbon number, along with the range in values among isomers. The maximum deviation in either direction from the average value does not exceed 2.2% in this carbon number range. This leads us to our second constraint; namely, we specify voo for an isoparaffin as
voo,isoparaffin (cm3/mol) ) (mvoo)n-alkane/misoparaffin
(6)
This constraint is equivalent to stating that the total volume of the isolated molecule is assumed to be independent of the branching. The segment number for the isoparaffin is computed from eq 3, whose only input is the topological measure WΘ. As with eq 2, this constraint is an implicit function of carbon number. For the n-alkanes, (mvoo)n-alkane is also well represented as a linear function of the carbon number
(mvoo)n-alkane(C#) (cm3/mol) ) 19.363 + 13.981C# (7) To test the accuracy of our parameter constraints for m and voo, we refit the pure-component parameter uo for our original set of paraffins, now using our formulas for m and voo as determined by eqs 2-7. In so doing, we have reduced the number of adjustable parameters in our equation of state from three to one. This removes some of the flexibility in the SAFT equation of state to fit the experimental data but provides a test of the validity of our parameter estimation rules for the segment number and volume. A poor fit resulting from the one-parameter regression suggests a problem with the constraints applied. As shown in Table 2, however, the average absolute deviation (AAD %) remains quite low for the one-parameter regression. The reduction in the number of parameters required to regress allows one to determine the last unknown (the segment energy) with less experimental data. For instance, knowledge of the normal boiling point or heat of vaporization of a species would be sufficient to determine the final parameter. Of course, the ideal situation is to be able to uniquely determine the remaining segment energy parameter solely as a function of molecular structure without requiring knowledge of specific physical properties. Unfortunately, this turns out to be a difficult task. The segment energy for the various isomers examined does not seem to obey a simple relationship with respect to a topological descriptor such as the Wiener number.
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2959 Table 2. Comparison of the Quality of Fit for Three- and One-Parameter LJ-SAFT Equations of State for Various Paraffins AAD %
paraffin group C4/C5 C6 C7 C8 C9 C10 n-alkanes C12< C# < C36 API project 42 isoparaffins
no. of species tested
three-parameter regression Psat Fliq
one-parameter regression Psat Fliq
5 5 9 18 35 75 10
0.2 0.3 0.3 0.3 0.2 0.2 1.1
0.8 1.1 1.3 1.7 0.3 0.2 1.5
0.6 0.4 0.3 0.7 0.7 0.8 1.5
1.4 1.4 1.8 2.9 0.7 0.8 1.8
43
1.6
0.6
3.4
2.1
Figure 3 shows a plot of uo (determined from the oneparameter regression) vs the Wiener number for a number of the C6-C10 isomers. For a given carbon number, as we move to compounds with lower Wiener number, the segment energy decreases slightly, followed by a sharp increase. There is a fair degree of scatter about the average value, however. This turns out to be quite problematic because the quality of fit obtained for vapor pressure is extremely sensitive to deviations in the segment energy. Numerous attempts were made to link the effective segment number to functions of other topological descriptors, but none could simultaneously provide a good fit to experimental data and yield a transparent relationship between the regressed value of the segment energy and a function of molecular structure. As a result, we turn to a simulated annealing approach in order to develop a quantitative structureproperty relationship (QSPR) for estimating uo for pure components. Simulated Annealing: Approach for QSPR Development. Simulated annealing is a computational procedure designed to find an optimal solution to a combinatorial problem. It is a technique that is particularly well suited to problems where it is not feasible to exhaustively search through every possible combination of variables that compose the range of outcomes. The technique was originally applied to study optimal layout configurations of circuit elements in computers,9 but the generality of the approach lends itself to easy migration to other problems. As examples, the technique has been applied to optimizing design and arrangement of heat exchanger networks,10 prediction of crystal structures of ionic compounds,11 and structural determination of proteins.12 In combinatorial optimization problems, the goal is usually to find an optimum value for a predefined objective function, quantifying the “goodness” of the solution. Common examples might include minimizing the cost of a project or constraining the physical space required for layout of a network of components. An iterative improvement scheme starts with an initial guess of the optimum solution and attempts to generate alternative solutions that improve the value of the objective function. Possible alternative configurations are usually generated through a random perturbation around the most recent estimate of the optimal configuration. This process continues until no further improved solutions are found. This approach tends to get stuck in locally optimal solutions. Addition of a Metropolis sampling scheme helps prevent the iterative scheme from getting trapped in a
Figure 3. Variation of segment energy regressed in a oneparameter fit with the Wiener number for isomers of paraffins of carbon number between 6 and 10: ([) n-alkanes, (4) C6, (0) C7, (O) C8, (]) C9, and (×) C10 isomers. Note that while a general trend of an initial decrease in uo followed by an increase with decreasing Wiener number exists within each isomer group, there is a considerable amount of scatter within the series.
locally optimum solution. A fictitious temperature is defined to allow for “uphill” moves in the objective function to escape getting trapped in local minima. When the temperature is slowly lowered in a controlled fashion during the search, solutions near the globally optimum solution may be reached at a fraction of the computational expense of methods that rigorously find the optimum solution. This approach is similar to the physical annealing of materials; starting with a liquid phase, dropping the temperature rapidly to quench the material (analogous to pure iterative optimization) leads to an amorphous glass trapped in a local minimum energy state. Slowly cooling the same material will lead to a crystal with a minimum of defects, close to its global energy minimum state.9 In the present case, we have a number of molecular descriptors of isoparaffins and would like to find some combination of these in a suitable functional form that yields a good prediction of the segment energy parameter. A measure of the quality of the correlation can be defined as the sum of the squared deviations between the calculated and the actual regressed value of the segment energy
E)
x
n
∑i (uo,icalc - uo,iactual)2
(8)
The functional form of the segment energy correlation is specified with a constant, n1 terms with a single independent variable Xi raised to a power ei, and ncross terms that contain combinations of the independent variables. n1
uo,calc ) uo,n-alkane + c0 +
∑ i)1
ncross
aiXei i +
e e bjX1,j X2,j ∑ j)1 1,j
2,j
(9)
The presence of the paired terms allows one to look for combinations of topological descriptors that “interact”, in the sense that the product of the descriptors may yield a better correlation than the individual terms can. As with eqs 2 and 6, this equation is a function of the carbon number; the term uo,n-alkane refers to the value of the n-alkane with the same carbon number as that of the isoparaffin of interest. The segment energy for
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Figure 5. Results of an optimized functional fit from the simulated annealing method for LJ-SAFT segment energies. The functional form (eq 9) was constrained to include three single variable terms and three cross terms. The squared correlation coefficient for data used in the fit (0) and test data (]) are shown for comparison.
Figure 4. Listing of topological descriptors employed in a simulated annealing fit of segment energy. A total of 23 descriptors were computed for each molecule, and values are shown for the case of 2,4-dimethyl-3-ethylhexane.
the n-alkanes can be obtained from the tabulated data in Appendix B, although for C# > 10 these values can be computed accurately through the relation
(uo)n-alkane (K) ) 279.60[1 - exp(-0.2737C#)]
(10)
The topological descriptors chosen as candidate independent variables in eq 9 are listed in Figure 4, along with values for a representative example, 2,4-dimethyl3-ethylhexane. Briefly, variables 2-7 refer to the Balaban index13 (Jx) and molecular connectivity indices (0χ, ...,3χcluster).14 Variables 8-12 count the total number of carbon atoms, and the number of each type of methyl subgroup (CH3, CH2, CH, and C). Variables 13-18 examine pairs of branch points; they count the number of instances where branched pairs occur on adjacent sites (Px,adj) or are separated by one intermediate site (Px,pr). Variable 19 counts the number of pairs of carbon sites in the molecule separated by three bonds and is often referred to as the polarity number.15 Variables 2023 represent ratios of various CHx-substituted groups to the total number of carbon atoms present in the molecule. Each of these descriptors is simple to calculate and conveys information about the size or connectivity of the molecule. In our simulated annealing procedure, a fixed number of single (n1) and cross terms (ncross) were chosen, and all exponents were initially set to 1.0. Independent variables are randomly selected from the 23 listed in Figure 4. Using this as a starting functional form, the parameters c0, {ai}, and {bj} are regressed and the quality of fit E (eq 8) is computed. From this starting point, modifications to the functional form are generated through a number of “moves”. Moves include shifting a single variable term to a new variable, shifting a cross term pair to a new pair of variables, and changing the exponent of a single variable to a new value. In each case, a variable (or term) of the appropriate type is selected at random, and a new independent variable (or term) is selected (again, at random) as a replacement. The new equation form is then used to derive the
parameters c0, {ai}, and {bj}. The new functional form is retained according to a Metropolis scheme; if the energy E is lower than the fit using the old form of the equation, the new functional form is accepted. If E is higher, then the move is accepted with probability exp(-∆E/T), where ∆E is the energy difference between the new and old functions. The temperature of the annealing process is initially set to a high value and is lowered in a controlled fashion until the acceptance rate of certain moves drops below a predefined threshold. The system should be near its global minimum if the temperature has been dropped gradually enough. We have restricted the permissible exponents to discrete values in the range 0.125 < ei < 3.0, with increments of ∆ ) 0.125 as the only allowed values. This was done in order to simplify the functional form search process and, more importantly, to avoid large exponents that may be dubious when using the equation to make extrapolative predictions about compounds with a high degree of branching. To test the correlation developed through the simulated annealing process, the 376 species in our data set were divided equally into two groups. The first set was used to create the equation and is referred to as the training set. The second was set aside and used to test the resulting equation on molecular species that were not included in the equation development. Additional details on the data can be found in Appendix A. The simulated annealing process was repeated several times for a fixed number of cross and single terms to ensure that the process would yield consistent results. Thereafter, the number of terms in the fit equation was increased and the process repeated until it was judged that additional terms did not measurably enhance the overall correlation. This limit was deemed to exist at three single terms and three cross terms. A typical correlation resulting from the simulated annealing process is shown in Figure 5. The actual value of the regressed segment energy parameter is plotted against the value predicted from eq 9. In this case, n1 and ncross were fixed at three terms. The specific variables used in the correlation, along with the coefficients and exponents, are summarized in Table 3. The agreement is quite good for both the training data set, from which the correlating equation was created, and the test data set. The AAD % in uo/k for species in the training set is 1.21 K, with a maximum deviation of 5.74
Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2961 Table 3. Parameter Summary for Segment Energy Correlation Corresponding to Eq 9 Constant -3.189
c0
Single Terms and Coefficients Xi fracCH2 P4,pr 3χ cluster
ai
ei
4.57 8.36 -5.83
1.75 0.88 0.63
Product Terms and Coefficients X1,j
X2,j
bj
e1,j
e2,j
# P3 nc,tot WΘ
WΘ fracc nc,tot
37.33 5.99 × 10-3 -6.53
0.50 2.50 3.00
2.63 0.25 1.13
Table 4. Summary of Boiling Point Predictions for Selected Groups of Paraffinsa uo regressed
paraffin group
no. of species tested
C6 C7 C8 C9 C10 C11 C12 API project 42b higher n-alkanes
5 9 18 35 75 50 50 43 10
uo estimated
|∆nbp|, K
max error, K
|∆nbp|, K
max error, K
0.5 0.6 0.6 0.5 0.3 1.3 0.5 1.5 1.0
0.6 0.7 1.1 1.4 1.8 2.6 1.8 8.8 3.9
1.7 2.0 1.4 1.9 2.0 2.1 1.8 2.9 0.9
4.0 6.0 3.4 8.7 10.6 8.5 6.6 7.2 1.8
a The average deviation and maximum discrepancy between experimental and computed values are given for the cases where the segment energy is regressed to available data and where uo is estimated according to eq 9 and constants from Table 3. b Boiling points were computed at P ) 1.33 kPa, where data were available.
K. These values for the test data set are 1.44 and 6.03 K, respectively, showing that the correlation has not overfit the training data set. As a test of the parameter estimation formulas, we summarize in Table 4 the predicted normal boiling points of a variety of paraffinic species. This comparison gives us a means to directly assess the impact of errors in estimating the LJ-SAFT parameters on thermodynamic properties. The average and maximum errors are reported for each group. In addition to the groups described in Table 1, 50 isomers of C11 and C12 were also selected for comparison. The comparison for the branched paraffins from API project 42 was made at a lower pressure (Psat ) 1.33 kPa) where experimental data are available. Generally, the average error in the boiling point is larger when the segment energy is estimated from eq 9, but the overall agreement is still very good. The maximum error in predicted boiling point is also higher when uo is estimated; these cases correspond to species for which eq 9 gives a poor estimate of uo. The average errors in boiling point for the cases where uo is regressed and estimated are 0.8 and 2.2 K, respectively. We also note that the AAD % for the vapor pressure and liquid density for the compounds in Table 2 are 6.7 and 1.6%, respectively, when the SAFT parameters are estimated. While the quality of the vapor pressure agreement is somewhat diminished, it is still quite good for a predictive equation. IV. Results and Discussion Application to Lube-Ranged Materials. Given the rules of eqs 2, 6, and 9, we are now in a position to
Figure 6. Estimated normal boiling points for the effective homologous series depicted in Figure 1. Symbols are as marked in Figure 1.
generate LJ-SAFT parameters for any saturated hydrocarbon with formula CnH2n+2, requiring only the fractional Wiener number, WΘ, and the additional topological descriptors listed in Table 3 as inputs. These parameters can then be used to estimate the thermodynamic properties of molecules with any branching character. This provides us with the means to explore the relationship between a specific branching structure and a resulting thermodynamic property. As an example, we consider the computation of the normal boiling point for a variety of “effective” homologous series. The normal boiling point of each of the series described in Figure 1 is shown as a function of carbon number in Figure 6. The species in the series corresponding to structures with a minimum Wiener number have been constrained to have the same normal boiling point as the n-alkane of the same carbon number; the reasons for this approximation are discussed in detail in Appendix A. The use of eq 9 to estimate the segment energy produces small deviations ( 12
(A-1)
This states that we equate the normal boiling point of the n-alkane with the normal boiling point of the
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Table 7. LJ-SAFT Parameters from a One-Parameter Fit to the Segment Energy AAD % species name
Tmin, K
Tmax, K
W
m
n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane nt-ridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane n-nonadecane n-eicosane n-docosane n-tetracosane n-hexacosane n-octacosane n-triacontane n-hexatriacontane n-tetratetracontane
200 229 254 278 299 320 331 373 389 404 418 431 443 455 467 479 402 419 435 449 463 498 535
292 331 365 397 425 452 476 520 540 559 577 594 611 624 639 653 642 664 685 704 722 769 818
10 20 35 56 84 120 165 286 364 455 560 680 816 969 1140 1330 1771 2300 2925 3654 4495 7770 14190
2.216 2.576 2.936 3.296 3.655 4.015 4.375 5.094 5.454 5.813 6.173 6.533 6.893 7.252 7.612 7.972 8.691 9.410 10.130 10.849 11.569 13.727 16.604
branched structure with the lowest Wiener number we can find, or in the limit WΘ f 1. This approximation involves several assumptions. It assumes that there is a unique molecular structure for which W is minimized. In fact, this is not the case; as the carbon number increases, the degree of degeneracy of values for indices such as the Wiener number increases.13,19 Thus, it is possible that multiple structures may have the same value of the minimum Wiener number. Another practical difficulty involves the search for the structure(s) in an isomer population that possess(es) the minimum Wiener number. As noted in part 1, the number of paraffin isomers grows dramatically with carbon number and far exceeds our ability to examine every possible candidate at lube-ranged molecular weights. We must then appeal to a search strategy that will efficiently hunt through the very large number of possible candidate structures and pick out the structure with the desired property, namely, the minimum Wiener number. Simulated annealing is well-suited for this task and was employed in the carbon number range 13 < C# < 50. A complete listing of the results are shown in Table 6, along with several examples of structures obtained from the simulated annealing procedure. It is clear that the structures obtained are very highly branched and may be physically unstable. The high degree of strain associated with adjacent tert-butyl groups, for example, would likely make such molecular structures unstable. Nonetheless, these structures, coupled with the approximation of eq A1, provide us with a limit (WΘ f 1) with which we can provide upper bounds on our data set. Appendix B: n-Alkane LJ-SAFT Parameters Presented in Table 7 are the pure species LJ-SAFT parameters from a one-parameter fit for the segment energy. These values constitute the reference values implied in eqs 2, 6, and 9 and may be used explicitly in place of the approximations of eqs 4, 7, and 10. The table includes the temperature range of the vapor pressure data used in the regression, the three LJ-SAFT parameters, and the quality of fit obtained as measured by AAD %.
νoo,
cm3/mol
33.968 34.652 35.169 35.573 35.897 36.163 36.386 36.736 36.877 37.000 37.109 37.206 37.293 37.371 37.442 37.506 37.619 37.714 37.796 37.867 37.930 38.077 38.214
uo/k, K
Psat
Fliq
231.71 241.60 248.42 253.94 258.19 261.57 264.67 268.75 270.86 272.03 273.16 274.28 274.46 275.41 276.86 277.58 279.02 279.69 280.12 280.43 280.57 280.25 278.78
0.2 0.4 0.5 0.1 0.1 0.2 0.2 0.2 0.3 0.5 0.5 0.5 2.0 0.5 0.9 1.0 1.9 1.7 1.7 1.8 2.6 5.7 10.4
1.9 1.1 1.2 1.0 1.4 1.0 1.2 1.8 1.1 2.4 1.5 2.0 1.6 2.1 2.4 1.8 1.8 1.4 1.4 1.8 2.0 1.9 1.4
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Received for review December 1, 2000 Revised manuscript received April 6, 2001 Accepted April 11, 2001 IE001044J
