Ind. Eng. Chem. Res. 1997, 36, 1895-1907
1895
General Equation for Correlating the Thermophysical Properties of n-Paraffins, n-Olefins, and Other Homologous Series. 2. Asymptotic Behavior Correlations for PVT Properties John J. Marano* and Gerald D. Holder Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
In this second in a series of three papers, asymptotic behavior correlations (ABCs) are presented for PVT-related properties: normal boiling and melting point, critical temperature, pressure, and volume, acentric factor, liquid molar volume, and vapor pressure. The theoretical basis for the asymptotic behavior of these properties is discussed. The correlations were developed using literature data for n-paraffins and n-olefins (1-alkenes) and give accurate and consistent predictions. They are preferable to existing correlations in most instances. For melting point and liquid density, comparisons are made with high molecular weight, linear polyethylenes. It is also demonstrated that the ABCs developed for n-paraffins and n-olefins can be used to estimate the properties of other higher carbon-number n-alkyl derivatives, if the properties are known for at least one lower carbon-number member of the homologous series.
Y ) Y∞ - ∆Y0 exp(-β(n ( n0)γ)
(1a)
with either a “lattice” or a “cell” model for chainlike molecules. These are exemplified by the lattice-fluid models of Kurata and Isida (1955) and Sanchez and Lacombe (1976, 1978) and the cell model of Flory, Orwoll, and Vrij (1964), which are discussed below. Lattice-fluid theory serves as the basis for the ABCs presented here. Derivations for the asymptotic behavior of PVT-related properties appear in the Appendix to this paper.
Y∞ ) Y∞,0 + ∆Y∞(n - n0)
(1b)
Lattice-Fluid Model
Introduction In part I of this series (Marano and Holder, 1997), the development of a generalized asymptotic behavior correlation (ABC) applicable to homologous series of compounds was described. The form of this correlation is
This equation applies equally well to type I properties, where Y∞ is a constant, and type II properties, where the asymptote Y∞ is a linear function of carbon number n. For any particular property, behavior of the asymptote is based on molecular theory and yields reasonable extrapolations to large carbon numbers. The exponential term is empirical and accurately represents lower carbon-number behavior. In this paper, the theoretical basis for Y∞ will be discussed, and eq 1 will be applied to correlate PVTrelated properties of n-paraffins and n-olefins (1-alkenes). These correlations are useful in a number of engineering applications in petroleum and synthetic (Fischer-Tropsch) fuels processing. For example, they can be used to generate the properties of carbonnumber-based pseudocomponents used in process calculations. Theory As discussed in part 1, the form of the asymptote used in developing an ABC must be deduced either from experimental data for high molecular weight, linear compounds, such as polyethylene, or from theory. Since saturation properties cannot be determined directly for high molecular weight polymers, equations of state (EOS) derived from statistical-mechanical models are used for this purpose. Two different approaches start * Corresponding author. E-mail address: marano@ petc.doe.gov. Present address: U.S. Dept. of Energy, Federal Energy Technology Center, P.O. Box 10940, Pittsburgh, PA 15236-0940. S0888-5885(96)00512-X CCC: $14.00
In the lattice-fluid theory, the fluid is considered to have a quasi-crystalline structure, with sites on the lattice occupied by either segments of the chain molecule (mers, as they are often referred to) or empty “holes.” Kurata and Isida developed their lattice model starting from the partition function for the canonical ensemble, whereas Sanchez and Lacombe started from the isothermal-isobaric ensemble. The resulting EOS from either ensemble are identical
1 1 P ˜V ˜ V ˜ ) ln - 1- s T ˜ V ˜ -1 V ˜T ˜
(
)
(2)
where s is the number of segments per molecule (s-mer) and P ˜, T ˜ , and V ˜ are the reduced pressure, temperature, and volume defined as
T ˜ ≡ T/T*; T* ≡ */k; * ≡ z/2
(3)
P ˜ ≡ P/P*; P* ≡ */v*
(4)
V ˜ ≡ V/V*; V* ≡ N(sv*)
(5)
and N is the number of molecules, v* is the close-packed mer volume, is the nonbonded, mer-mer interaction energy, * is the total interaction energy per mer, and z is the coordination number of the lattice. Kurata and Isida have estimated that the value of z is within the range of 4-6, which correspond to a diamond lattice and a simple cubic lattice. Equation 2 does not satisfy a simple corresponding states principle except in the limit of polymeric liquids (s large). © 1997 American Chemical Society
1896 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Starting with the lattice-fluid EOS and a similar expression for chemical potential, equations can be derived for critical properties, vapor pressure, and normal boiling and melting points as described in the Appendix. It should be noted that the Kurata and Isida expression for critical temperature is incorrect due to the substitution of an erroneous expression for as a function of temperature; however, this does not qualitatively affect the asymptotic behavior. Lattice-fluid theory predicts that critical temperature and pressure are both type I properties and that in the limit critical pressure approaches zero. Both these limits are consistent with experimental data for the highest carbonnumber compounds measured and are used here for developing ABCs. The critical volume in the limit is proportional to s3/2. Kurata and Isida also make the assertion that the number of chain segments is related to carbon number by s ) n2/3, which is not in agreement with recent measured data for critical volumes of n-paraffins (see below). If s ) n, the agreement is improved; however, the limiting value for critical density is zero. Flory Cell Model In the cell model of Flory, Orwoll, and Vrij (henceforth referred to as Flory theory), the mer can move only within a limited volume of space due to the close presence of neighboring mers (bonded and nonbonded); this limited volume formed by neighboring mers is called a cell. The canonical-ensemble partition function is formulated in terms of the free volume of the cell and holes are not explicitly introduced into the model. The resulting EOS is 1/3
V ˜ P ˜V ˜ 1 ) 1/3 T ˜ ˜T ˜ V ˜ -1 V
(6)
where P ˜, T ˜ , and V ˜ are again reduced properties. P ˜ and V ˜ are defined as before; however, T ˜ is redefined as
T ˜ ≡ T/T*; T* ≡ */ck; * ≡ ζη/2v*
(7)
and v* is now the hard-core mer volume, η/v* is the mean intermolecular energy per contact pair, c is a correction to the number of external degrees of freedom per mer (c < 1), and ζ is the mean number of external contacts per mer. Equation 6 satisfies the principle of corresponding states as developed by Prigogine et al. (1957) for chain molecules. Flory theory has found some utility as a starting point for developing EOS for predicting vapor-liquid equilibria in mixtures containing polymers and solvents (e.g., see Chen et al., 1990 references contained therein). Even though there is striking similarity between the definitions for the reduced properties in eqs 2 and 6, the functional dependences for the repulsive term in the EOS are quite different. While the parameters in both models must be determined from empirical fitting of either saturation or density data, Flory theory is somewhat more empirical in the sense that there is an unspecified dependence of c, v*, and ζη on s. Tsonopoulos and Tan (1993) have shown that these parameters can be expanded as power series in 1/s and then adjusted to obtain accurate predictions for the critical properties of n-paraffins. Our own calculations, however, have shown liquid-density predictions made with these parameters and the EOS are not very good, with errors exceeding 24% for polyethylenes at 150 °C.
Tsonopoulos and Tan developed critical property correlations for n-paraffins using two different sets of data. The first set contains the critical density values recommended by Ambrose (which includes values measured by Anselme et al. (1990)) and the second set, those of Steele. These data conflict with Steele’s data, suggesting a linear dependence for critical volume on carbon number, and Ambrose’s values, indicating a superlinear dependence. Ambrose’s recommendations were used in both sets of data for critical temperature and pressure (see Tsonopoulos and Tan (1993) for a discussion of these data). Different truncations for the power series in 1/s were used with each set of data, leading to quite different behavior for the critical pressure and density correlations at intermediate carbon numbers. However, the asymptotic behavior for both sets of correlations is the same. Critical temperature and pressure are predicted by Flory theory to be type I properties in agreement with lattice-fluid theory. However, the critical pressure approaches a nonzero limit. Flory theory always predicts that critical volume becomes linear, and thus the critical density constant, at large enough carbon numbers. The critical density correlation based on Steele’s data is monotonic with carbon number, whereas that based on Ambrose’s values predicts a maximum at intermediate carbon numbers. Lattice-fluid theory also predicts a maximum in critical density; however, it predicts zero limiting values for both critical pressure and density. Insufficient experimental data are available, and sufficient data may never be available, to determine which theory is more realistic. Molecular Simulation Recently, molecular simulation techniques have been developed to study PVT behavior of long-chain paraffins to linear polyethylenes. Of particular interest are the simulations of de Pablo et al. (1992, 1993). Siepmann et al. (1993) have conducted such “computer” experiments for a series of n-paraffins, C5 to C8, C10, C16, C24, C28, and C48, in order to “measure” the critical properties of the higher carbon-number homologs. Their results are in qualitative agreement with the experimental data of Anselme et al. (1990) and tend to support theories which predict critical volume has a greater-than-linear dependence on carbon number. Below, the critical values reported for C24 and C48 are compared with predictions made with ABCs. Additional molecular simulation should be useful in deducing the behavior of other properties of practical interest and establishing bounds on the theoretical models and property correlations. Results: Normal Boiling Point Results based on the ABCs developed for normal boiling point of n-paraffins and n-olefins were presented in part 1. The superiority of eq 1 over other proposed ABCs was demonstrated. Normal boiling point is a type I property. Table 1 list the ABC parameters for normal boiling point, and Figure 1 presents a comparison of these correlations with data. Older data for C31 to C40 n-paraffins have been included in Figure 1 (API Project 44, 1953). These compounds are unstable at their normal boiling points, resulting in lower measured temperatures. The normal boiling point correlations developed for n-paraffins and n-olefins were used to estimate the
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1897 Table 1. ABC Parameters for Boiling and Melting Points of n-Paraffins and n-Olefins Y ) Y∞ - (Y∞ - Y0) exp(-β(n - n0)γ) normal boiling point (K) n-paraffin n-olefin CNs
3-30
3-30
pts ref n0 Y0 Y∞ β γ RMSE AD AAD %AAD
28 28 3, 35, 36 3, 35, 36 -1.487 453 -1.340 265 -164.93 1091.11 0.153 505 0.602 490 0.0416 0.0631 0.0361 -0.0551 0.1569 0.2912 0.0362 0.0613
normal melting point (K) n-paraffin 9-21 (odd), 22-40, 43, 44, 46, 50, 52, 54, 60, 62, 64, 66, 67, 70, 82, 94, 100 41 3, 9, 35, 36 0.340 979 -6 288 460 418.07 8.929 364 0.069 040 6 0.0435 0.0624 0.2194 0.0634
Figure 1. Asymptotic behavior correlations for normal boiling Point of n-paraffins and n-olefins. Table 2. Prediction of Boiling Points of Isoparaffins, n-Alkylcyclohexanes, and n-Alcohols from Single Data Point and ABC
CNs pts ref n0 RMSE AD AAD %AAD a
isoparaffin
n-alkylcyclohexane
3-19 1 (C3)a 6, 20 -2.202 315 0.1834 0.2633 0.6093 0.1405
3-20 1 (C3) 6 -7.738 951 1.5375 -6.0294 6.0295 0.9953
n-alcohol 3-12, 16, 20 1 (C8) 6, 21, 45 -5.452 246 -4.455 702 4.9947 3.6526 15.2085 -4.9968 15.2971 9.5910 3.1195 2.1668 1 (C3)
In Cn, n refers to the CN of straight chain; e.g., iso(C3) is i-C4H10.
transition II (K) n-paraffin
transition III (K) n-paraffin
10-40
10-32 (even), 36
9-31 (odd), 34, 35, 36
31 3, 6, 35, 36 2.081 202
13 9, 35, 36 1.169 704 -3 873 512 418.07 8.763 524 0.060 887 5 0.1017 0.0005 0.3088 0.1009
15 9 0.340 979 -379.78 418.07 0.574 102 0.403 091 0.2720 0.0059 0.8912 0.3193
n-olefin
0.0446 -0.0956 0.2241 0.0735
properties of other higher carbon-number n-alkyl derivatives: isoparaffins, n-alkylcyclohexanes, and nalcohols. These compounds were selected because they show differences in size of the functional group present and in polarity. The normal boiling points for the C3 compounds: 2-methylpropane (isobutane), propylcyclohexane, and n-propanol, were used to calculate values for n0 in eq 1 for the respective series of compounds. All other parameters were kept the same as determined for n-paraffins and n-olefins. Table 2 gives the results from the prediction of all other members in the different series for which data could be found. This comparison is shown in Figure 2. It can be seen that this estimation procedure, based on an “effective” carbon number n n0, performs very well for isoparaffins and n-alkylcyclohexanes. The comparison is less satisfactory for n-alcohols; however, the predictions for higher carbonnumber n-alcohols can be much improved if n-octanol (C8) data is used to determine n0. As seen in Figure 2, n-propanol and n-octanol lie on opposite sides of an inflection point in the data for n-alcohols. This inflection is not present in the data for the other homologous series. Normal Melting Point Correlation of normal melting point data is not as straightforward as that of other properties because at lower carbon numbers, n-paraffins melt from different
Figure 2. ABC predictions for normal boiling point of isoparaffins, n-alkylcyclohexanes, and n-alcohols.
1898 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Figure 4. Solid-liquid phase diagram for n-paraffins. Figure 3. Asymptotic behavior correlations for normal melting point of n-paraffins and n-olefins.
crystalline phases. Broadhurst (1962a) has elaborated on the different liquid-solid/solid-solid transitions that have been observed. For all carbon numbers between 1 and 10, and for even carbon numbers between 12 and 20, n-paraffins melt from a triclinic (t) crystalline phase. For odd carbon numbers between 11 and 19, and for all carbon numbers between 21 and 43, n-paraffins melt from a hexagonal (h) crystalline phase. Above a carbon number of 43, melting occurs from an orthorhombic (o) phase. The orthorhombic to liquid transition (o-l) is also observed in samples of highly-linear polyethylenes, in impure n-paraffins (i.e. n-paraffin mixtures), and for most n-olefins. Solid-solid transitions observed in n-paraffins include orthorhombic to hexagonal (o-h), triclinic to hexagonal (t-h), monoclinic to hexagonal (m-h), and monoclinic to orthorhombic (m-o) transitions. As a result of the various solid-liquid transitions, odd and even carbon-numbered n-paraffins below C20 fall on two separate melting point versus carbon number curves. The even-numbered t-l, h-l, and o-l transitions fall on one curve (I), and the odd-numbered t-l, along with the t-h, m-h, and m-o transitions, fall on another (II). The o-h transition has a distinct phasetransition curve (III). ABCs were developed for all the observed solid-phase transition curves. The transition I curve for n-paraffins corresponds to the n-olefin melting point curve (o-l), and these sets of ABC parameters were optimized simultaneously. Normal melting point is a type I property, and all ABC parameters are listed in Table 1. Results are shown in Figure 3 for carbon numbers up to 100. n-Paraffins with carbon numbers less than 9, and n-olefins with less than 10, were not included in determining the parameters for the melting point correlation. This improved the accuracy of the correlations for higher carbon numbers and is in line with Broadhurst’s observation (1962a) that the shorter paraffins (C1-C9) form solid structures, which, while triclinic, do not fit into the chainlike pattern of the longer paraffins. The parameters for the transition II and III curves were obtained by modifying selected parameters in the correlation for transition I. The n-paraffin solid-liquid phase diagram is shown in Figure 4. The value obtained for Tm∞ from the ABC is 144.9 °C (418.1 K). This is similar to the extrapolated value of 141.2 °C obtained by Broadhurst (1962b) and agrees well with data obtained for polyethylenes. Linear polyethylenes undergoing nonequilibrium phase transitions typically have a melting point between 132 and 135 °C, with a rather narrow melting range, whereas branched polyethylenes melt at lower temperatures around 112 °C, with broader ranges (Raff, 1968). However, Chiang and Flory (1961) have measured
Figure 5. Asymptotic behavior correlations for critical temperature of n-paraffins and n-olefins.
melting points as high as 138.5 °C for narrow carbonnumber fractions of high molecular weight, highly-linear polyethylenes (Marlex-50, Phillips). Richardson et al. (1963) have concluded that the most probable value for Tm∞ is within the range of 140-145 °C, which is where the ABC value falls. Critical Temperature and Pressure Unlike normal boiling and melting point, the availability of critical property data for n-paraffins and n-olefins is quite limited. The origin and reliability of some of the data present in the literature is also questionable. The C11+ paraffins and the C8+ olefins are unstable at their critical temperatures. n-Paraffins tend to decompose, whereas n-olefins polymerize at elevated temperatures. Methods which involve quick response times are necessary in order to measure critical properties of these unstable compounds. Recent measurements have been made by Teja and co-workers for critical temperature, pressure, and density using rapid-heating and flow apparatuses (n-paraffins: Anselme et al. (1990) and Rosenthal and Teja (1989); n-olefins: Gude et al. (1991)). These critical values are considered to be the best currently available and are used here as the basis for developing ABCs. However, rather large uncertainties still remain for the higher carbon-number compounds. Critical temperature and pressure both exhibit type I behavior. The critical temperature data encompass n-paraffins from C1 to C18 and n-olefins from C2 to C10 and C12. The reported uncertainties are low up to C13 for n-paraffins and C9 for n-olefins and are on the order of 1-2 °C for the higher carbon-number compounds. The ABC parameters determined from these data are given in Table 3 and the correlations are shown in Figure 5. Included in the figure are C12+ n-olefin data from the API Technical Data Book (1992) for comparison. These values are slightly lower than the data of Gude et al.
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1899 Table 3. ABC Parameters for Critical Temperature and Pressure of n-Paraffins and n-Olefins Y ) Y∞ - (Y∞ - Y0) exp(-β(n - n0)γ) critical temperature (K) n-paraffin CNs pts ref n0 Y0 Y∞ β γ RMSE AD AAD %AAD
n-olefin
3-18 3-10, 12 16 9 45 1, 19 0.896 021 0.980 154 127.89 1020.71 0.198 100 0.629 752 0.0773 0.2444 0.0193 -0.0966 0.2784 0.5947 0.0475 0.1220
critical pressure (bar) n-paraffin
n-olefin
3-18 3-8, 10, 12 16 8 45 19 -3.625 581 -3.039 461 1336.74 0 2.111 827 0.258 439 0.0459 0.9817 -0.0027 0.9540 0.1400 1.2028 0.7274 1.0251
(1991). Comparison of critical properties predicted by the ABCs with those from group contribution methods was presented in part 1. It should be noted that the errors associated with critical temperature are larger than for normal boiling point. The lack of higher carbon-number data, in particular for the n-olefins, and the rather large uncertainties for the existing data all tend to limit the uniqueness of the asymptotic extrapolation, whether made with the ABC or by other methods such as those outlined under Theory. Predictions made with the ABC for C24 and C48 n-paraffins were compared with those obtained from molecular simulations by Siepmann et al. (1993). They report values for C24 and C48 of 800 ( 10 and 930 ( 10 K, respectively. The ABC values are in reasonable agreement: 807.3 and 925.8 K. This gives some confidence the correlations extrapolate fairly well, at least up to C48. The value obtained for Tc∞ from the ABC is 1020.7 K, which is somewhat lower than obtained from correlations based on Flory theory. The critical temperature and normal boiling point ABCs cross at a carbon number of 110.5. The critical pressure data for C3 through C18 nparaffins and C3 through C8, C10, and C12 n-olefins were used to obtain the ABC parameters given in Table 3. The reported uncertainties in the data are (0.2 bar up to C15 and range between 0.5 and 1.6 bar for the higher carbon-number n-paraffins. On the basis of the results from lattice-fluid theory, the critical pressure at infinity was set to zero. Following the recommendations of Gray et al. (1989), the critical pressure at the carbon number of the intersection of the critical temperature and normal boiling point ABCs was constrained to 1.01 bar during parameter optimization to maintain consistency between the different correlations. The resulting correlations are shown in Figure 6. The n-olefin data are not as smooth as that reported for n-paraffins. Data from the API Technical Data Book (1992) for C9+ n-olefins are also shown in Figure 6. The C15, C17, and C19 data are clearly lower than the data of Gude et al. (1991). Critical Volume and Compressibility The recommended values for critical density of Teja and co-workers for C3 through C18 n-paraffins and C3 through C10 n-olefins were used to calculate critical volumes. The reported uncertainties for these data are (0.005 g/cm3. As mentioned earlier, these data are at odds with earlier data and the data reported by Steele but are in agreement with theory and results from
Figure 6. Asymptotic behavior correlations for critical pressure of n-paraffins and n-olefins.
Figure 7. Asymptotic behavior correlations for critical volume of n-paraffins and n-olefins. Table 4. ABC Parameters for Critical Volume and Acentric Factor of n-Paraffins and n-Olefins Y2/3 ) ∆Y∞(n - n0) - ∆Y0 exp(-β(n - n0)γ) γ
Y ) ∆Y0 + β(n - n0) critical volume (cm3/g‚mol) n-paraffin CNs pts ref n0 ∆Y0 ∆Y∞ β γ RMSE AD AAD %AAD
n-olefin
3-18 3-10 16 8 45 19 -53.520 023 -53.081 049 218.71 2.379 19 1.140 26 × 10-4 2.192 58 1.0151 0.7905 0.1528 -0.0999 3.1518 1.8384 0.6390 0.5197
for
for
Vc
ω acentric factor
n-paraffin
n-olefin
3-18 3-15, 17, 18 16 15 5 5, 35 -23.608 415 -23.174 122 -6.5597 3.383 261 0.208 770 0.0021 0.0025 -0.0003 0.0003 0.0067 0.0071 1.4053 1.7901
molecular simulation. The earlier data suggest critical density is a type I property, and thus critical volume is a type II property. On the basis of lattice-fluid theory, critical volume is proportional to n3/2; therefore, Vc2/3 is a type II property and is used to develop the ABCs. Parameters appear in Table 4, and results are shown in Figure 7. The discrepancy is apparent between the correlations and both Steele’s data and the data for C12+ n-olefins (API Technical Data Book, 1992), which appear linear. Siepmann et al. (1993, report values for C24 and C48 of 0.20 464 ( 0.009 and 0.19 486 ( 0.014 g/cm3, respectively, on the basis of molecular simulations. The ABC values are in excellent agreement: 0.2054 and 0.1948 g/cm3. In order to compare the consistency of the critical property ABCs, the critical compressibility was calculated from the data and correlation results. This comparison is shown in Figure 8. Clearly, the correla-
1900 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Figure 8. Critical compressibility of n-paraffins and n-olefins calculated from ABCs.
tions perform poorly at both low and high carbon numbers. They also predict a crossover for n-paraffins and n-olefins at about C8. Whether this is real or not is difficult to determine because the difference in values between n-paraffins and n-olefins is much smaller for critical compressibility than for the other critical properties, and the experimentally-derived compressibilities are somewhat erratic. On the basis of the C12+ data, it can be seen that if a linear relation holds for critical volume, critical compressibility falls much faster with increasing carbon number. Critical compressibility is used for correlating a number of properties, such as Lennard-Jones collision diameters and saturated-liquid densities (Rackett equation). In the limit, the ABCs yield a value of zero for critical compressibility due to the representation of critical pressure as an exponential function. This is at odds with lattice-fluid theory which predicts a limiting value of 1/3. For these applications, it is much preferred to develop the ABC directly from the data of interest. For example, starting with a consistent set of saturatedliquid density data, it would be possible to develop an ABC for the Rackett parameter (see Kontogeorgis et al. (1995) for an outline of this approach). Finally, the values for n0 determined for critical volume and acentric factor (see below) are uncharacteristically large. Analysis of the results of the regressions used to determine n0 indicate smaller values give almost as good a match to the data, and these large values appear to be a result of the rather limited amount of critical property data available for higher carbonnumber compounds. Further results from molecular simulation would be valuable in determining more meaningful values for the ABC parameters. Acentric Factor
Figure 10. Asymptotic behavior correlations for liquid molar volume of n-paraffins and n-olefins.
easily be shown that the ABC for the acentric factor has the following form
ω ) ∆ω0 + β(n - n0)γ
(9)
where ∆ω0, β, and γ are the ABC parameters to be determined from data regression. These parameters are given in Table 4. Results are shown in Figure 9, where it can be seen that much of the data above C15 are erratic. This is not surprising since the calculation of values for the acentric factor is dependent on having accurate values for critical pressure and temperature. Extrapolated values of ω based on eq 9 are in excellent agreement with the estimated values for C21 through C30 n-paraffins given in the API Technical Data Book (1992) and are in reasonable agreement with the C20+ n-paraffin values presented by Magoulas and Tassios (1990) based on eq 8 directly and extrapolated values for vapor pressure and critical pressure. Their values for ω increase more rapidly with carbon number. Liquid Molar Volume
The acentric factor is defined as
ω ) -log(Pvap,r)Tr)0.7 - 1.000
Figure 9. Asymptotic behavior correlations for acentric factor of n-paraffins and n-olefins.
(8)
where Pvap,r is the reduced vapor pressure and Tr is the reduced temperature, both relative to their critical-point values. On the basis of the ABCs for Tc and Pc previously described and the ABC for Pvap to be discussed shortly, it is possible to calculate values for ω directly from eq 8. However, for some applications, such as EOS-based vapor-liquid equilibrium calculations, it is simpler to use an ABC developed explicitly for ω. If it is assumed the ABC parameters for Pvap at Tr ) 0.7 and Pc (Pvap at Tr ) 1) are the same, then it can
Liquid molar volume is a type II property and is a function of both pressure and temperature. The pressure dependence, however, is quite weak, especially at or below the normal boiling point. For example, the volume increase from 20 to 250 °C is roughly 20% for n-C30H62, whereas the volume decrease from 1 to 25 bar is only about 0.4%. A large amount of data for liquid density/molar volume is available in the literature spanning a wide range of temperatures and very high carbon numbers. The ABCs for liquid molar volume were developed primarily from data at atmospheric pressure; however, some saturation values were included if the temperature was close to the normal boiling point of the compound. Table 5 contains the temperature-independent ABC parameters for liquid
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1901 Table 5. ABC Parameters for Liquid Molar Volume and Vapor Pressure of n-Paraffins and n-Olefins Y ) Y∞,0 + ∆Y∞(n - n0) - ∆Y0 exp(-β(n ( n0)γ) n + n0
for
V L,
n - n0
molar volume (cm3/g‚mol) n-paraffin CNs pts ref n0 ∆Y0 Y∞,0 ∆Y∞ β γ RMSE AD AAD %AAD
for
ln Pvap
ln vapor pressure (bar)
n-olefin
n-paraffin
3-40, 44, 64, 94 3-40 156 82 3, 4, 14, 15, 21, 29, 3, 4, 44, 48 37, 42, 46, 49 -1.388 524 -1.061 318 see Table 6 0 see Table 6 5.519 846 0.057 063 2 0.1668 0.1201 -0.0198 0.0293 0.4752 0.3086 0.1225 0.1548
3-20, 24, 28 114 5
n-olefin 3-20 106 5
1.126 231 1.281 405 see Table 6 2.7271 see Table 6 0.619 226 0.416 321 0.0100 0.0072 0.0002 0.0002 0.0213 0.0167 0.6433 0.5470
molar volume. The temperature-dependent parameters are given in Table 6. Figure 10 shows the liquid molar volume of nparaffins and n-olefins at three representative temperatures. The data for C17+ n-paraffins and C19+ n-olefins at 20 °C are for supercooled liquids. The ABCs do a reasonable job of predicting saturated-liquid volumes above the normal boiling point as long as the temperature is not too near the critical point of the compound. It should also be noted that while molar volume increases with increasing carbon number at low pressures, near the critical point, the volume increases rapidly with a decrease in carbon number. The ABCs reproduce this behavior, however, only qualitatively. The ABCs reproduce all other data extremely well, with errors typically less than 0.5% for both n-paraffins and n-olefins and percent absolute average deviations (%AADs) of 0.12 and 0.15, respectively. The percent maximum absolute deviations (%MADs) are between 1.5 and 2.0 and occur for the C3 compounds at 0 °C. This is not surprising since these compounds are saturated
liquids well above their critical temperature. Plots showing the error distributions for liquid molar volume as a function of carbon number and temperature were presented in part 1. Liquid molar volumes predicted by the ABCs can be converted to densities (F ) M/V). In the limit of large carbon number, the density approaches a finite value (type I property). These values can be compared to measured densities of polyethylene melts. This comparison is made in Table 7 for three linear polyethylenes and three branched polyethylenes. The samples studied by Olabisi and Simha (1975) were a linear polyethylene (LPE) and a branched polyethylene (BPE) both from the NBS data bank and a high molecular weight, linear polyethylene (HMLPE, M > 5×106) obtained from Allied Chemical Co. At low temperatures, the HMLPE densities compare favorably with those predicted from the ABC, as do the LPE densities at high temperatures. Surprisingly, BPE densities match the ABC over the entire temperature range better than those for the other polyethylenes. The source of the polyethylenes studied by Chung (1971) was not identified. His LPE densities also match the ABC very well, whereas his BPE densities clearly do not match as well. The %AAD for all LPEs is 0.54, and the %MAD is 0.94 for the HMLPE at 200 °C. The %AAD for all BPEs is only 0.61, which indicates the ABC is generally applicable to both linear and branched polyethylenes. Also included in Table 7 are density predictions made using the correlation of Richardson et al. (1963). This correlation was developed from data for two highly-linear polyethylenes, Marlex 50 and Super Dylan, over the temperature range of 140-180 °C. The agreement between this equation and the ABC is excellent. The ABC for liquid molar volume developed from n-paraffin and n-olefin data was used to estimate the densities of C18 n-alkyl derivatives using only data for the corresponding C8 homologs. As was done for normal boiling point, the lower carbon-number data were used to calculate values of n0 for the different homologous series. These values were then used to estimate the
Table 6. Temperature-Dependent ABC Parameters for Liquid Molar Volume and Vapor Pressure ∆Y ) A + BT + CT2 + DT3
for
∆Y ) A + B/T + C ln T + DT2 + E/T2
VL for
ln Pvap
molar volume (cm3/g‚mol) A B C D E
∆Y0
∆Y∞
8592.30 -85.7292 0.280 284 -4.484 51 × 10-4
12.7924 0.015 062 7 -1.307 94 × 10-5 1.596 11 × 10-8
n-paraffin n-olefin
ln vapor pressure (bar) ∆Y0
∆Y∞
-5.75509 -7.565 68 0.085 773 4 -1.419 64 × 10-5 2.672 09 × 10 5
15.8059 -1496.56 -2.17342 7.277 63 × 10-7 378 76.2
temp range of data
temp range of data
0-300 °C 0-150 °C
0-300 °C 0-300 °C
Table 7. ABC Predictions for Densities (g/cm3) of Polyethylenes
temp (°C) 140 160 180 190 200 220
NBS LPE ref 30
NBS BPE ref 30
Allied HMLPE ref 30
Chung LPE ref 13
Chung BPE-A ref 13
Chung BPE-A ref 13
Richardson et al. ref 38
ABC for liq molar volume
0.7897 0.7775 0.7654
0.7856 0.7741 0.7629
0.7827 0.7695 0.7565
0.7756
0.7785
0.7795
0.784 0.774 0.763
0.7597
0.7622
0.7639
0.7536
0.7517
0.7438 0.7438
0.7462
0.7477
0.783 0.772 0.761 0.756 0.751 0.740
1902 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 8. Prediction of Densities of C18 Compounds from C8 Data homologous series 1-aminoalkanes C8 C18 1-bromoalkanes C8 C18 1-chloroalkanes C8 C18 1-iodoalkanes C8 C18 1,m-alkanethiols 1,8-C8 1,18-C18 1-alkanoic acids C8 C18 1-alkanols C8 C18 m-alkenes (cis) 4-C8 9-C18 m-alkenes (trans) 4-C8 9-C18 1-alkynes C8 C18 2-alkynes C8 C18
n0 predicted -1.527 866 -2.028 991 -1.823 010 -2.437 523 -2.019 468 -0.563 138
density density reporteda predicted (g/cm3) (g/cm3)
% error
0.7826 0.8618
0.8224
-4.5679
1.1122 0.9848
0.9922
0.7493
0.8738 0.8641
0.8687
0.5350 Figure 11. Asymptotic behavior correlations for vapor pressure of n-paraffins and n-olefins.
1.3297 1.0994
1.1096
0.9269
0.8433 0.8475
0.8532
0.6717
0.9688 0.9408
0.9194
Table 9. Comparison of Boiling Point Data of n-Paraffins and n-Olefins with Predictions from Vapor Pressure ABC data at 10, 40, 100, 400, 760, 1500 mmHg
-2.2724 n-paraffin
-1.073 448 -0.962 627 -1.054 264 -0.494 100 -0.338 243
data at 0.5, 1, 2, 5, 10 mmHg
0.8270 0.8124
0.8165
0.5108
0.7212 0.7916
0.7932
0.2016
0.7141 0.7916
0.7894
-0.2753
0.7461 0.8025
0.8066
0.5067
0.7596 0.8061
0.8063
0.0264
a
All densities at 20 °C except 1-octadecanol, 59 °C; 2-octadecyne, 30 °C. Source of data: Handbook of Chemistry and Physics, 56th ed., 1975-1976.
densities for the C18 compounds. Eleven different homologous series were considered in the comparison. Results are shown in Table 8 and are quite acceptable. The %AAD and %MAD for the 11 data sets are 1.0 and 4.6, respectively. If the data for 1-aminoalkanes and 1-alkanoic acids are excluded, the %AAD is only 0.49. Vapor Pressure As shown in the Appendix, logarithm of vapor pressure is a Type II property and is a function of temperature. The high-order data regressions available in the API Technical Data Book (1992) were used to generate data for natural logarithm of vapor pressure as a function of temperature. Equations are available for C1 through C20, C24, and C28 n-paraffins and C2 through C20 n-olefins. The temperature-independent ABC parameters for the vapor pressure are given in Table 5, and the temperature-dependent parameters are given in Table 6. Caldwell and Van Vuuren (1986) developed a correlation for the vapor pressure asymptote of n-paraffins over the temperature range of 175-300 °C. They report a root mean square error (RMSE) for the logarithm of vapor pressure of 0.0905. The ABC with its greater number of parameters has a RMSE of 0.0100 for n-paraffins over a wider range of 0-300 °C. Magoulas and Tassios (1990) also describe a procedure for extrapolating vapor pressures to higher carbon numbers. Their approach is to develop ABCs of the KreglewskiZwolinski type (see part 1) for the boiling point at various pressures. For a given carbon-number n-
n-olefin
CNs
4-20
4-20
pts ref
102 3 35, 36@760 0.0045 0.3042 0.0739 0.1055
102 3 35, 36@760 0.0115 0.3685 0.0902 0.1202
RMSE AD AAD %AAD
n-paraffin
n-olefin
13-20, 23, 24, 26, 28, 32, 35, 36 75 4
13-17
0.1483 1.0864 0.2238 0.4378
0.6476 0.8798 0.2367 0.4686
25 4
paraffin, the ABCs are used to calculate boiling points over a narrow range of pressures, which in turn are used to estimate Antoine coefficients. The Antoine equation is then used to calculate the vapor pressure at the temperature of interest. Clearly, the direct approach of developing a vapor pressure ABC is much simpler. Figure 11 shows the vapor pressure at three representative temperatures in the range correlated with the ABCs. Plots showing the error distributions for logarithm of vapor pressure as a function of carbon number and temperature are presented elsewhere (Marano, 1996). The ABCs reproduce the data extremely well except near 1 bar (ln Pvap ) 0), where the percent errors can be quite large. The relative error for ln Pvap is typically less than 0.05 for both n-paraffins and nolefins, though some outlying points do exist. To further test the vapor pressure ABCs, the correlations were used to calculate the boiling points for n-paraffins and n-olefins reported in the API Project 44 and 42 tables (1953 and 1966, respectively) and the ultralow pressure data of Morecroft (1964) for C19 and C27 n-paraffins. For C4 through C20 over the pressure range of the API 44 tables, 10-1500 mmHg, the ABCs do exceptionally well as can be seen from Table 9. The comparison is only slightly worse with C3 data, but the overprediction is quite large for C2 data (not reported). Much of the data used for this comparison are outside the temperature range used to develop the correlations (0-300 °C), ranging from -81 to 379 °C, and the performance of the ABCs are not negatively affected. The API Project 42 data are for low pressures, 0.5-10 mmHg, and include a number of high molecular weight n-paraffins: C23, C24, C26, C32, C35, and C36, not used in developing the ABCs. Table 9 also contains the results of this comparison. The errors are larger for this data set but are still acceptable. The comparison with Morecroft’s data is given in Table 10. The largest deviations occur at the lowest pressures. The %AADs
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1903 Table 10. Comparison of Boiling Point Data of C19 and C27 n-Paraffins with Predictions from Vapor Pressure ABC
n-paraffin C19 C27
a
pressure reported (mmHg)
temperature reporteda (K)
temperature predicted (K)
% error
1.41 × 10-4 5.34 × 10-4 1.57 × 10-3 2.76 × 10-5 9.32 × 10-5 2.08 × 10-4 1.33 × 10-3 6.27 × 10-3 5.00 × 10-2
306.2 318.2 328.2 351.3 362.7 374.2 391.7 408.6 434.8
306.9 318.3 328.3 354.1 364.5 371.8 390.4 408.0 435.3
0.2347 0.0413 0.0548 0.8063 0.5117 -0.6151 -0.3134 -0.1327 0.1344
Source of data: Morecroft (1964).
are 0.11 and 0.42 for C19 and C27, respectively. That the ABCs perform so well in predicting these data, much of which is outside the range of carbon numbers and temperatures used in their development, is evidence of their extrapolative ability. As a final example of the ability of the ABCs developed from n-paraffin and n-olefin data to be used for other homologous series, the vapor pressure ABC was used to predict the boiling points of C18 alkyl derivatives using data for the corresponding C8 homologs. Results are shown for 13 different homologous series in Table 11. The values for n0 were calculated from the C8 data at 760 mmHg, with the exception of 1,8-alkanediol where data were only available at 20 mmHg. The values for n0 were then used to calculate boiling points
Table 11. Prediction of Boiling Points of C18 Compounds from C8 Data homologous series 1-aminoalkanes C8
n0 predicted -1.113 526
C18 1-bromoalkanes C8 C18 1-chloroalkanes C8
-2.101 757
-1.223 038
C18 1-iodoalkanes C8
-3.338 485
C18 1,m-alkanethiols 1,8-C8 1,18-C18 1,m-alkanediols 1,8-C8 1,18-C18 alkanals C8
-2.020 094
-7.413 477 -0.730 759
C18 1-alkanoic acids C8 C18 1-alkanols C8 C18 m-alkenes (cis) 4-C8 9-C18 m-alkenes (trans) 4-C8 9-C18 1-alkynes C8
-4.072 604
-1.799 133
1.246 947
1.254 533
1.144 536
C18 2-alkynes C8 C18 a
0.647 749
pressure reported (mmHg)
temperature reporteda (K)
temperature predicted (K)
% error
760 10 760 10
452.8 336.4 622.0 472.7
335.3 619.2 472.1
-0.3191 -0.4431 -0.1125
760 10 10
474.0 350.5 483.2
352.2 482.6
0.5113 -0.1086
760 15 760 10
455.2 351.2 621.2 472.2
345.1 620.5 473.3
-1.7333 -0.0973 0.2441
760 5 760 10
498.7 359.7 656.2 496.2
358.7 645.1 495.2
-0.2545 -1.6841 -0.1828
760 15 1-2
472.3 359.2 461.2
359.0 441.1
-0.0542 -4.3387
20 2
445.2 483.7
495.7
2.4969
760 20 760 12
444.2 345.2 534.2 485.7
341.9 614.4 472.3
-0.9521 15.0245 -2.7475
760 23 15
512.5 413.2 505.2
401.4 512.8
-2.8395 1.5176
760 19 15
467.6 371.2 483.7
360.1 489.5
-2.9852 1.2094
760 10 9
395.7 287.2 435.2
289.8 442.9
0.9317 1.7882
760 10 9
395.5 286.9 435.2
289.7 442.8
0.9824 1.7646
760 10 760 15
398.4 292.9 586.2 453.2
292.0 589.4 456.1
-0.3026 0.5560 0.6496
760 15
411.2 457.2
462.0
1.0613
Source of data: Handbook of Chemistry and Physics, 56th ed., 1975-1976.
1904 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
of C18 compounds at 760 mmHg and both C8 and C18 compounds at various other pressures as low as 1 mmHg. For predicting C8 boiling points the %AAD and %MAD are 1.08 and 2.99, respectively. Excluding the outlying point for octadecanal at 760 mmHg, the %AAD and %MAD for the C18 boiling points are 1.24 and 4.33, respectively. Conclusions Asymptotic behavior correlations have been developed for normal boiling and melting point, critical temperature, pressure, and volume, acentric factor, liquid molar volume, and vapor pressure of n-paraffins and n-olefins. These correlations are useful for engineering applications in a number of areas, including petroleum and synthetic fuels processing. The form of the asymptote used in the correlations is based on predictions made with lattice-fluid theory. It has been demonstrated the ABCs yield accurate and consistent predictions and extrapolate well to higher carbon numbers. For critical temperature and density, the predicted values are in general agreement with the results from molecular simulations. In part 1, the ABCs for Tc, Pc, and Vc were shown to be superior to general group contribution methods. For the other properties, the ABCs represent the only reliable method for estimating property values for high molecular weight homologous compounds. In the case of vapor pressure, the ABC performs surprisingly well outside the temperature range it was developed for. The limiting values for melting point and liquid density are also in reasonable agreement with measurements for high molecular weight, linear polyethylenes. It has also been shown the ABCs for normal boiling point, liquid molar volume, and vapor pressure can be used to calculate “effective” carbon numbers for other homologous series of compounds. Predicted property values for these other series are reasonably accurate. It is believed that this approach is generally applicable to other properties as well. Part 3, the final in this series, presents ABCs developed for thermal and transport properties of n-paraffins and n-olefins: ideal-gas enthalpy and free energy of formation, ideal-gas heat capacity, enthalpy of vaporization, liquid heat capacity, liquid viscosity, thermal conductivity, and surface tension. Lattice theories for these properties are also described and used to establish asymptotic behavior. Nomenclature c ) correction to the number of external degrees of freedom per mer E ) internal energy k ) Boltzmann constant (1.3806 × 10-16 erg/(molecule K)) M ) molecular weight n ) carbon number N ) Avogadro’s number (6.0222 × 1023) P ) pressure, bar Q ) canonical-ensemble partition function s ) number of segments or mers in chain molecule S ) entropy T ) temperature, K v ) volume per molecule v* ) close-packed or hard-core volume per mer V ) volume or molar volume, cm3/(g‚mol) Y ) physical property correlated z ) coordination number of the lattice Z ) compressibility
Greek Letters β, γ ) correlating parameters ) nonbonded, mer-mer interaction energy ζ ) mean number of external contacts per mer η/v* ) mean intermolecular energy per contact pair per λ ) heat of vaporization per molecule µ ) chemical potential per molecule µ0 ) additive constant in expression for µ F ) density, g/cm3 ω ) acentric factor Ω ) system degeneracy Subscripts b ) at normal boiling point c ) at critical point g ) gas phase l ) liquid phase L ) as a liquid, nominally at 1 atm m ) at normal melting point 0 ) at effective carbon number of zero r ) reduced property based on critical property s ) solid phase vap ) of vaporization ∞ ) as carbon number approaches infinity Superscripts and Accents ˜ ) reduced property in EOS * ) scaling parameter in EOS Notation in Tables and Figures CNs ) those components whose reported values were used in ABC pts ) the number of reported values used in ABC ref ) literature citation for data used in ABC in order of importance RMSE ) root mean square error, [∑(yi - xi)2]1/2/p AD ) average deviation, ∑(yi - xi)/p AAD ) absolute average deviation, ∑|yi - xi|/p %AAD ) percent absolute average deviation, 100 ∑|(yi xi)/yi|/p; for |yi| < 1, yi set to 1 %MAD ) percent maximum absolute deviation, 100 |(yi xi)/yi|max yi ) reported value xi ) correlated value p ) number of reported values correlated data, used ) data used to develop correlation data, not used ) data not used to develop correlation data sat. ) data for saturated liquid, not used to develop correlation
Appendix. Results from Lattice-Fluid Theory In order to predict the limiting PVT behavior of long chain molecules, it is necessary to develop statisticalmechanical expressions for pressure P and chemical potential µ as a function of the number of segments s in the molecule. P and µ are related to the canonicalensemble partition function Q by
P ) kT
|
∂ ln Q ∂V N,T
(A1)
and
µ ) -kT
|
∂ ln Q ∂N V,T
(A2)
and Q is related to the number of molecules N, volume V, and energy E of the system by
Q(N,V,T) ) Ω(N,V) e-E/kT
(A3)
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1905
Figure 12. Two-dimensional lattice fluid showing distribution of hexamers and lattice holes.
where Ω is the system degeneracy: the number of configurations available to the N molecules, each of which occupies s sites (an s-mer), and N0 vacant holes in the lattice fluid. An example of a two-dimensional lattice fluid is shown in Figure 12. Flory’s approximation (Flory, 1942) is used to express Ω as a function of N and N0
( ) (
z-1 Ω) e
(s-1)N
)
z 2(z - 1)
N
(N0 + sN)N0+N N0N0NN
(A4) S ) k ln Ω
Clearly, N0 + sN is the total number of sites, NT, and is equal to V/v* where v* is the close-packed mer volume. The lattice potential energy is approximated (Sanchez and Lacombe, 1976) as
( )
E ) -NT(z/2)
sN NT
2
) -*(sN)2/NT
kT sv* v* * sv* 2 ln 1 + (s - 1) v* v v v* v
[ (
]
)
( )
(A6)
[ (sv*v ) - s ln(1 - sv*v ) - (s - 1)] sv* 2s*( ) (A7) v
µ ) µ0(s,T) + kT ln
where v is the volume per molecule, equal to V/N. Equation A6 is the EOS for the lattice fluid and can be rewritten in the form given by eq 2. In general, eq A6 cannot be explicitly solved for either v or T. However for v large, eq A6 reduces to the ideal-gas law: P ) kT/ vg. While eq A6 does not satisfy a simple corresponding states principle, it can be shown in the limit as s f ∞, v f sv*/P∞-1(T) ≡ sv∞(P,T); that is, molecular volume is proportional to s. In the lattice-fluid model, it is not necessary to specify the relationship between s and n. For polymersolvent mixtures, it is convenient to relate s to the size of the solvent molecule. However, the best fit of PVTproperty data with n is obtained with s ) n. Internal Energy and Entropy. On the basis of eq A5, the internal energy per molecule is given by 2
E/N ) -*s v*/v
(A8)
In the limit as s f ∞, E/N f -*sv*/v∞. Therefore, the molar internal energy of the system is asymptotically linear in s.
(A9)
On the basis of eq A4, the entropy per molecule is given by
((
) (
) ( ))
v sv* v* S )k sln 1 - ln N v* v v
(A5)
After substitution of V/v* for NT, P and µ may be evaluated from eqs A1 through A5
P)-
It is important to note, however, that the above formulation for Q, eqs A3-A5, only considers external (i.e. intermolecular) modes of motion. The high-frequency internal (i.e. intramolecular) modes are assumed to be unaffected by neighbors in the lattice and are neglected. The intramolecular potentials associated with the external modes are disregarded altogether and are treated as translational motions. Thus, eq A8 is not a complete expression of the internal energy of the system. However, it is clear that for a chainlike molecule of repeating segments, the potential energy associated with internal vibrational and rotational modes of motion will also be proportional to s in the limit of long chains. That the above treatment is still only approximate is clear from examining the derivative of E/N with respect to temperature at constant v (i.e. constant volume heat capacity). Equation A8 gives a value of zero for this derivative which is clearly at odds with both the value for an ideal gas, 3/2k, and the value for a classical crystalline solid, 3k. This is a general limitation of both lattice and cell models and results from a lack of any direct temperature dependence on the intermolecular potential. The entropy of the system is given by the well-known Boltzmann relationship:
+
(
k((s - 1) ln(z - 1) - 1) + k ln
)
z (A10) 2(z - 1)
In the limit as s f ∞, S/N f ks((1 - v∞/v*) ln(1 - v∞/v*) + ln(z - 1) - 1). Therefore, the molar entropy of the system is asymptotically linear in s. It follows from their definitions that all other thermodynamic energy functions, enthalpy, free energy, etc., are also asymptotically linear in s. Critical Properties. Application of the critical-point criteria
|
∂P )0 ∂V T
(A11)
and
|
∂2P ∂V2
)0
(A12)
T
yields the following relationships for critical volume, temperature, and pressure as functions of s:
1 sv* ) vc (1 + s1/2)
(A13)
kTc 2s ) * (1 + s1/2)2
(A14)
(
)
v*Pc 1 2s1/2 - 1 ) ln 1 + 1/2 kTc 2s s
(A15)
In the limit as s f ∞, vc f v*s3/2, Tc f 2*/k, and Pc f 0. The critical compressibility Zc ) Pcvc/kTc f 1/3, which is clearly too high (see Figure 8).
1906 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Vapor Pressure, Boiling Point, Heat of Vaporization. The criteria for vapor-liquid equilibrium are
Pg ) Pl
Applying expressions for µs and µl, eqs A23 and A7, respectively, combining with eq A6 written for the liquid, and solving for T gives
(A16) kTm ) *
and
(A17)
µg ) µl
In theory, eqs A6 and A7 may be substituted into eqs A16 and A17, resulting in two simultaneous equations which may be solved for the independent variables, vg and vl. If the expression for vg is then substituted back into eq A6, the vapor pressure as a function of temperature will be obtained. Unfortunately, these equations are highly nonlinear and cannot be solved analytically for vg and vl. If we assume the gas phase is ideal (P small, vg large), however, an analytical expression can be obtained for vapor pressure:
( )
ln Pvap ) ln
( )
v* s* sv* 2 kT + s(s - 1) + vl vl kT vl 2s* sv* (A18) kT vl
( )
If it is further assumed the number of holes in the liquid phase is negligible, then vl can be replaced by sv*. This assumption will be reasonable at low temperatures. Equation A18 then reduces to
s* kT + (s - 1) ln Pvap ) ln sv* kT
( )
(A19)
In the limit as s f ∞, ln Pvap f s - s*/kT, and Tb f */k. Thus, ln Pvap is asymptotically linear in carbon number and Tb approaches a finite limit. For an ideal gas, the Clapeyron equation is λ ) (kT2){(d ln Pvap)/dT}. Differentiating eq A19 yields
λ ) kT + s*
(A20)
Thus, the heat of vaporization is asymptotically linear in carbon number. For further details on the above derivations, see: Kurata and Isida (1955), Nakanishi et al. (1960), and Sanchez and Lacombe (1976, 1978). Melting Point. For a solid phase, it can be assumed there are no vacant holes in the lattice; then, N0 ) 0 and NT ) sN ) V/v*. The expressions given above for Ω and E, eqs A4 and A5, reduce to
Ω)
(z -e 1)
(s-1)N
(1/2)NsN
E ) -*sN
(A21) (A22)
On the basis of eq A2, the chemical potential in the solid phase is
µs ) µ0(s,T) - s*
(A23)
The criterion for phase equilibrium between solid and liquid is
µs ) µl
(A24)
(
)
( )
sv* P 2sv* sv* 2 -s -1 vl * vl v* sv* ln + s(s - 1) - (s - 1) vl vl s
( )
(A25)
Then, at low pressure and in the limit as s f ∞, Tm f (1 - v*/vl∞)*/k. Thus, Tm approaches a finite limit. This result assumes the temperature dependence of vl is negligible. This same assumption is implicit in the substitution of sv* for vl in the derivation of eq A19. This latter substitution is not appropriate in this derivation since vs and vl are of the same magnitude. The quantity v*/vl∞ may be evaluated from the limiting values obtained from the ABCs for Tm and Tb: Tm/Tb ) 0.3832 and therefore v*/vl∞ ) 0.6168. The above treatment assumes the chain molecules are arranged randomly on the lattice, which can only truly be representative of an amorphous-like state. A more detailed analysis is required for a solid exhibiting the crystalline states associated with n-paraffins. To our best knowledge, the above formulation for the melting point of chain molecules has not previously been reported. Literature Cited (1) Ambrose, D. Vapor-Liquid Critical Properties; Technical Report N.P.L. 107; National Physics Laboratory: U.K., 1980. (2) Anselme, M. J.; Gude, M.; Teja, A. S. The Critical Temperatures and Densities of The n-Alkanes from Pentane to Octadecane. Fluid Phase Equilib. 1990, 57, 317-326. (3) API Research Project 44, Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; Carnegie Press: Pittsburgh, PA, 1953. (4) API Research Project 42, Properties of Hydrocarbons of High Molecular Weight; American Petroleum Institute: New York, 1966. (5) API Technical Data BooksPetroleum Refining, 2nd ed. 1970; American Petroleum Institute: Washington DC, 1976 revision. (6) API Technical Data BooksPetroleum Refining, 5th ed. 1992; American Petroleum Institute: Washington, DC, 1992. (7) Beaton, C. F.; Hewitt, G. F. Physical Property Data for The Design Engineer; Hemisphere Publishing: New York, 1989. (8) Broadhurst, M. G. An Analysis of The Solid Phase Behavior of The Normal Paraffins. J. Research Natl. Bur. Stand. 1962a, 66A, 241-249. (9) Broadhurst, M. G. Extrapolation of The Orthorhombic n-Paraffin Melting Properties to Very Long Chain Lengths. J. Chem. Phys. 1962b, 36, 2578-2582. (10) Caldwell, L.; Van Vuuren, D. S. On The Formation of The Liquid Phase in Fischer-Tropsch Reactors. Chem. Eng. Sci. 1986, 41, 89-96. (11) Chen, F.; Fredenslund, A.; Ramussen, A. Group-Contribution Flory Equation of State for Vapor-Liquid Equilibria in Mixtures with Polymers. Ind. Eng Chem. Res. 1990, 29, 875-882. (12) Chiang, R.; Flory, P. J. Equilibrium between Crystalline and Amorphous Phases in Polyethylene. J. Appl. Polymer Sci. 1971, 15, 1277-1281. (13) Chung, C. I. Compressibility of Polyethylene Melts at Low Pressure. J. Appl. Polymer Sci. 1971, 15, 1277-1281. (14) Doolittle, A. K.; Peterson, R. H. Preparation and Physical Properties of A Series of n-Alkanes. J. Am. Chem. Soc. 1951, 73, 2145-2151. (15) Doolittle, A. K.; Doolittle, D. B. Compressions of Liquids: III. Temperature and Molecular Weight Dependence of The Hudleston Parameters for The n-Alkanes. AIChE J. 1960, 6, 157162. (16) Flory, P. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1942, 10, 51-61.
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1907 (17) Flory, P. J.; Orwoll, R. A.; Vrij, A. Statistical Thermodynamics of Chain Molecule Liquids. I. An Equation of State for normal Paraffin Hydrocarbons. J. Am. Chem. Soc. 1964, 86, 35073514. (18) Gray, R. D.; Heidman, J. L.; Springer, R. D. Characterization and Property Prediction for Heavy Petroleum and Synthetic Liquids. Fluid Phase Equilib. 1989, 53, 355-376. (19) Gude, M. T.; Rosenthal, A. S.; Teja, A. S. The Critical Properties of 1-Alkenes from 1-Pentene to 1-Dodecene. Fluid Phase Equilib. 1991, 70, 55-64. (20) Hadden, S. T. New Correlation for The Specific Heat of Liquid Hydrocarbons. Hydrocarbon Process. 1966, 45, 137-142. (21) Handbook of Chemistry and Physics, 56th ed., 1975-1976; CRC Press: Cleveland, OH, 1975. (22) Kontogeorgis, G. M.; Fredenslund, A.; Tassios, D. P. Chain Length Dependence of The Critical Density of Organic Homologous Series. Fluid Phase Equilib. 1995, 108, 47-58. (23) Kurata, M.; Isida, S. Theory of Normal Paraffin Liquids. J. Chem. Phys. 1955, 23, 1126-1131. (24) Magoulas, K.; Tassios, D. Thermophysical Properties of n-Alkanes from C1 to C20 and Their Prediction for Higher Ones. Fluid Phase Equilib. 1990, 56, 119-140. (25) Marano, J. J. Property Correlation and Characterization of Fischer-Tropsch Liquids for Process Modeling. Ph.D. Dissertation: University of Pittsburgh, Pittsburgh, PA, 1996. (26) Marano, J. J.; Holder, G. D. General Equation for Correlating the Thermophysical Properties of n-Paraffins, n-Olefins, and Other Homologous Series 1. Formalism for Developing Asymptotic Behavior Correlations. Ind. Eng. Chem. Res. 1997, 36, 1887-1894. (27) Morecroft, D. W. Vapor Pressures of Some High Molecular Weight Hydrocarbons. J. Chem. Eng. Data 1964, 9, 488-490. (28) Nakanishi, K.; Kurata, M.; Tamura, M. Physical Constants of Organic Liquids, A Nomograph for Estimating Physical Constants of Normal Paraffins and Isoparaffins. J. Chem. Eng. Data 1960, 5, 210-219. (29) National Research Council. International Critical Tables of Numerical Data, Physics, Chemistry and Technology, 1st ed.; McGraw-Hill: New York, 1926; 4th impression, Vol. I, Table C. (30) Olabisi, O.; Simha, R. Pressure-Volume-Temperature Studies of Amorphous and Crystallizable Polymers. I. Experimental. Macromolecules 1975, 8, 206-210. (31) de Pablo, J. J.; Laso, M.; Siepmann, J. I.; Suter, U. W. Continuum-Configurational-Bias Monte Carlo Simulations of Long-Chain Alkanes. Mol. Phys. 1993, 80, 55-63. (32) de Pablo, J. J.; Laso, M.; Suter, U. W. Simulation of Polyethylene Above and Below The Melting Point. J. Chem. Phys. 1992, 96, 2395-2403. (33) Prigogine, I.; Bellemans, A.; Naar-Colin, C. Theorem of Corresponding States for Polymers. J. Chem. Phys. 1957, 26, 751755.
(34) Raff, R. A. V. Polyethylene. Encyclopedia of Polymer Science and Technology, Plastics, Resins, Rubbers, Fibers; John Wiley & Sons: New York, 1968; Vol. 8, pp 276-332. (35) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987. (36) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases & Liquids, 3rd ed.; McGraw-Hill: New York, 1977. (37) Reinhard, R. R.; Dixon J. A. Tetranonacontane. J. Org. Chem. 1965, 9, 1450-1453. (38) Richardson, M. J.; Flory, P. J.; Jackson, J. B. Crystallization and Melting of Copolymers of Polymethylene. Polymer 1963, 4, 221-236. (39) Rosenthal, D. J.; Teja, A. S. The Critical Properties of n-Alkanes Using A Low-Residence Time Flow Apparatus. AIChE J. 1989, 33, 1829-1834. (40) Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys. Chem. 1976, 80, 2352-2362. (41) Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145-1156. (42) Schaer, A. A.; Busso, C. J.; Smith, A. E.; Skinner, L. B. Properties of Pure Normal Alkanes in The C17-C36 Range. J. Am. Chem. Soc. 1955, 77, 2017-2018. (43) Siepmann, J. I.; Karborni, S.; Smit, B. Simulating The Critical Behavior of Complex Fluids. Nature 1993, 365, 330-332. (44) Steele, W. V.; Chirico, R. D. Thermodynamic Properties of Alkenes. J. Phys. Chem. Ref. Data 1993, 22, 377-430. (45) Teja, A. S.; Lee, R. J.; Rosenthal, D.; Anselme, M. Correlation of The Critical Properties of Alkanes and Alkanols. Fluid Phase Equilib. 1990, 56, 153-169. (46) Templin, P. R. Coefficient of Volume Expansion for Petroleum Waxes and Pure n-Paraffins. Ind. Eng. Chem. 1955, 48, 154161. (47) Tsonopoulos, C.; Tan, Z. The Critical Constants of Normal Alkanes from Methane to Polyethylene II. Application of The Flory Theory. Fluid Phase Equilib. 1993, 83, 127-138. (48) Vargaftik, N. B. Tables on The Thermophysical Properties of Liquids and Gases, 2nd ed. (English trans.); John Wiley & Sons: New York, 1975. (49) Warth, A. H. The Chemistry and Technology of Waxes; Reinhold Publishing: New York, 1947; Chapter 1, Table 1.
Received for review August 15, 1996 Revised manuscript received January 6, 1997 Accepted January 6, 1997X IE960512F
X Abstract published in Advance ACS Abstracts, March 1, 1997.
