A General Equation for Correlating the Thermophysical Properties of n

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Ind. Eng. Chem. Res. 1997, 36, 2399-2408

2399

A General Equation for Correlating the Thermophysical Properties of n-Paraffins, n-Olefins, and Other Homologous Series. 3. Asymptotic Behavior Correlations for Thermal and Transport Properties John J. Marano* and Gerald D. Holder Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261

In this final paper of the series, asymptotic behavior correlations (ABCs) are presented for thermal properties (ideal-gas enthalpy and free energy of formation, ideal-gas heat capacity, enthalpy of vaporization, and liquid heat capacity) and for transport properties (liquid viscosity, thermal conductivity, and surface tension). The theoretical basis for 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 specific heat, viscosity, thermal conductivity, and surface tension, comparisons are made with high molecular weight, linear polyethylenes and other long-chain molecules. Introduction In part 1 of this series (Marano and Holder, 1997a), the development of a generalized asymptotic behavior correlation applicable to homologous series of compounds was described. The form of this correlation is

Y ) Y∞ - ∆Y0 exp(-β(n ( n0)γ)

(1a)

Y∞ ) Y∞,0 + ∆Y∞(n - n0)

(1b)

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, the 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. Part 2 (Marano and Holder, 1997b) applied eq 1 to correlate PVT-related properties. In this paper, the final in the series, the theoretical basis for Y∞ will be discussed and eq 1 will be applied to correlate thermal and transport properties of n-paraffins and n-olefins (1alkenes). 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. Lattice-Fluid Theory In the appendix to Part 2, it was shown that latticefluid theory predicts all thermodynamic energy functions: internal energy, enthalpy, entropy, free energy, etc., are asymptotically proportional to carbon number. It directly follows that ideal-gas enthalpy and free energy of formation, ideal-gas heat capacity, and liquid * Corresponding author. E-mail address: marano@ fetc.doe.gov. Present address: U.S. Department of Energy, Federal Energy Technology Center, P.O. Box 10940, Pittsburgh, PA 15236-0940. S0888-5885(96)00513-1 CCC: $14.00

heat capacity are type II properties. It was also shown that enthalpy of vaporization for the lattice fluid can be expressed as

∆Hvap ) ∆En + RT

(2)

This equation assumes the vapor phase can be approximated as an ideal gas. Thus, at conditions where this assumption holds, enthalpy of vaporization at fixed temperature is also a type II property. In fact, theory predicts it is linear for all carbon numbers. The enthalpy of vaporization at the normal boiling point, however, is a linear combination of a type II property ∆En and a type I property Tb. In the appendix to this paper, lattice-fluid theory is extended to predict the thermodynamic energy functions for a liquid-gas interface. These properties are directly related to surface tension, and it is shown that surface tension is a type I property. Limiting Behavior of Viscosity In the appendix, it is also shown that a modification to Eyring rate theory predicts the logarithm of viscosity is a type II property. This theory is in many ways similar to lattice-fluid theory in that it assumes the existence of holes within a quasi-crystalline liquid lattice. Resistance to flow is the result of activation barriers for the movement of segments of molecules into and out of these holes. However, unlike other properties considered, the predicted asymptotic behavior does not agree with observed behavior for high molecular weight polyethylenes. Measurements for polyethylenes and other linear polymers indicate that, at a critical carbon number (actually a critical range of carbon numbers), a transition occurs in the flow behavior of polymer melts. The rate of increase in viscosity with carbon number declines, and the onset of non-Newtonian behavior is observed. Above the critical carbon number, polymer melts and solutions are pseudoplastic, or shear thinning, materials: their apparent viscosities decrease as shear force is applied. The critical carbon number for nparaffins/linear polyethylenes has been reported by © 1997 American Chemical Society

2400 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 1. ABC Parameters for Ideal-Gas Properties of n-Paraffins and n-Olefins

Y ) ∆Y∞(n - n0) enthalpy of formation/RT0 (T0 ) 298.15 K) CNs Pts ref n0 ∆Y∞ RMSE AD AAD % AAD

n-paraffin

n-olefin

6-20 15 23 -2.111 890

Gibbs free energy of formation/RT0 (T0 ) 298.15 K) n-paraffin

n-olefin

6-16, 18 12 23 3.977 209

6-20 15 23 6.045 009

6-16, 18 12 23 -4.402 804

0.0031 0.0000 0.0088 0.0164

0.0066 0.0000 0.0169 0.1431

n-paraffin

n-olefin

6-20 6-16, 18 120 96 23 23 -0.284 370 0.226 568 see Table 2 0.0127 0.0116 0.0051 -0.0110 0.0465 0.0371 0.1217 0.1087

3.3931

-8.3206 0.0033 0.0000 0.0108 0.0093

ideal-gas heat capacity/R

0.0054 0.0000 0.0149 0.0280

Table 2. Temperature-Dependent ABC Parameters for Ideal-Gas and Liquid Heat Capacity

∆Y ) A + BT + CT 2 + DT3 ideal-gas heat capacity/R ∆Y 0 A B C D

liquid heat capacity/R

∆Y∞

∆Y 0

∆Y∞

-0.091 905 5 0.011 308 0 -6.379 20 × 10-6 1.406 05 × 10-9

-58.0001 0.330 453 0 -5.860 37 × 10-4 3.243 82 × 10-8

0.017 811 8 0.021 419 4 -3.445 32 × 10-5 2.003 73 × 10-8

temp. range of data (°C) n-paraffin n-olefin

ideal-gas heat capacity/R

liquid heat capacity/R

0-300 0-300

0-250 0-100

Schreiber et al. (1963) to be 286. The transition can be explained with the concept of chain entanglement. As the carbon number of the fluid molecules increases, the molecules become more and more entwined, such that any long-range motion of one molecule requires the cooperative motion of other molecules, and flow becomes more viscous. In the critical region, a transition occurs in the mechanism for viscous transport. Bulk flow below the critical carbon number is predominantly due to the motion of individual molecules. Above, bulk flow is predominantly due to motion of entangled chains. Entangled molecules can be visualized as soft spheres undergoing dilation and compression due to the applied shear force. In the critical region, the spheres begin to rotate and roll with the flow. This same concept explains the pseudoplastic behavior observed above the critical carbon number. Since the entangled chains are not held together by permanent bonds, as the rate of shear increases, the chains tend to slip past one another, eventually becoming disentangled.

Figure 1. Asymptotic behavior correlations for ideal-gas enthalpy and Gibbs free energy of formation at 25 °C of n-paraffins and n-olefins.

Results: Ideal-Gas Properties The ideal-gas properties of interest (enthalpy and Gibbs free energy of formation and heat capacity) are necessary for establishing the reference state for calculation of real-fluid enthalpies, entropies, etc., and for formulating phase and chemical equilibrium. In addition, real fluids in many instances may be accurately approximated as ideal gases. The ABC parameters for these ideal-gas properties are listed in Table 1. All three properties exhibit type II behavior. The reference temperature for the formation properties is 25 °C. The coefficients for the temperature-dependent polynomials used for ideal-gas heat capacity are given in Table 2, and the correlations are shown in Figures 1 and 2. Above C5, the ideal-gas properties are linear with carbon number. In fact, a linear approximation works

Figure 2. Asymptotic behavior correlations for ideal-gas heat capacity of n-paraffins and n-olefins.

well down to C3. Therefore, the ABC was simplified, and only the coefficients for the asymptote were determined. The data for ideal-gas heat capacity were generated from pure-compound heat-capacity polynomials presented by Reid et al. (1987). The ABC is in good agreement with the linear equation for the specific heat of polyethylene over the range of 100-300 °C proposed

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2401 Table 3. ABC Parameters for Enthalpy of Vaporization and Liquid Heat Capacity of n-Paraffins and n-Olefins

Y ) Y∞,0 + ∆Y∞(n - n0) - ∆Y0 exp(-β(n ( n0)γ)

n+n0

for Cp,L/R,

n - n0

for ∆H/RT0

enthalpy of vaporization/RT0 at 25 °C (T0 ) 298.15 K) CNs Pts ref n0 ∆Y0 Y∞,0 ∆Y∞ β γ RMSE AD AAD % AAD

n-paraffin

n-olefin

6-20 15 1 0.112 756

6, 7, 8 3 29 0.389 302

0 1 1.995 16 0.0072 0.0000 0.0166 0.0567

0.0686 0.0000 0.1058 0.7478

at normal boiling point (T0 ) 298.15 K) n-paraffin

n-olefin

3-20 3-16, 20 18 15 1, 3 3, 29 -8.051 557 -7.967 319 3080.98 812.14 0 1.293 274 0.015 618 5 0.0111 0.0307 -0.0065 0.0149 0.0343 0.0961 0.1885 0.5807

Figure 3. Asymptotic behavior correlations for enthalpy of vaporization at 25 °C and the normal boiling point of n-paraffins and n-olefins.

by Maloney and Prausnitz (1974). The percent absolute average deviation (% AAD) and percent maximum absolute deviation (% MAD) are 0.44 and 1.18, respectively. The higher-order temperature dependence used here enables the ABC to be applicable over a wider temperature range than the equation of Maloney and Prausnitz. Enthalpies of Vaporization Enthalpy of vaporization data are generally available at 25 °C and the normal boiling point. Only scattered values can be found at other temperatures and often are based on inadequate interpretation of vapor-pressure data. For these reason, ABCs were only prepared for 25 °C and the normal boiling point. Table 3 contains the ABC parameters for these properties. Comparisons with the data are shown in Figure 3. The basis for the asymptotic behavior used in the correlations for enthalpy of vaporization is eq 2. This equation predicts type II behavior with an intercept of RT. Examination of all the data available for nparaffins indicates a higher-order temperature dependence, a probable result of nonideal-gas behavior at higher temperatures. Even so, as shown in Figure 3, the ABC matches the data extremely well at 25 °C. As was done for the ideal-gas properties, only the coefficients for the asymptote were determined. The behavior of enthalpy of vaporization at the normal boiling point is quite different. Type I behavior is approached even though the ∆En term in eq 2 should

liquid heat capacity/R n-paraffin 3-18 36 4, 10, 11, 28, 29 1.153 418

0.0480 -0.0310 0.1163 0.3027

n-olefin 3-14, 16-18 23 3, 4, 10, 11, 26, 28, 29 1.523 496 see Table 2 0 see Table 2 0.183 717 0.753 795 0.1168 0.0286 0.2087 0.5922

Figure 4. Asymptotic behavior correlations for liquid heat capacity of n-paraffins and n-olefins.

be dominant over RTb at large carbon numbers. Again, this can be interpreted as a result of nonideal-gas behavior. As described in part 2, the normal boiling points of n-paraffins and n-olefins approach their critical temperatures at higher carbon numbers, and, thus, the volume change upon vaporization approaches zero. As a result, enthalpy of vaporization at the normal boiling point approaches a finite limit. Substitution of the ideal-gas law into the Clapeyron equation, the basis for eq 2, is invalid at these conditions, and eq 2 does not describe the observed behavior correctly. Liquid Heat Capacity The data available for liquid heat capacity of pure n-paraffins and n-olefins are limited to lower carbon numbers and moderate temperatures. Data for nparaffins at 1 atm are only available up to 150 °C and for n-olefins up to 100 °C. In order to extend the applicability of any correlation developed, saturatedliquid heat capacities for n-C10 were used up to 250 °C. This is a modest extrapolation since the normal boiling point of n-C10 is 174.2 °C, and the critical temperature is 344.6 °C. The temperature-independent ABC parameters based on these data are given in Table 3 and the temperature-dependent parameters in Table 2. In Figure 4, comparisons with the data are shown for three representative temperatures: 25, 100, and 250 °C. Plots showing the error distributions for liquid heat capacity as a function of carbon number and temperature are presented elsewhere (Marano, 1996). The errors are typically less than 1%; however, there are some out-lying points in the n-olefin data set.

2402 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 4. ABC Predictions for Specific Heats (J/(g K)) of Polyethylenes temp (°C) 40 110 140 180

Marlex 50 (ref 31)

DuPont PE (ref 31)

amorphous PE (ref 32)

ABC for liq. heat capacity

2.453 2.554

2.297 2.461 2.531 2.624

2.35 2.55 2.61 2.68

2.598 2.688

The ABCs developed from the n-paraffin and n-olefin data were extrapolated to obtain limiting values for specific heat (Cp,L/M). These were compared with the values for polyethylene (PE) correlated by Wunderlich and Dole (1957) and Wunderlich and Baur (1970). Results of this comparison are given in Table 4. Marlex 50 (Phillips) is highly linear, the DuPont PE contains moderate branching, and the amorphous PE is highly branched. The degree of branching is evident from the lowest temperature at which specific heats could be measured for the various melts. The melting point of completely linear polyethylene is expected to be between 140 and 145 °C (part 2). The comparison with Marlex PE is excellent with an average error of 0.4%, and the comparison is still reasonable for the branched PEs with errors between 2 and 4%. Liquid Viscosity In the appendix, it was shown that Eyring rate theory predicts the asymptote for the natural logarithm of viscosity is linear with carbon number. That is, logarithm of viscosity is a type II property. As was discussed above, this relationship should hold at least up to a carbon number of 286. Above this carbon number, the mechanism for viscous transport changes with the onset of non-Newtonian behavior, and the linear relationship is no longer valid. Considerable data exist for viscosity of pure n-paraffins, with measured values available for carbon numbers as high as 64 and 94. The ABC parameters developed from these data are given in Tables 5 and 6. Figure 5 shows liquid viscosity at three representative temperatures in the range correlated with the ABCs. Plots showing the error distributions for liquid viscosity as a function of carbon number and temperature are presented elsewhere (Marano, 1996). The errors obtained are larger than those with some of the other ABCs. In particular, the error is large at low temperatures and high carbon numbers, which corresponds to high viscosities. Relative errors are typically less than 0.05 for logarithm of viscosity outside this region. Doolittle and Peterson (1951) have shown the maximum error in viscosity per unit temperature inaccuracy occurs for the most viscous compounds at any temperature. This is due to the strong temperature dependence of viscosity. They also indicate impurities, isomers, and n-paraffins of slightly differing carbon numbers are more prevalent in higher molecular weight samples and result in additional uncertainty in measured values for viscosity. The effect of branching can be seen from the comparison presented in Table 7, where the predictions from the ABC for n-paraffins are compared with data for several branched long-chain paraffins. The branched compounds are 3-ethyltetracosane (PSU 109), hexamethyltetracosane (PSU 223), and 11,20-di-n-decyltriacontane (PSU 59). Viscosities for these compounds were measured as part of API Project 42 (1966). PSU 109 and PSU 223 both have a C24 backbone; PSU 109

contains one ethyl side chain and PSU 223 six methyl side chains. The error in viscosity is about 5% for PSU 109 at 37.8 and 98.9 °C, and 1.0% at 60 °C. For PSU 223, the error falls from about 3% at 37.8 °C to 1% at 98.9 °C, and is about 8% at 0 °C. Thus, for the same carbon-number backbone, the presence and frequency of branching of small side chains do not appear to have a large effect at low-to-moderate viscosities, with any effect much more prevalent at high viscosities. PSU 59 has a C30 backbone and two C10 side chains. Over the temperature range from 37.8 to 98.9 °C, the errors are much larger, dropping from about 260 to 70%. Thus, the length of the side chain has a critical effect on viscosity, with this effect again more pronounced at high viscosities. It is worth noting that the ABC underpredicts the viscosities of PSU 223 and overpredicts those of PSU 59. Predictions made with the viscosity ABC were compared to measured values (Gormley and co-workers, 1995) for a low molecular weight, highly-branched, commercial polyethylene. The number-average carbon number for this polyethylene is 69.7 and the weight average is 125, indicating the polyethylene has a broad molecular weight distribution. The ABC estimated viscosities, calculated from both carbon number averages, are given in Table 8. The number average underpredicts the measured viscosities by a large amount, whereas the weight average compares more favorably, with errors of roughly 2 and 11% at 130 and 150 °C, respectively. This result is in agreement with the recommendations of Mendelson (1968) that the correct average to use lies between the weight and nexthigher average, approaching the latter as the distribution becomes broader. Figure 6 shows a comparison between predictions from the viscosity ABC and the empirical correlation of Schreiber et al. (1963). This correlation is based on data (also shown) for a fractionated linear polyethylene (Phillips) and predicts a linear dependence for logarithm of Newtonian viscosity (zero-shear limit) versus logarithm of molecular weight (weight average). This dependence has been shown to hold for other polymers as well. Data for higher carbon-number n-paraffins have been included in Figure 6. Where the two curves touch identifies the critical carbon number for linear polyethylene. This number is roughly 450 and is higher than the value reported by Schreiber of 286, based on linear extrapolation of the data of Doolittle and Peterson (1951). Clearly, the Schreiber equation should not be extrapolated much below 450 and the ABC not much above. In agreement with the earlier discussion of the limitations of Eyring rate theory, the ABC overpredicts the Newtonian viscosity above the critical carbon number, evidence of the shift in mechanism. Also shown in Figure 6 are measured viscosities for four other polyethylenes (Dettre and Johnson, 1966). These are as follows: Allied Chemical A-C PE 6, a highly-branched polyethylene with 5 methyl groups per 100 carbon atoms and a weight-average molecular weight of 2000; Bakelite DYLT, also highly branched (5.2 methyl groups) with a molecular weight of 7000; an experimental DuPont PE (3.5 methyl groups) with a molecular weight of 340 000; and DuPont Alathon 7050, highly linear (0.2 methyl groups) with a molecular weight of 67 000. The first two polyethylenes have carbon numbers less than the critical value. Similar to the results for PSU 223, the ABC underpredicts the viscosities for these branched polyethylenes. The Du-

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2403 Table 5. ABC Parameters for Liquid Viscosity, Thermal Conductivity, and Surface Tension of n-Paraffins and n-Olefins

Y ) Y∞,0 + ∆Y∞(n - n0) - ∆Y0 exp(-β(n - n0)γ) ln viscosity (cP) CNs Pts ref n0 ∆Y0 Y∞,0 ∆Y∞ β γ RMSE AD AAD % AAD

thermal conductivity (W/m K)

surface tension (dyn/cm)

n-paraffin

n-olefin

n-paraffin

n-olefin

3-20, 23, 24, 26, 28, 32, 35, 36, 44, 64, 94 123 1, 2, 4, 8, 24, 29 -2.293 981 see Table 6 57.8516 see Table 6 2.476 409 0.011 211 7 0.0097 0.0015 0.0284 2.2655

3-20 56 1, 2, 4 -1.706 413

3-20 82 1, 4, 10, 11 -1.201 270

0.0097 -0.0044 0.0254 2.2492

0.0004 0.0002 0.0008 0.6170

3, 5-8, 10, 12, 14, 16, 18, 20 19 1, 4, 10, 11 -0.364 059 see Table 6 see Table 6 0 1.241 494 0.235 832 0.0016 -0.0018 0.0032 2.3730

n-paraffin

n-olefin

3-20 3, 5-20 64 48 1, 4, 15, 16, 29 3, 4, 11, 15, 29 0.264 870 0.236 483 see Table 6 see Table 6 0 2.511 846 0.201 325 0.0251 0.0171 0.0436 -0.0445 0.0810 0.0581 0.4118 0.2542

Table 6. Temperature-Dependent ABC Parameters for Liquid Viscosity, Thermal Conductivity, and Surface Tension

∆Y ) A + B/T + C ln T + DT2 + E/T2

∆Y or Y ) A + BT + CT2 ln viscosity (cP) A B C D E

for ln µL

for λL, σ

thermal conductivity (W/(m K))

surface tension (dyn/cm)

∆Y0

∆Y∞

∆Y0

Y∞,0

∆Y 0

Y∞,0

-602.688 77 866.8 198.006 -4.180 77 × 10-5 -2.494 77 × 10 6

0.029 019 6 -241.023 0.044 095 9 -1.848 91 × 10-7 56561.7

0.069 095 5 0.001 730 44 0

0.212 451 -4.103 23 × 10-5 0

627.213 -0.882 888 0.002 681 88

73.8715 -0.177 123 1.545 17 × 10-4

temp. range of data (°C) n-paraffin n-olefin

ln viscosity (cP)

thermal conductivity (W/(m K)

surface tension (dyn/cm)

0-300 0-150

0-300 0-100

0-150 0-100

Table 7. Comparison with Viscosities (cP) of Some Branched Long-Chain Compounds temp (°C)

Figure 5. Asymptotic behavior correlations for liquid viscosity of n-paraffins and n-olefins.

Pont PEs lie above the critical carbon number. The viscosity of the DuPont linear PE is in fair agreement with the equation of Schreiber and co-workers, whereas the branched PE is well below the predicted value. This may be evidence of non-Newtonian behavior at the conditions used for this measurement which were not identified. Liquid Thermal Conductivity Very limited data are available for liquid thermal conductivity of n-paraffins, and especially n-olefins, for which data are only available up to 100 °C. It is also clear on examination of these data that considerable scatter exists. As Jamieson (1979) has commented, most investigators claim accuracy for their individual thermal conductivity measurements better than what is apparent when the data from all investigations are

37.8 60.0 98.9 0.0 20.0 37.8 60.0 98.9 37.8 60.0 98.9

PSU 109, CN ) 26 (ref 2)

PSU 223, CN ) 30 (ref 2)

PSU 59, CN ) 50 (ref 2)

ABC for viscosity

43.96 19.60 7.013

9.668 5.116 2.347 108.3 34.86 16.20 7.808 3.234 158.0 47.78 11.88

9.176 5.170 2.480 117.5 36.36 16.68 7.837 3.204

Table 8. ABC Predictions for Viscosities (cP) of Low Molecular Weight Polyethylene temp (°C)

commercial PE (ref 12)

130 150

71.9 46.3

ABC for viscosity #ave. CN ) 67.9 wt. ave. CN ) 125 12.1 7.9

69.7 34.8

plotted together. In the measurement of the thermal conductivity of a liquid, it is extremely difficult to eliminate or properly account for the effects of natural convection. In the appendix it is shown, based on analogy with the kinetic theory of gases, that liquid thermal conductivity is a type I property. The limiting value for large carbon numbers is a function of temperature. The temperature-independent ABC parameters obtained appear in Table 5 and the temperature-dependent

2404 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

Figure 6. Prediction of melt viscosity of polyethylenes at 190 °C.

Figure 8. Asymptotic behavior correlations for surface tension of n-paraffins and n-olefins. Table 10. ABC Predictions for Surface Tensions (dyn/ cm) of Polyethylenes temp (°C)

LPE of MW ) 67 000 (ref 30)

140 180

28.8 26.5

140 180 140 180 Figure 7. Asymptotic behavior correlations for liquid thermal conductivity of n-paraffins and n-olefins. Table 9. ABC Predictions for Thermal Conductivities (W/(m K)) of Polyethylenes temp (°C) 105-140 130-170 140

Marlex PEs (ave.) (ref 13)

HDPE (ref 27)

LDPE (ref 27) 0.19

0.24 0.26

ABC for thermal conductivity 0.197-0.195 0.196-0.194 0.195

parameters in Table 6. Figure 7 is a comparison of the ABCs with the data at three temperatures in the range correlated. The n-paraffin error is generally less than 2%, with an average of 0.62%. The error associated with the n-olefin data is considerably larger than that for the n-paraffin data and averages 2.4%. This is believed to be a result of the inaccuracy of the n-olefin data used. Plots showing the error distributions for liquid thermal conductivity as a function of carbon number and temperature (Marano, 1996) show all the data are more scattered around the ABC than with other ABCs previously discussed. The ABCs do an excellent job of “smoothing” the existing data. Table 9 contains a comparison of the ABC with thermal conductivity data for several polyethylenes. Shown are data for Marlex PEs, which are highly linear; a high-density polyethylene (HDPE), also linear; and a low-density polyethylene (LDPE), which is branched. The effect of branching is to lower the thermal conductivity. Polybutene, where branching is a regular feature of the repeating polymer unit, has a thermal conductivity of just 0.12 W/(m K) at 115 °C (Boggs et al., 1955), much lower than the values reported for polyethylenes in Table 9. Average carbon numbers were reported for the Marlex PEs and ranged between 9 000 and 14 000. The ABC values fall between the values reported for HDPE and LDPE. The average error for all PEs is about 16%. This error is much larger than the errors obtained for other polyethylene properties excluding viscosity. The source of discrepancy is difficult to determine given the uncertainty in the data and in fact

BPE of MW ) 7 000 (ref 30)

BPE of MW ) 2 000 (ref 30)

ABC for surface tension 27.1 25.3

27.3 24.6

27.0 25.2 26.5 24.1

26.3 24.5

may be a result of this uncertainty. Data for higher molecular weight n-paraffins would be particularly beneficial in improving and verifying the ABCs. The ABCs still provide the most reasonable method of extrapolating the existing data to higher carbon numbers, if not all the way to the polyethylene limit. Surface Tension An excellent compilation of surface tension data for a large number of compounds has been performed by Jasper (1972). Still, reliable data are limited to pure n-paraffins and n-olefins with carbon numbers less than 20. More importantly, data are limited to 150 °C for n-paraffins and 100 °C for n-olefins. Surface tension, like thermal conductivity, is a temperature-dependent type I property. ABC parameters for surface tension are also reported in Tables 5 and 6. In Figure 8, comparisons with the data are shown for three representative temperatures: 20, 100, and 150 °C. The errors are generally less than 1%, with averages of 0.41 and 0.25% for n-paraffins and n-olefins, respectively. These are of the same order of magnitude as obtained for thermal conductivity of n-paraffins. Plots showing the error distributions for surface tension as a function of carbon number and temperature (Marano, 1996) show the errors to be more systematic with temperature; however, a higher-order temperature correlation could not be justified based on the limited temperature range covered. There is very little difference between surface tensions of n-paraffins and n-olefins at any carbon number. Comparisons are made with various polyethylenes in Table 10. Correlations available for both linear and branched polyethylenes (Wu, 1974) were used to calculate values at 140 and 180 °C. The presence of branching reduces surface tension. The average error for the LPE is 4.9% and is 1.5% for the BPEs. As with thermal conductivity, the ABC appears to predict surface tensions lying between those of linear and branched polyethylene, but closer to the latter. The branched

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2405 Table 11. Comparison with Surface Tensions (dyn/cm) of Some Other Long-Chain Compounds temp C26H54 C60H122 n-C10H21OH C30H61OH ABC for (°C) (ref 14) (ref 14) (ref 14) (ref 14) surface tension 70 100 180

26.3 24.1 18.3

110 150 180 100 250 100 250

26.2 23.9 19.1 24.5 22.2 20.4

26.5 24.2 22.8 23.0 12.0

16.6 6.6 25.7 15.1

24.6 17.3

PEs, however, are of relatively low molecular weight, and, thus, their mixture properties may be different than those of a pure n-paraffin of the same molecular weight. Table 11 compares the predictions made with the ABC for n-paraffins to measured values for other related compounds. The first two compounds were reported to be C26 and C60 n-paraffins. Their surface tensions were measured in 1923 before pure samples of synthetic n-paraffins were available. While the comparison for C26 is reasonable, the C60 data are clearly lower, most likely the result of a mixture of closely-related compounds in the original sample. The comparison with the n-alcohol data shows that at higher carbon numbers the value for surface tension converges for different homologous series. It also demonstrates that the presence of n-alcohols (or other compounds) in impure samples can have a detectable effect on the measured value for surface tension even at relatively-high carbon numbers. Conclusions Asymptotic behavior correlations have been developed for thermal properties (ideal-gas enthalpy and free energy of formation, ideal-gas heat capacity, enthalpy of vaporization at 25 °C and the normal boiling point, and liquid heat capacity) and for transport properties (liquid viscosity, thermal conductivity, and surface tension). 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 these correlations is based on predictions made with lattice-fluid theory for the thermal properties and surface tension and with Eyring rate theory for viscosity. The theoretical basis for the behavior of thermal conductivity is less well-established and is based on analogy with the kinetic theory of dilute monatomic gases. It has been demonstrated that the ABCs yield accurate and consistent predictions and extrapolate well to higher carbon numbers, with the exception of viscosity for which non-Newtonian behavior of high molecular weight polyethylenes is not predicted. They represent the only reliable method for estimating property values for high molecular weight homologous compounds. While the ranges of temperature covered by the data were in some cases limited, the correlations have been developed in such a way to ensure reasonable extrapolations up to at least 300 °C. Comparisons of the ABCs for liquid heat capacity, thermal conductivity, and surface tension with data for polyethylene are reasonable for both linear and branched polyethylenes, as long

as branching is not too severe. Any discrepancies appear due to limited data for long-chain n-paraffins and n-olefins. The correlations reported here, and those for PVT properties reported in part 2 of this series, were specifically developed for pure n-paraffins and n-olefins. It was demonstrated in part 2 how the ABCs could be used to determine “effective” carbon numbers for other homologous series based on data for a single compound in a given series. It can also be assumed that the ABCs will give reasonable predictions for mixtures containing narrow ranges of carbon numbers, if an average carbon number is used. However, the properties of mixtures with broad carbon-number distributions may not be accurately predicted. This was demonstrated for viscosity of low molecular weight polyethylene. Which carbonnumber average is appropriate, or what property mixing rule should be used, depends on the composition of the mixture in a yet undefined way. Since knowledge of mixture properties is of fundamental importance to a number of applications, further correlation development is needed. The prediction of mixture properties is the subject of a paper to be published by the authors in the near future. Nomenclature A ) total surface energy c ) correction to the number of external degrees of freedom per segment Cp ) heat capacity at constant pressure, J/(g mol K) cˆ v ) heat capacity at constant volume per segment C ˆ v ) heat capacity at constant volume per molecule E ) total internal energy f ) fraction of lattice sites g0 ) Gibbs free energy per segment G ) total or molar Gibbs free energy, J/(g mol K) G ˆ 0 ) Gibbs free energy per molecule h ) Planck constant (6.624 ×10-27 erg s) h0 ) enthalpy per segment H ) total or molar enthalpy, J/(g mol) k ) Boltzmann constant (1.3806 × 10-16 erg/(molecule K)) l ) mean free path M ) molecular weight n ) carbon number N ) Avogadro’s number (6.0222 × 1023) P ) pressure r ) jump frequency R ) ideal-gas constant (8.3144 J/(g mol K)) s0 ) entropy per segment S ) total entropy T ) temperature, K T0 ) reference temperature (298.15 K) u ) velocity v ) volume per molecule v* ) close-packed volume per segment V ) total volume Y ) physical property correlated z, z′, z′′ ) coordination numbers of the lattice Greek Letters β, γ ) correlating parameters δ ) distance between lattice planes  ) nonbonded, segment-segment interaction energy ∆Ε ) nonbonded interaction energy per segment, J/(g mol) λ ) thermal conductivity, W/(m K) µ ) viscosity, cP ν ) frequency of interaction between nearest neighbors σ ) surface tension, dyn/cm τ ) shear stress ω ) component of the system degeneracy

2406 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

gradient dux/dy develops and the frequency of jumps increases. This can be explained by the distortion of the energy barrier under the applied stress τyx, which can be approximated as q

q

-∆G ˆ ) -∆G ˆ0 ( Figure 9. Two-dimensional lattice model for flow of simple-fluid molecules.

Ω ) system degeneracy Subscripts b ) at normal boiling point L ) as a liquid, nominally at 1 atm 0 ) at effective carbon number of zero S ) of the surface T ) total number vap ) of vaporization x, y, z ) lattice coordinates ∞ ) as carbon number approaches infinity q ) of activation

where the + sign indicates movement over the barrier in the positive x-direction and the - sign indicates movement in the opposite direction. For the fluid in motion, the frequencies of forward and backward jumps are given by

rf )

kT -∆Gˆ 0q/kT τyxδzδxδ/2kT e e h

(A-3)

rb )

kT -∆Gˆ 0q/kT -τyxδzδxδ/2kT e e h

(A-4)

Notation in Tables and Figures CNs ) those components whose reported values were used in ABC Pts ) 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 Liquid Viscosity. The asymptotic behavior of liquid viscosity may be derived by extension of the rate theory of Eyring and co-workers (Powell et al., 1941; Bird et al., 1960, section 1.5) to long-chain molecules. This theory resembles the lattice-fluid theory developed in part 2 for prediction of asymptotic behavior of PVT and thermal properties and below for prediction of surface tension. Molecules of the fluid are postulated to be located at fixed lattice sites, with empty “holes” located throughout the lattice. A simple fluid, with each molecule occupying only a single site in the lattice, is depicted in Figure 9. For the fluid at rest, a molecule is largely confined to vibrations within the cage formed by its nearest neighbors. The random jump of molecules into adjoining holes is controlled by an activation energy barrier of height ∆G ˆ 0q over the distance δ. The frequency of jumps per molecule r is given by

kT -∆Gˆ 0q/kT e h

(A-2)

The net velocity is then just the distance traveled per jump multiplied by the net frequency of forward jumps, and the velocity gradient can be approximated as

Superscripts

r)

τyxδzδxδ 2

(A-1)

If a shearing force is applied in the x-direction, a velocity

-

dux δx ) (rf - rb) dy δy

(A-5)

Combination of eqs A-3-A-5 yields

-

(

)(

)

τyxδzδxδ dux δ kT -∆Gˆ 0q/kT ) e 2 sinh dy δy h 2kT

(A-6)

which is the Eyring model for non-Newtonian behavior. For simple molecules, the argument of sinh is small compared to unity and eq A-6 reduces to Newton’s law of viscosity, with µ defined by

µ)

δyh

e∆Gˆ 0/kT

δzδx2δ

(A-7)

Now, a chain molecule, will occupy n adjoining sites of the lattice. Since the n segments of the molecule are attached, any jump by a segment of the molecule requires the coordinated motion of all n segments. Thus, n activation barriers must be surmounted and n distances δ traversed. That is

∆G ˆ 0q ) ng0q,

δ ) nδx

(A-8,9)

where g0q is the free energy of activation per segment. Assuming δx ) δy ) δz, then δzδx3/δy ) v*, the closepacked segment volume. Substitution of eqs A-8 and A-9 into eq A-7 gives for the natural logarithm of µ

ng0q h + nv* kT

( )

ln µ ) ln

(A-10)

The free energy of activation can be expressed as the sum of enthalpic and entropic terms: g0q ) h0q - Ts0q. Then, in the limit as n f ∞, ln µ f -ns0q/k + nh0q/kT. Thus, ln µ is asymptotically linear in carbon number. The mechanism for viscous transport described above is in many ways similar to what occurs in vaporization of a liquid, where similar bonds are being broken. The similarity is evident for long-chain molecules from the expression derived in part 2 for vapor pressure. In the limit as s f ∞, ln Pvap f n - n*/kT.

Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2407

Liquid Thermal Conductivity. Molecular theories for liquid thermal conductivity of chain molecules are less well-developed than lattice-fluid or the Eyring rate theories. They rely primarily on analogies with the kinetic theory for dilute monatomic gases (Bird et al., 1960, section 8.3). This theory predicts

1 ˆ u λ) C j l v v y

(A-11)

where v is the molecular volume, C ˆ v is the constantvolume heat capacity per molecule, u j y is the average absolute velocity in the y-direction, and l is the mean free path. In order to extend this equation to a lattice fluid, λ can be considered as the product of three factors: l/v which is replaced by l/v*, the number of segments per unit area, C ˆ v which is replaced by cˆ v, the heat capacity per segment, u j y/l which is replaced by ν, the frequency of interaction between nearest neighbors, and l which is replaced by δ, the lattice spacing. cˆ v may be approximated as 3kc, where 3k is the value obtained for a classical monatomic crystalline solid, and c is a correction factor. δ may be approximated as v*1/3. Substitution into eq A-11 yields for the lattice fluid

λ ) 3νkc/v*1/3

(∂G ∂A )

T,P

(A-13)

where A is the surface area and GS is the Gibbs free energy per unit area which is related to the other surface energy functions by

GS ) HS - TSS ) ES + PVS - TSS

(A-14)

Thus, if the asymptotic behavior of the energy functions ES, VS, and SS can be determined, the asymptotic behavior of σ will also be known. Consider the surface to be defined as a fixed fraction of the total sites in the lattice fluid, that is, NS ) fSNT. Assuming the chain molecules are not oriented at the surface and, further, the distribution of holes at the surface is the same as in the bulk fluid, then

( ) nN NT

2

(A-15)

where  is the nonbonded, segment-segment interaction energy and z′ is the coordination number of a surface lattice site. For a simple cubic lattice, z ) 6 and z′ ) 5, and for a face-centered-cubic lattice, z ) 12 and z′ ) 8. Equation A-15 is analogous to eq A-5 of part 2. Since the area of the surface is simply A ) NSv*2/3, substitution into eq A-15, along with the expressions: NT ) V/v* and N ) V/v, yields

n2 E ) -Av*4/3(z′/2) 2 v

(A-16)

Therefore,

ES )

n2 ∂E ) -v*4/3(z′/2) 2 ∂A T,P v

( )

(A-17)

In the limit as n f ∞, v f nv∞ and ES f -(v*4/3/v∞2)(z′/2). Thus, the internal energy per unit surface area approaches a finite limit. It also follows from V ) NSv* that

(A-12)

Only ν and c in eq A-12 are dependent on n. As discussed in part 2 in relation to Flory theory, c can be expanded as a power series in 1/n. The frequency will have different values depending on whether a segmentsegment interaction is between bonded or nonbonded pairs. A weighted average can be used for ν. The fraction of bonded pairs in the lattice is given by 2(n 1)/zn, where z is the coordination number of the lattice. For large n, this reduces to 2/z, and the fraction of nonbonded pairs is 1 - 2/z. Based on the above analysis, then liquid viscosity approaches a finite limiting value. Since the thermal conductivity of a liquid is mainly the result of vibrations, the presence of holes in the lattice has been completely neglected in this formulation. Thus, the above result is equally valid for both liquids and solids. Differences between the properties of the two will be a result of different values for c, z, and ν and the result of possible preferred orientations in the solid (i.e., crystalline) phase. Surface Tension. The lattice-fluid theory developed in the Appendix of part 2, can be extended to surface properties. Surface tension is related to the Gibbs free energy of the surface by

σ ) GS )

E ) -NS(z′/2)

VS )

∂V (∂A )

T,P

) v*1/3

(A-18)

Using the same assumptions as Flory (1942) in his derivation of the degeneracy Ω for the bulk fluid, the surface degeneracy ΩS can be expressed as n

1

ΩS ) ln

+ N!

n

ln ∑∑ωij ∑ N i)1 j)1

(A-19)

where

ωij )

1 NS (N - nN)j-i+1z′′(z′′ - 1)j-i-1NTi-j 2 NT T

(A-20)

where z′′ is the coordination number between surfacesurface sites. The quantity NT - nN is just N0, the total number of holes in the lattice. For both simple and facecentered-cubic lattices, z′′ ) 4. Again, it is assumed the chain molecules are not oriented at the surface, and the distribution of holes at the surface is the same as that in the bulk fluid. The double summation in eq A-19 results from configurations where not all segments of any molecule reside on the surface. However, for large n, the logarithm of the double summation may be approximated by the logarithm of its largest term with no loss of accuracy. Applying Stirling’s approximation then yields

ΩS )

( ) ( z′′ - 1 e

(n-1)N

)

( )

N(N + nN)N0+N N N z′′ 0 S N N 0 N 2(z′′ - 1) N0 N T (A-21)

This expression is the same as eq A-4 of part 2 except z has been replaced by z′′ and the term involving NS/NT has been added. The entropy of the surface is given by the Boltzmann relationship:

S ) k ln ΩS

(A-22)

Substitution of eq A-21, along with A ) NSv*2/3, into eq A-22 yields

2408 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997

SS )

kN kv*1/3 ∂S ) ) ∂A T,P A f Sv

( )

(A-23)

Since v is asymptotically linear with n, it follows that, in the limit as n f ∞, SS f 0. Based on eqs A-13 and A-14 and the results for ES, VS, and SS, surface tension σ approaches a finite limit for large n. Literature Cited (1) API Research Project 44. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; Carnegie Press: Pittsburgh, PA, 1953. (2) API Research Project 42. Properties of Hydrocarbons of High Molecular Weight; American Petroleum Institute: New York, 1966. (3) API Technical Data BooksPetroleum Refining, 5th ed.; American Petroleum Institute: Washington, DC, 1992. (4) Beaton, C. F.; Hewitt, G. F. Physical Property Data for The Design Engineer; Hemisphere Publishing: New York, 1989. (5) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960. (6) Boggs, J. H.; Sibbitt, W. L. Thermal Conductivity Measurements of Viscous Liquids. Ind. Eng. Chem. 1955, 47, 289-293. (7) Dettre, R. H.; Johnson, R. E., Jr. Surface Properties of Polymers. I. The Surface Tensions of Some Molten Polyethylenes. J. Colloid Interface Sci. 1966, 21, 367-377. (8) Doolittle, A. K.; Peterson, R. H. Preparation and Physical Properties of a Series of n-Alkanes. J. Am. Chem. Soc. 1951, 73, 2145-2151. (9) Flory, P. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1942, 10, 51-61. (10) Gallant, R. W.; Yaws, C. L. Physical Properties of Hydrocarbons, Volume I; 2nd ed.; Gulf Publishing: Houston, 1992. (11) Gallant, R. W.; Yaws, C. L. Physical Properties of Hydrocarbons, Volume III; Gulf Publishing: Houston, 1993. (12) Gormley, R. J. U.S. Department of Energy, Pittsburgh Energy Technology Center, personal communication, 1996. (13) Hansen, D.; Ho, C. C. Thermal Conductivity of High Polymer. J. Polym. Sci. 1965, A3, 659-670. (14) Jamieson, D. T. Thermal Conductivities of Liquids. J. Chem. Eng. Data 1979, 24, 244-246. (15) Jasper, J. J. The Surface Tension of Pure Liquid Compounds. J. Phys. Chem. Ref. Data 1972, 1, 841-1009. (16) Korosi, G.; Kovats, E. sz. Density and Surface Tension of 83 Organic Liquids. J. Chem. Eng. Data 1981, 26, 323-332. (17) Maloney, D. P.; Prausnitz, J. M. Thermodynamic Properties of Liquid Polyethylene. J. Appl. Polym. Sci. 1974, 18, 27032710. (18) Marano, J. J. Property Correlation and Characterization of Fischer-Tropsch Liquids for Process Modeling. Ph.D. Dissertation, University of Pittsburgh, Pittsburgh, PA, 1996.

(19) Marano, J. J.; Holder, G. D. A General Equation for Correlating the Thermophysical Properties of n-Paraffins, nOlefins, and Other Homologous Series. 1. Formalism for Developing Asymptotic Behavior Correlations. Ind. Eng. Chem. Res. 1997a, in press. (20) Marano, J. J.; Holder, G. D. A General Equation for Correlating the Thermophysical Properties of n-Paraffins, nOlefins, and Other Homologous Series. 2. Asymptotic Behavior Correlations for PVT Properties. Ind. Eng. Chem. Res. 1997b, in press. (21) Mendelson, R. A. Melt Viscosity. Encyclopedia of Polymer Science and Technology, Plastics, Resins, Rubbers, Fibers, Volume 8; John Wiley & Sons: New York, 1968; pp 587-620. (22) Powell, R. E.; Roseveare, W. E.; Eyring, H. Diffusion, Thermal Conductivity, and Viscous Flow of Liquids. Ind. Eng. Chem. 1941, 33, 430-435. (23) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987. (24) Reinhard, R. R.; Dixon J. A. Tetranonacontane. J. Org. Chem. 1965, 9, 1450-1453. (25) Schreiber, H. P.; Bagley, E. B.; West, D. C. Viscosity/ Molecular Weight Relation in Bulk PolymerssI. Polymer 1963, 4, 355-364. (26) Steele, W. V.; Chirico, R. D. Thermodynamic Properties of Alkenes. J. Phys. Chem. Ref. Data 1993, 22, 377-430. (27) Tomlinson, J. N.; Kline, D. E.; Sauer, J. A. Effect of Nuclear Radiation on The Thermal Conductivity of Polyethylene. SPE Trans. 1965, 5, 44-48. (28) TRC (Thermodynamic Research Center). TRC Thermodynamic TablessHydrocarbons; The Texas A&M University: College Station, TX, 1987; revision. (29) Vargaftik, N. B. Tables on The Thermophysical Properties of Liquids and Gases, 2nd ed. Edition (English trans.); John Wiley & Sons: New York, 1975. (30) Wu, S. Interfacial and Surface Tensions of Polymers. J. Macromol. Sci. 1974, C10, 1-73. (31) Wunderlich, B.; Dole, M. Specific Heat of Synthetic Polymers. VII. Low Pressure Polyethylene. J. Polym. Sci. 1957, 24, 201-213. (32) Wunderlich, B.; Baur, H. Heat Capacities of Linear High Polymers. Adv. Polym. Sci. 1970, 7, 151-368.

Received for review August 15, 1996 Revised manuscript received February 5, 1997 Accepted February 5, 1997X IE9605138

X Abstract published in Advance ACS Abstracts, April 15, 1997.