Ind. Eng. Chem. Res. 1997, 36, 1887-1894
1887
General Equation for Correlating the Thermophysical Properties of n-Paraffins, n-Olefins, and Other Homologous Series. 1. Formalism for Developing Asymptotic Behavior Correlations John J. Marano* and Gerald D. Holder Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261
In this first in a series of three papers, a formal treatment is presented for the development of property correlations for homologous series of compounds. The theoretical basis for asymptotic behavior is discussed, and the methodology used to regress parameters is described. The equations developed are quite general and can be extended to other properties or homologous series. Parts 2 and 3 of this series present correlations developed for n-paraffins and n-olefins (1-alkenes) to predict PVT related properties, normal boiling and melting points, critical temperature, pressure, and volume, acentric factor, liquid molar volume, and vapor pressure (part 2); thermal properties, ideal-gas enthalpy and free energy of formation, ideal-gas heat capacity, enthalpy of vaporization, and liquid heat capacity (part 3); transport properties: liquid viscosity, thermal conductivity, and surface tension (part 3). It is demonstrated that these correlations are accurate, consistent, and yield reasonable extrapolations. They are preferable to existing correlations in most instances. Introduction
Y ) Y∞ - (Y∞ - Y0) exp(-βn2/3)
The determination and prediction of the properties of homologous compounds is of importance in a number of research and industrial areas. High molecular weight n-paraffins are present in petroleum, and pure nparaffins are useful as model compounds for the development and testing of correlations for predicting the properties of petroleum fractions (Gray et al., 1989). n-Paraffins, n-olefins, and n-alcohols are all found in synthetic oils derived from the Fischer-Tropsch synthesis. Further process development of this technology requires properties of these fluids to be known or estimated with reasonable accuracy. Finally, properties of n-paraffins and derivatives are of importance to those industries manufacturing paraffin-based waxes and various linear polymers. Group contribution methods may be applied to the prediction of the properties of homologous compounds (e.g., see Chapter 2, Reid et al., 1987). In general, these methods fail, predicting unreasonable behavior when extrapolated to high carbon numbers (Gasem et al., 1993). Theoretical equations which appear to predict asymptotic behavior correctly typically perform poorly at lower carbon numbers. A different approach is necessary to predict behavior over a wide range of carbon numbers and in the limit of infinite carbon number. A class of equations which will be referred to as asymptotic behavior correlations or ABCs can be attributed to Kreglewski and Zwolinski (1961). Numerous variations of their K-Z correlation have appeared in the literature applied to a variety of properties and homologous compounds. The K-Z correlation can be written as * 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)00511-8 CCC: $14.00
(1)
where Y is the property value at carbon number n, Y0 the property pseudo-value at n ) 0, Y∞ the limiting property value as n becomes very large, and β an adjustable parameter. In their original work, Kreglewski and Zwolinski applied eq 1 to boiling point, heat of vaporization, and Antoine coefficient B, for n-paraffins, n-olefins and n-alkylbenzenes with n g 3. Predictions based on eq 1 (Kudchadker and Zwolinski, 1966) for the normal boiling of n-paraffins up to n ) 100 have been included in the API Research Project 44 tables (TRC, 1987). Later work extended the correlations for n-paraffins to critical temperature and pressure (Kudchadker et al., 1968). Equation 1 contains three adjustable parameters, Y0, Y∞, and β, which are evaluated by fitting data for lower carbon numbers. A number of investigators have updated the original parameters using newer experimental data (see Table 1). The two-thirds power of n appearing in the exponential term was borrowed from an empirical modification to the lattice theory proposed by Kurata and Isida (1955). While eq 1 works well, Teja et al. (1990) have demonstrated the advantage of adjusting this value to improve the fit to experimental data. More recently, Gasem and co-workers at Oklahoma State University (OSU) have proposed ABCs for critical properties of n-paraffins (Gasem and Robinson, 1985; Gasem et al., 1993). The name asymptotic behavior correlationsABCsfor these types of correlations can be attributed to Gasem. The form of the OSU correlation is
YR ) Y∞R - (Y∞R - YRo ) exp(-Rβ(n - 1))
(2)
where the variables have the same meaning as in eq 1, except that Y0 is the property value at n ) 1 and R is an additional adjustable parameter. Gasem and coworkers adopted the form for eq 2 from the field of biometrics. It has the capability of representing a © 1997 American Chemical Society
1888 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 1. ABC Parameters for Critical Temperature and Pressure of n-Paraffins Kudchadker et al. (1968) eq 1 n0 Tc0 Tc∞ R β γ n0 Pc0 Pc∞ R β γ
Tsonopoulos (1987) eq 1
Teja et al. (1990) eq 3
57.41 961.
Critical Temperature (K) 0 48.24 -141.93 959.98 1143.8
0.208 545
0.211 145
76.561 0
Critical Pressure (bar) 0 75.1710 66.0003 0 8.4203
0.280 45
0.274 281
diverse range of functional shapes from exponential to sigmoidal. A drawback of eq 2, however, is that YR is undefined for negative values of Y with noninteger values of R. Unfortunately, when Y0 and R are treated as adjustable parameters, Y0 can converge to zero causing the error-minimization procedure to fail. More recently, Shaver et al. (1991, 1992) have generalized the OSU correlation under the name scaled variable reduced coordinate (SVRC) correlation to correlate purefluid and mixture saturation properties. Both eqs 1 and 2 are subsets of the generalized ABC proposed below. Table 1 summarizes parameters for these equations recommended by various investigators to predict critical temperature and pressure. While four adjustable parameters are adequate when considering n-paraffins alone, at least one additional parameter is useful when considering n-paraffins and n-olefins together. In addition, eqs 1 and 2 can only be applied to properties which converge to a limiting value. Not all thermophysical properties of interest exhibit this type of behavior. A Generalized ABC The purpose of the ABC is to allow the properties of higher carbon-number homologs to be estimated by extrapolation from known property values of their lower carbon-number relatives. One expects this procedure to be more accurate and consistent than generalized group contribution methods (Gasem et al., 1993). While the accuracy of the ABC for reproducing known property values of lower carbon-number members of a series is a measure of the quality of the correlation, it is not necessarily indicative of the quality of any extrapolated values. Properties which can be measured for high molecular weight, linear polyethylene are used as a check of the correlation. However, since samples of linear polyethylene always contain a range of carbon numbers and some branching is always present, these values are not directly substituted for Y∞ in any correlation. It can be seen from Table 1 that estimated limiting values can be quite different depending on the form for the ABC. In many cases, the properties of interest may not be measurable in the limit. For example, the higher carbon-number paraffins (roughly n > 10) decompose well below their critical temperature (Anselme et al., 1990). However, estimates of critical temperature and pressure are still needed for VLE and other thermodynamic calculations. Therefore, it is required that the ABC predict “reasonable” values upon extreme limits of extrapolation. For most physical properties, this
0.303 158 0.469 609
0.196 383 0.890 006
Gasem et al. (1993) eq 2
190.54 926.14 2.2265 0.0245
42.044 0.1 0.0136 1.0318
this work eq 3 0.896 021 127.89 1020.71 0.198 100 0.629 752 -3.625 581 1336.74 0 2.111 827 0.258 439
precludes the existence of maxima and minima at higher carbon numbers. The forms presented here always yield monotonic extrapolations. In addition, for absolute quantities such as critical pressure, extrapolated values should remain non-negative. A desire for reasonable extrapolations also necessitates that the predicted property values exhibit consistency. The need for consistency between different homologous series was first recognized by Kreglewski and Zwolinski (1961). They demonstrated that nparaffin, n-olefins, and n-alkylbenzenes all have the same limiting boiling point. What they and others have not recognized is that, in approaching this limit, the difference between the property values for different homologs should monotonically decrease. For example, the difference between the critical temperature of an n-paraffin and n-olefin of the same carbon number decreases with increasing n. This implies interrelationships between the correlation parameters used for different homologous series. These relationships are valuable when correlating properties for a series of compounds for which little actual experimental data exist. For the properties of interest in this investigation, two types of behavior can be identified. These will be referred to here as types I and II behavior. Type I properties, exemplified by normal boiling and melting points, are properties which approach a finite value for large carbon numbers. In the limit, the addition of more carbon units to the chain has no effect on the property. For this to occur, effects of any functional groups present must become negligible for large n. As would be expected, the limiting value is the same for all n-alkyl compounds and their derivatives. Equations 1 and 2 were developed for type I properties. Type II properties are additive in nature. In the limit, each additional carbon unit contributes a fixed increment to the property value. Thus, rather than being a fixed value, the limit is instead linear with carbon number. Since each increment contributes to the whole, it should be expected that these properties will have different limiting values for different homologous series, reflecting any different functional groups present. However, in the limit, the value for the increment should be the same for all n-alkyl compounds and derivatives. On the basis of the repetitive chemical structure of homologous compounds, it can be reasonably argued that molar volume, enthalpy, entropy, and related properties of state should exhibit type II behavior. Since molecular weight is also linear with carbon number (type II behavior), it follows that mass-based properties
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1889
will exhibit type I behavior (i.e., the ratio of two type II properties has a finite limit). However, mass-based properties can also exhibit maxima and minima (e.g., critical volume (Anselme et al., 1990)). More complex behavior is also possible but often reduces to type II under reasonable assumptions. Where experimental data are lacking, theoretical models based on equations of state or molecular simulation are useful for deducing the form of the asymptote for complex properties. A goal of this work was to develop a universal form for the ABC which can be used for a wide range of different thermodynamic and transport properties, exhibiting both type I and type II behavior. Examination of eqs 1 and 2 suggests the following more general form for an ABC
Y ) Y∞ - ∆Y0 exp(-β(n ( n0)γ)
(3a)
Y∞ ) Y∞,0 + ∆Y∞(n - n0)
(3b)
n g n0, β > 0, γ > 0 For type I properties, Y∞ is a constant, that is, ∆Y∞ ) 0 and ∆Y0 ) Y∞ - Y0. There are then five adjustable parameters: n0, Y0, Y∞, β, and γ. For type II properties, the asymptote Y∞ is a linear function of carbon number n, and there are six adjustable parameters: n0, ∆Y0, Y∞,0, ∆Y∞, β, and γ. The Y in eq 3 may represent a property function (e.g., YR); thus, eqs 1 and 2 can both be represented in the form of eq 3. Since eq 3 contains more adjustable parameters than either eq 1 or 2, it should be expected that it can be used to obtain very accurate correlations for most thermophysical property data. It can also represent sigmoidal curves. This can be seen by determining the point of inflection of eq 3. For type I behavior, the carbon number for which the inflection occurs is given by
n ) n0 +
(γ βγ- 1)
1/γ
(4)
Clearly, γ must be greater than 1 for an inflection to occur. The first constraint (n g n0) on eq 3 is required since the term (n - n0)γ is undefined for n < n0 with fractional values of γ. For most properties, the best value for n0 was determined to be less than 1, so that the ABC is defined for all n-paraffins and n-olefins. The last two constraints in eq 3 (β > 0 and γ > 0) ensure asymptotic behavior as n approaches infinity. For eq 3, sufficient criteria for convergence of properties of n-paraffins and n-olefins are given by
∆Y0(p) ) ∆Y0(o) Y∞,0(p) ) Y∞,0(o) ∆Y∞(p) ) ∆Y∞(o)
(5)
β(p) ) β(o) γ(p) ) γ(o) where p and o signify n-paraffin and n-olefin, respectively. It is also necessary to specify the sign to be used in the term (n ( n0)γ. This sign is taken as negative for all properties except those which increase with increasing carbon number and exhibit positive curva-
ture (i.e., liquid molar volume and heat capacity). For these properties, the plus sign ensures that the difference between predicted n-paraffin and n-olefin values decreases with increasing carbon number. Due to the constraints given by eq 5, the correlations for n-paraffins and n-olefins will only have different values for n0; all other parameters will be the same. The term n - n0 can be considered the effective carbon number for different homologous series and reflects the differences due to the functional groups present in these compounds. The n0 can also be considered a measure of the deviation of n-paraffins and n-olefins from an ideal (-CH2-)n series for which n0 would be zero. For type II properties, n0, ∆Y0, and Y∞,0 are not independent parameters if γ ) 1. For this case, variation of n0 corresponds to a translation of the property value along the n-axis. In general, it was observed that there was a significant dependence between n0 and Y∞,0 even when γ was not 1. Therefore, theoretical arguments and experimental observations were used in establishing values for Y∞,0. Data Sources Numerous sources were consulted to obtain property data for pure n-paraffins and n-olefins (1-alkenes). Major data compilations used include: The API Technical Data BooksPetroleum Refining (5th ed., 1992; 2nd ed., 1970), API Research Project 44 tables (1953; TRC, 1987), API Research Project 42 tables (1966), The Thermophysical Properties of Liquids and Gases, second ed. (Vargaftik, 1975), The Properties of Gases & Liquids (fourth ed., 1987; third ed., 1977), and Physical Property Data for The Design Engineer (Beaton and Hewitt, 1989). In addition, a large number of other sources were consulted for individual properties and/or compounds. Sources for all data used, in order of preference, are identified in the tables reporting the ABC parameters for each property considered (see parts II and III). Data were screened to ensure only experimentally determined values were considered for individual compounds (with the exceptions of vapor pressure and idealgas heat capacity, where high-order regressions were used to generate values). Interpolation and extrapolation of experimental data are used as necessary for properties which are temperature dependent. Experimental methods employed in measuring data were reviewed when available and appropriate. Most data measured before 1940 were excluded due to the lack of high-purity, high molecular weight samples prior to API Projects 42 and 44. For the correlations developed, C1 and C2 data (and occasionally data for higher carbon numbers) were not considered when fitting data, since in general, this improved the accuracy of the correlation for the remaining compounds. Other data were excluded as well, if they did not appear to be consistent with data for adjacent compounds in a series. Methodology The adjustable parameters in eq 3 were determined by minimizing the error between predicted and reported property values. The objective function used for minimization was the root mean square error (RMSE). Other statistical properties are of interest in judging how well the ABCs match data. These are the average deviation (AD), which provides an indication of bias, average absolute deviation (AAD), and percent average absolute deviation (%AAD). When n-paraffin and n-
1890 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 2. Effect of Number of Parameters on ABCs for Normal Boiling Point of n-Paraffins
TbR ) Tb∞R - (Tb∞R - Tb0R) exp(-Rβ(n - n0)γ)
cantly reduced. For this reason, the ABC proposed here is preferred; it produces good asymptotic values and is consistent across homologous series. The correlations obtained from the parameters in columns 6 and 7 of Table 3 are shown graphically in Figure 1.
general normal bp (K)
3 parameter
4 parameter
5 parameter
OSU 4 parameter
n0 Tb0 Tb∞ R β γ RMSE AD AAD %AAD
0 0.47 1069.78 1 0.116 584 2/ 3 0.0326 0.0000 0.1378 0.0306
0 5.76 1056.37 1 0.114 673 0.677 050 0.0279 0.0000 0.1025 0.0205
-0.548 608 -55.14 1079.14 1 0.129 010 0.641 755 0.0249 0.0000 0.0831 0.0153
1 112.35 971.73 1.75485 0.017 280 2 1 0.0472 0.0002 0.2115 0.0455
olefin data were optimized simultaneously, the sum of the RMSEs was used as the objective function. For some properties where n-olefin data were limited or of questionable accuracy, it was necessary to weight the n-paraffin data more heavily in order to obtain reasonable ABC parameters. Proper scaling was also necessary due to the rather large differences in absolute value for some of the parameters. The determination of optimal parameters for eq 3 poses a significant mathematical problem given the number of parameters and constraints and the nonlinear form of the ABC. Newton’s method was used to perform all multiple-parameter optimizations simultaneously. These calculations were carried out using a standard spreadsheet software package. Table 2 contains a comparison of the values obtained for the ABC parameters when using different combinations of adjustable parameters. Normal boiling point was chosen for comparison since accurate data are available through a carbon number of 30. All the correlations give acceptable accuracy. For the general ABC, there is an improvement in the error with the introduction of more adjustable parameters. The threeparameter correlation which is equivalent to the K-Z correlation does surprisingly well. This is due to the fact that the two-thirds power of γ is very close to the value of 0.677 obtained using four adjustable parameters. This is not the general case, as can be seen from the critical pressure correlation in Table 1. The OSU correlation, even though it contains four parameters, performed the worst for Tb. Also worth noting, the first three correlations give similar limiting values for Tb, whereas the OSU value is significantly lower. Therefore, scaling of Tb by a factor of R may not be appropriate. From Table 2, it is not clear that there is significant benefit to using five adjustable parameters versus four or even three when optimizing data for a single homologous series. However, when data from another series are included in the parameter optimization, the value of five parameters is clear as shown in Table 3. As expected, the best fit is obtained when the n-olefin and n-paraffin data are optimized separately (columns 2 and 3). However, different values are obtained for the ABC parameters: Tb0, Tb∞, β, and γ, and the two correlations are inconsistent. When Tb∞, β, and γ are constrained to be equal (five adjustable parameters in all), the error increases significantly. By adding two additional parameters, n0(p) and n0(o), and applying all of the constraints identified in eq 5 (6 adjustable parameters), the errors for both n-paraffins and n-olefins are signifi-
Functions of Temperature When the property of interest is a function of temperature, parameter optimization is not as straightforward as described above. For properties such as molar volume or vapor pressure, semiempirical equations with many adjustable parameters are typically used to regress property versus temperature data. The coefficients which result are not unique, and their individual values are highly dependent on the range of temperature data used in the regression and the distribution of individual data within this range. For this situation, the above optimization procedure performs poorly. The individual coefficients cannot be represented by a smooth ABC. An alternative parameter estimation procedure was developed. Consider the following general temperaturedependent property:
Y ) A1f1(T) + A2f2(T) + ... + Arfr(T)
(6)
At any given temperature, eq 3 should also be valid for the property as a function of carbon number:
Y ) Y∞,0(T) + ∆Y∞(T)(n - n0) ∆Y0(T) exp(-β(n ( n0)γ) (7) Further, if it is assumed that Y∞,0, ∆Y∞, and ∆Y0 can be expressed by the same temperature dependency as given in eq 6, that is
Y∞,0 ) A1∞,0f1(T) + A2∞,0f2(T) + ... + Ar∞,0fr(T) ∆Y∞ ) A1∞f1(T) + A2∞f2(T) + ... + Ar∞fr(T) ∆Y0 ) A10f1(T) + A20f2(T) + ... + Ar0fr(T)
(8)
then the carbon-number dependence of A1, A2, ..., Ar in eq 3 can be expressed by the ABCs:
A1 ) A1∞,0 + ∆A1∞(n - n0) - ∆A10 exp(-β(n ( n0)γ) A2 ) A2∞,0 + ∆A2∞(n - n0) - ∆A20 exp(-β(n ( n0)γ) l Ar ) Ar∞,0 + ∆Ar∞(n - n0) - ∆Ar0 exp(-β(n ( n0)γ) (9) The adjustable parameters in eq 9 are n0, A1∞,0, ∆A1∞, ∆A10, A2∞,0, ∆A2∞, ∆A20,..., Ar∞,0, ∆Ar∞, ∆Ar0, β, and γ. As before, different values for n0 are determined for n-paraffins and n-olefins. For consistency, all other parameters are kept equal. The large number of adjustable parameters can be reduced for type II properties by employing the restriction that the asymptotes at different temperatures converge to a point when extrapolated. This experimental observation forms the basis for the infinite point on Cox charts used for estimating vapor pressures of homologous series of compounds (section 6-3, Reid et al., 1977). Therefore, only A1∞,0 can be nonzero (f1(T) ) 1), and A2∞,0, ..., Ar∞,0 ) 0.
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1891 Table 3. Effect of Number of Parameters on ABCs for Normal Boiling Point of n-Paraffins and n-Olefins
Tb ) Tb∞ - (Tb∞ - Tb0) exp(-β(n - n0)γ) 4 parameter
5 parameter
normal bp (K)
n-paraffin
n-olefin
n-paraffin
n0 Tb0 Tb∞ β γ RMSE AD AAD %AAD
0 5.76 1056.37 0.114 673 0.677 050 0.0279 0.0000 0.1025 0.0205
0 12.10 1013.00 0.109 537 0.711 703 0.0465 0.0000 0.1883 0.0383
0 13.85
0.0851 0.1007 0.3881 0.0829
6 parameter
n-olefin
n-paraffin
n-olefin
0 7.94
-1.487453
-1.340265
1029.33 0.111 470 0.698 096 0.0963 -0.1140 0.4184 0.0925
0.0416 0.0361 0.1569 0.0362
-164.93 1091.11 0.153 505 0.602 490 0.0631 -0.0551 0.2912 0.0613
Table 4. ABC Parameters for Liquid Molar Volume Evaluated from Single and Multiple Temperature Data
VL ) ∆VL∞(n - n0) - ∆VL0 exp(-β(n + n0)γ) molar volume from data at 20 °C at 20 °C (cm3/g‚mol) n-paraffin n-olefin
Figure 1. Asymptotic behavior correlations for normal boiling point of n-paraffins and n-olefins.
Since the form of eq 6 is empirical and usually involves polynomials in T, the real possibility exists that the ABC parameters will converge to unsuitable values. That is, the temperature behavior of the asymptote will contain maxima, minima, or inflection points. In order to prevent this from occurring, additional constrains were introduced when correlating properties which are temperature dependent. It is required that the first and second derivatives of the asymptote with respect to temperature not change sign within the temperature range of interest. In passing, it should be noted that equations of the form of eq 6 are notorious for predicting unreasonable estimates when extrapolated outside the range of data with which they were developed. The ABC provides a more sound approach for estimating property values for a given carbon-number compound outside the range of data available for that compound. Since the ABC is well behaved within a known temperature range, it can be expected that as long as property data for some carbon-number compounds at the desired temperature were used in developing the correlation, the ABC will give reasonable values for other carbonnumber compounds in the series at this temperature. Thus, the extrapolation is being performed on the basis of the carbon number, not the temperature. For temperature-dependent properties, the ABC parameter optimization was carried out by minimizing the average RMSE for a specified number of temperature intervals within the range of interest. The temperature range used was 0-300 °C. Typically, data were available at or near the following temperatures: 0, 20 or 25, 50, 100, 150, 200, 250, and 300 °C; however, data was not always available for all properties over the entire range. The range of applicability for a correlation is identified in the tables containing the parameters for eq 8, which appear in parts II and III. The ABC parameters obtained for a given property are not totally unique. Different values for the param-
n0 ∆VL0 ∆VL∞ β γ RMSE AD AAD %AAD
-1.799 935 -1.473 026 -154 09.2 16.4146 7.141 348 0.135 577 0.0086 0.0070 0.0002 -0.0002 0.0440 0.0345 0.0150 0.0132
from all data 0-300 °C n-paraffin n-olefin -1.388 524 -1.061 318 -3750.07 16.4862 5.5198 46 0.0570 632 0.0218 0.0251 0.0471 -0.0009 0.1046 0.1238 0.0337 0.0472
Figure 2. Asymptotic behavior correlations for liquid molar volume of n-paraffins and n-olefins.
eters at any temperature are obtained depending whether the optimization is performed only using data for this temperature or using data over the entire temperature range. This is illustrated in Table 4 which contains the ABC parameters obtained for liquid molar volume at 20 °C. The first set of parameters were based only on data at 20 °C. The second set were calculated from eq 8 (cubic in T) and are based on the entire set of data from 0-300 °C. The values for ∆Y∞ are roughly the same in both cases, whereas the values for n0, ∆Y0, β, and γ are different. As expected, the correlation performs better with the data set restricted. This effect, however, can be minimized by using high-order polynomials to represent temperature behavior. The error for the second set of ABC parameters given in Table 4 is well within an acceptable range. The correlations obtained from the parameters in columns 4 and 5 of Table 4 are shown graphically in Figure 2 at three representative temperatures. Figures 3 and 4 show the ABC error distributions for the n-paraffin and n-olefin data, respectively. The ABCs
1892 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 5. Comparison of ABC with API Correlations for Critical Temperature n-paraffins data (K) RMSE AD AAD %AAD C5 C10 C15 C50 C100 Tc∞
469.7 617.7 707.8
n-olefins
ABC
API 4A1.1 (Ambrose)
API 4A2.1
0.0773 0.0193 0.2784 0.0475 469.4 618.0 707.9 931.3 995.8 1020.7
0.0862 0.0637 0.2805 0.0533 469.2 617.9 708.0 954.2 1051.8 Tb∞ ) 1091.1
0.6552 -1.3328 1.9925 0.3587 470.1 618.1 705.2 935.1 1023.3 1089.8
data (K)
ABC
API 4A1.1 (Ambrose)
464.7 616.4
0.2444 -0.0966 0.5947 0.1220 465.9 616.1 706.7 931.0 995.8 1020.7
0.2095 -0.3996 0.5507 0.1065 465.9 616.2 707.5 954.1 1051.8 Tb∞ ) 1091.1
API 4A2.1 1.8340 4.3136 4.9040 0.8872 467.3 624.3 720.6 936.1 1024.0 1089.8
Table 6. Comparison of ABC with API Correlations for Critical Pressure n-paraffins data (bar)
Figure 3. ABC error distribution for liquid molar volume of n-paraffins.
ABC
API 4A1.1 (Ambrose)
RMSE 0.0459 AD -0.0027 AAD 0.1400 %AAD 0.7274 33.69 33.53 C5 C10 20.99 21.12 C15 14.79 14.90 C50 3.63 C100 1.21 Pc∞ 0
0.0708 0.1046 0.2456 1.3450 33.43 21.06 15.28 5.19 2.67 0
n-olefins data (bar)
ABC
API 4A1.1 (Ambrose)
0.9817 0.9540 1.2028 1.0251 35.27 35.83 22.18 22.13 15.46 3.69 1.22 0
0.1705 -0.3012 0.4293 1.5464 35.58 21.84 15.67 5.24 2.68 0
Table 7. Comparison of ABC with API Correlations for Critical Volume n-paraffins
Figure 4. ABC error distribution for liquid molar volume of n-olefins.
data (cm3/ g‚mol)
ABC
API 4A1.1 (Ambrose)
0.7905 -0.0999 1.8384 0.5197 290 589 934 3577 6945 ∝n3/2
2.1189 -2.3649 4.5588 1.1039 296 571 847 2775 5530 ∝n
Comparisons with Group Contribution Methods As previously mentioned, an ABC should perform better than group contribution methods. To demonstrate this point, the ABCs developed for critical temperature, pressure, and volume (part II) were compared to the methods recommended in the API Technical Data Book (1992), procedure 4A1.1, which are attributed to Ambrose (Chapter 2, Reid et al., 1987). The older, empirical correlation for critical temperature, API procedure 4A2.1, has also been included in this comparison. The data used to make these comparisons are the recommend values of Teja et al. (1990) for n-paraffins and of Gude et al. (1991) for n-olefins. These data are for n-paraffins up to C18, but only up to C10 or C12 for n-olefins. The comparisons appear in Tables 5-7. It can be seen from Table 5 that the ABC and API 4A1.1 methods give comparable accuracy for critical temperature. Any differences can easily be attributed to the differences in data sets employed in developing the correlations. API 4A2.1, however, predicts critical temperature values with errors an order of magnitude worse. This is expected since structural information on a compound is only introduced in this correlation through the correlating parameters: normal boiling point and specific gravity. Major differences between
the correlations are apparent only when predictions for higher carbon numbers are compared, as can be seen for C50 and C100. In API 4A1.1, critical temperature is correlated to normal boiling point through the use of structural parameters developed for organic compounds with a variety of different functional groups. The critical temperature approaches the normal boiling point in the limit of large carbon numbers. For API 4A2.1, the limiting value also is essentially the limiting value for the normal boiling point, whereas in the ABC, the limiting value is treated as an adjustable parameter. There is no intuitive or theoretical reason to believe the critical temperature and normal boiling point should converge in the limit of high carbon numbers. In fact, Magoulas and Tassios (1990) have determined (using the K-Z correlation) the limiting saturation temperature as a function of pressure. Their results indicate this convergence occurs at a much lower pressure, approximately 10 mmHg. Above this pressure, the locus of critical temperature and boiling point intersect. The application of API procedures 4A1.1 and 4A2.1 are both limited by the need to have available values for normal boiling point (and specific gravity for 4A2.1).
314 624 966
17.751 -49.851 51.114 5.7741 316 591 867 2795 5550 ∝n
ABC
RMSE AD AAD %AAD C5 C10 C15 C50 C100 Vc∞
reproduce the data extremely well, excluding the C3 data (this will be discussed further in part II). Plots of error distributions for all the temperature-dependent ABCs developed in parts II and III may be found elsewhere (Marano, 1996).
1.0151 0.1528 3.1518 0.6390 314 618 966 3608 6975 ∝n3/2
n-olefins
data API 4A1.1 (cm3/ (Ambrose) g‚mol)
293 584
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1893 Table 8. Thermophysical Properties Correlated with ABCs part II Tb Tm Tc Pc Vc ω VL Pvap
normal boiling point (I) normal melting point (I) critical temperature (I) critical pressure (I) critical volume (II) acentric factor (derived from I) liquid molar volume (II) vapor pressure (II)
For n-paraffins and n-olefins, these data are available up to C30. For other compounds, these data may not be available even up to C20. For the predictions for C50 and C100 given in Table 5, the ABCs developed for normal boiling point (Table 4) and molar volume (see part II) were used to estimate these properties. The predictions for critical pressure are given in Table 6. Neither correlation, the ABC or API 4A1.1 performs as well for critical pressure as for temperature; however, the ABC results are clearly superior to the API method based on RMSE and the other statistical criteria presented in Table 6. Both correlations predict a limiting value of zero. The predictions for critical volume are given in Table 7. The results from the two correlations are surprisingly different. The reason for this discrepancy is that the correlations predict different types of asymptotic behavior. The API 4A1.1 method predicts the critical molar volume to be linear in carbon number n and is based on earlier experimental data which support this proportionality. The newer data of Anselme et al. (1990, included in Teja’s recommendations) indicate a higher-order dependence on n. Lattice theories and molecular simulations tend to support the validity of Anselme’s results (see part II). A proportionality of n3/2 was used for the ABC for critical molar volume on the basis of lattice theory. Obviously, this relationship works better for correlating Anselme’s data. A more detailed discussion of the chain-length dependence of critical volume is given by Kontogeorgis et al. (1995). Conclusions A new, general form for an ABC has been developed which can be used to accurately correlate the thermophysical properties of n-paraffins, n-olefins, and other homologous series of compounds. This equation is applicable to properties exhibiting a wide range of behavior and has been adapted to predict temperaturedependent properties. A methodology for regressing the parameters for the ABC has been described and leads to correlations which give accurate and consistent predictions and which yield reasonable extrapolations. The ABCs presented in parts II and III of this series are useful for engineering applications in a number of areas, including petroleum and synthetic fuels processing. The ABCs for critical properties of n-paraffins and n-olefins have been shown to be superior to the methods recommended in the API Technical Data Book (1992) based on group contribution methods. For other properties, the ABCs represent the only reliable method for estimating property values for high molecular weight homologous compounds. Parts II and III of this series present the ABCs developed for PVT, thermal, and transport properties. The specific properties correlated are listed in Table 8. Also identified are the types of behavior (I or II) demonstrated by the various properties. Parts II and
part III CpId HFId GFId ∆Hvap° ∆Hb Cp,L µL λL σ
ideal-gas heat capacity (II) enthalpy of formation (II) gibbs free energy of formation (II) enthalpy of vaporization at 25 °C (II) enthalpy of vaporization at Tb (I) liquid heat capacity (II) liquid viscosity (II) liquid thermal conductivity (I) surface tension (I)
III also contain discussions of the theoretical basis for asymptotic behavior and predictions for the properties of high molecular weight linear polyethylene. Acknowledgment The authors are grateful to the U.S. Department of Energy, Pittsburgh Energy Technology Center for providing much of the resources used in this investigation. J.J.M. also expresses his gratitude to Burns and Roe Services Corporation and the University of Pittsburgh for their financial support. Nomenclature A1, ..., Ar ) temperature coefficients in eq 5 n ) carbon number P ) pressure, bar T ) temperature, K V ) volume, cm3/(g‚mol) Y ) physical property correlated Greek Letters R, β, γ ) correlating parameters Subscripts b ) at normal boiling point c ) at critical point L ) as a liquid 0 ) at effective carbon number of zero ∞ ) as carbon number approaches infinity Notation in Tables and Figures 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 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
Literature Cited 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. API Research Project 44, Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; Carnegie Press: Pittsburgh, PA, 1953. API Research Project 42, Properties of Hydrocarbons of High Molecular Weight; American Petroleum Institute: New York, 1966. API Technical Data BooksPetroleum Refining, 2nd ed., 1970; American Petroleum Institute: Washington, DC, 1976 revision.
1894 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 API Technical Data BooksPetroleum Refining, 5th ed., 1992; American Petroleum Institute: Washington, DC, 1992. Beaton, C. F.; Hewitt, G. F. Physical Property Data for The Design Engineer; Hemisphere Publishing: New York, 1989. Gasem, K. A. M.; Robinson, R. L., Jr. Prediction of Phase Behavior for CO2 + Heavy Normal Paraffins Using Generalized-Parameter Soave and Peng-Robinson Equations of State. Presented at the AIChE Annual Meeting, Houston, TX, March 26, 1985. Gasem, K. A. M.; Ross, C. H.; Robinson, R. L., Jr. Prediction of Ethane and CO2 Solubilities in Heavy Normal Paraffins Using Generalized-Parameter Soave and Peng-Robinson Equations of State. Can. J. Chem. Eng. 1993, 77, 805-816. 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. 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. 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. Kreglewski, A.; Zwolinski, B. J. A New Relation for Physical Properties of n-Alkanes and n-Alkyl Compounds. J. Phys. Chem. 1961, 65, 1050-1052. Kudchadker, A. P.; Zwolinski, B. J. Vapor Pressures and Boiling Points of Normal Alkanes, C21 to C100. J. Chem. Eng. Data 1966, 11, 253-255. Kudchadker, A. P.; Alani, G. H.; Zwolinski, B. J. Critical Constants of Organic Substances. Chem. Rev. 1968, 68, 723-725. Kurata, M.; Isida, S. Theory of Normal Paraffin Liquids. J. Chem. Phys. 1955, 23, 1126-1131. 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.
Marano, J. J. Property Correlation and Characterization of Fischer-Tropsch Liquids for Process Modeling. Ph.D. Dissertation, University of Pittsburgh, Pittsburgh, PA, 1996. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases & Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases & Liquids, 4th ed.; McGraw-Hill: New York, 1987. Shaver, R. D.; Robinson. R. L., Jr.; Gasem, K. A. M. A Framework for The Prediction of Saturation Properties: Vapor Pressures. Fluid Phase Equilib. 1991, 64, 141-163. Shaver, R. D.; Robinson. R. L., Jr.; Gasem, K. A. M. A Framework for The Prediction of Saturation Properties: Liquid Densities. Fluid Phase Equilib. 1992, 78, 81-98. 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. TRC (Thermodynamic Research Center). TRC Thermodynamic TablessHydrocarbons; The Texas A&M University: College Station, TX, 1987 revision. Tsonopoulos, C. Critical Constants of Normal Alkanes from Methane to Polyethylene. AIChE J. 1987, 33, 2080-2083. Vargaftik, N. B. Tables on The Thermophysical Properties of Liquids and Gases, 2nd ed. (English trans.); John Wiley & Sons: New York, 1975.
Received for review August 15, 1996 Revised manuscript received January 6, 1997 Accepted January 6, 1997X IE960511N
X Abstract published in Advance ACS Abstracts, February 15, 1997.
