4898
Ind. Eng. Chem. Res. 1998, 37, 4898-4902
A Simple Method of Estimating the Enthalpy of Petroleum Fractions H. M. Moharam, A. M. Braek, and M. A. Fahim* Department of Chemical Engineering, University of Kuwait, P.O. Box 5969, 13060 Safat, Kuwait
A simple method, requiring the characterization factor as the only input, is proposed to predict the enthalpies of the liquid and vapor petroleum fractions over a wide range of temperatures and pressures. This method showed a comparable accuracy in the prediction of the petroleum fraction enthalpy with much more complicated predictive methods in the literature. The proposed method gave overall deviations of (2.48% when tested on the data for liquid fractions and (1.23% when tested on the data for vapor fractions. Introduction A reliable method for the estimation of process stream enthalpies of petroleum fractions is of high importance in engineering calculations encountered in the petroleum industries. Several equations of state are being used for this purpose on a routine basis in thermodynamic property generators and in process design and simulation software. The enthalpy can be evaluated by substituting the equation of state in the following relation:
H-H RT
ID
)Z-1+
∂P ∫∞ [T(∂T )ν - P] dν
1 RT
ν
Table 1. Comparison of the Results of the Present Correlation with Previous Methods for the Data Used in Developing the Liquid Enthalpy Correlation Kw
γ
N
PR
% av abs dev SRK PRSV API 1.12 1.79 0.90 0.22
1.19 1.60 0.46 0.19
3.24 2.16 0.19 0.33
Jet Naphtha 2.99 0.41 2.37 0.47 0.95 1.50 0.39 0.15
1.01 3.57 1.10 0.36
0.69 0.38 0.45 0.10
2.44 1.27 1.07 0.30
Aromatic Naphtha 5 22.76 26.88 20.32 3 24.04 28.00 18.72 12 22.64 26.09 19.53 20 3.47 4.05 2.93
0.62 1.19 0.71 0.13
1.51 2.26 2.14 0.30
2.85 3.29 3.19 2.78 0.43
1.49 0.50 1.45 1.13 0.16
3.10 2.10 1.72 1.30 0.29
High-Boiling Naphtha 5 6.29 3.86 4.05 5 5.19 2.93 3.79 5 4.81 2.71 3.81 15 1.09 0.63 0.78
0.40 0.52 0.73 0.11
2.52 0.95 1.23 0.31
4 4 5 13
Kerosine 3.70 1.42 3.92 1.79 2.92 0.91 0.81 0.32
1.68 2.41 1.78 0.45
1.22 2.01 1.54 0.37
1.70 1.76 2.34 0.45
0.14 0.69
19 3 22
Fuel Oil 7.91 9.68 9.34 11.66 0.78 0.97
7.41 7.54 0.68
2.79 1.81 0.21
3.31 6.07 0.43
4.83
24 24
Gas Oil 3.21 5.54 3.21 5.54
3.67 3.67
1.53 1.53
2.29 2.29
4.09
3.21
1.14
1.97
0.41 2.07 4.14
11.48 0.804
0.21 0.69 2.76
10.5
0.852
1.38 3.45 6.89
12.1
0.739
0.69 2.07 3.79 9.65
Low-Boiling Naphtha 6 5.81 3.23 5 5.03 2.68 4 3.41 1.62 13 4.47 2.42 28 0.67 0.36
12.1
0.762
0.34 1.03 4.14
11.8
0.809
0.21 0.55 1.03
11.68 0.86
11.8
0.848
present
Alaska Naphtha 5 1.92 0.59 5 1.83 2.13 7 0.64 1.86 17 0.26 0.27
11.63 0.777
(1)
Several comparison studies have been published in the literature that compare enthalpy predictions using different methods. Unfortunately, most of these studies deal with pure components. The most frequently used methods in simulation software among these methods are the equations of state of Soave-Redlich-Kwong (Soave, 1972), Peng and Robinson (1976), and PengRobinson-Stryjek-Vera (Stryjek and Vera, 1986), and the method of Lee and Kesler (1975). Tarakad and Danner (Technical Data BooksPetroleum Refining, 1976) have shown that the SRK equation often gives large errors in the critical region and that the Lee and Kesler correlation is, in general, more reliable for a wide range of hydrocarbons. The work by Tarakad and Danner was undertaken to provide the basis for selecting the methods to be recommended in the third edition of the Technical Data BooksPetroleum Refining (1976). Toledo and Reich (1988) compared the enthalpy prediction methods for pure fluids and defined mixtures. They concluded that the Plocker-Lee-Kesler and the GCEOS equations of state were significantly better than the SRK and PR equations in estimating the enthalpies of single-phase nonpolar fluids. They also concluded that the SRK equation shows a better overall performance than the PR equation; however, both show large deviations at high pressures. Manavis et al. (1994) compared the accuracy of predicting the gas-phase enthalpy using 15 generalized equations of state. They concluded that for light and medium hydrocarbons, the Lee-Kesler method is more accurate than the other methods. Petroleum fractions are complex mixtures whose physical and chemical properties vary considerably with their composition. For this reason, developing an en-
P, MPa
4 5 7 16
155
3.69
thalpy correlation accounting for all the compositional details is difficult. A simple estimation method requiring little or no input data is often preferred over complex but more accurate correlations. Practical property prediction methods for undefined petroleum fractions
10.1021/ie9800997 CCC: $15.00 © 1998 American Chemical Society Published on Web 11/14/1998
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4899 Table 2. Comparison of the Results of the Present Correlation with Previous Methods for the Data Used in Developing the Vapor Enthalpy Correlation API gravity
Kw
P, MPa
N
PR
% av abs dev SRK PRSV API
present
50.5 11.63 0.21 1.38
Alaska Naphtha 13 0.93 0.78 5 0.17 1.39 18 0.72 0.95
0.45 0.87 0.57
1.14 0.4 0.93
0.66 0.46 0.60
44.4 11.48 0.14 0.69 1.38
Jet Naphtha 6 1.5 1.06 7 1.03 0.59 4 1.05 1.51 17 1.20 0.97
0.87 0.09 0.77 0.53
1.46 1.1 2.76 1.62
0.75 0.54 1.28 0.79
34.5 10.5
59.9 12.1
54.2 12.1
43.5 11.8
overall
Aromatic Naphtha 0.69 6 23.96 26 2.07 7 25.37 27.81 3.45 2 39.22 42.02 15 26.65 28.98
21.6 1.6 22.73 1.59 35.75 15.57 24.01 3.46
Low-Boiling Naphtha 0.00 3 1.06 2.76 1.03 10 1.68 3.26 2.41 7 2.38 4 3.27 5 2.79 4.43 25 2.02 3.64
0.78 1.27 12.37 2.55
2.54 2.86 3.4 3.58 3.12
0.18 0.9 2.06 2.14 1.39
1 1.89 1.45 1.41 1.56
High-Boiling Naphtha 0.00 7 0.21 1.79 1.33 0.34 4 0.71 2.51 1.81 1.03 5 0.97 2.59 1.83 16 0.57 2.22 1.61
0.16 0.94 1.73 0.85
0.31 0.48 0.4 0.38
1.77 1.89 1.08 1.63
2.86 3.1 2.6 2.89
2.2 2.17 1.31 1.94
4.70
1.80
1.31
0.14 0.34 0.55
5 8 5 18 109
Kerosine 0.63 2.35 0.93 2.49 0.22 1.75 0.65 2.25 4.63
5.83
are most conveniently based on commonly available parameters such as the characterization factor, which depends on the boiling temperature and specific gravity. The purpose of the present work is to propose simple and reliable correlations to calculate the enthalpies of the liquid and vapor petroleum fractions and to study the reliability in comparison with several equations of state that are widely used today in practical work for a wide range of conditions.
Table 3. Comparison of the Results of the Present Correlation with Previous Methods for Liquid Petroleum Fractions Kw
γ
P, MPa
HL ) [0.03181T + 0.00001791Kw4.693]2.2916
(2)
PR
% av abs dev SRK PRSV API
present
11.63 0.777
0.21 0.69 1.38 2.76 2.86 5.52 9.65
Alaska Naphtha 5 1.96 0.67 5 1.42 1.01 6 0.83 1.45 6 1.82 2.41 3 1.42 1.91 11 0.59 1.6 11 0.57 1.43 47 1.06 1.50
11.48 0.804
0.14 0.41 1.03 1.38 2.07 9.65
Alaska Naphtha 12 3.53 0.87 5 2.64 0.3 3 2.31 0.92 2 2.56 1.18 5 1.26 1.43 26 2.25 0.47 53 2.50 0.69
0.95 0.92 1.27 1.78 0.92 0.61 0.83
1.16 0.61 0.51 0.74 0.69 0.77 0.82
3.52 1.97 0.45 0.29 0.72 2.92 2.52
10.5
0.852
0.69 2.07 2.59 4.14 9.65
Aromatic Naphtha 5 22.94 27.31 20.76 6 22.4 26.21 19.27 4 23.48 27.35 19.22 3 51.33 55.86 44.27 20 22.54 26.23 20.28 38 24.94 28.83 21.97
0.44 0.91 0.61 21.21 0.82 2.37
2.9 1.05 1.77 24.71 2.59 4.05
12.1
0.739
0.21 1.03 1.38 2.41 2.76 3.1 3.27 3.45 4.14 5.52 6.89
Low-Boiling Naphtha 5 6.79 4.01 3.52 5 5.69 3.21 3.22 3 5.95 3.4 3.8 2 3.86 1.83 2.85 2 3.84 1.9 3.44 3 3.88 2.04 4.01 4 3.66 1.86 3.8 4 3.64 1.84 3.61 5 3.6 1.79 3.1 8 3.81 1.97 2.82 17 5.08 2.83 2.81 58 4.68 2.54 3.19
1.19 0.94 0.44 0.31 1.2 2.56 2.6 2.13 0.92 0.3 0.98 1.13
3.43 2.9 2.53 1.9 2.27 3.02 2.92 2.45 1.37 0.69 2.24 2.22
12.1
0.762
0.21 0.48 0.69 1.38 1.65 2.07 6.89 9.65
High-Boiling Naphtha 9 6.18 3.62 3.52 3 6.23 3.74 4.35 4 5.68 3.31 3.92 4 4.99 2.73 4.07 2 5.73 3.2 4.93 3 4.88 2.59 4.33 5 4.93 2.81 3.74 22 5.17 2.87 3.09 52 5.41 3.06 3.58
0.66 0.73 0.67 0.89 1.42 1.36 1.01 0.67 0.79
2.34 2.12 1.55 0.78 1.52 1.27 1.38 2.03 1.83
11.8
0.809
0.14 0.34 0.48 0.69 9.65
6 5 4 4 19 38
Kerosine 3.71 1.37 3.48 1.31 3.69 1.53 3.16 1.05 3.25 1.17 3.39 1.25
1.38 1.56 2.07 1.78 1.29 1.47
1.01 1.15 1.64 1.41 1.5 1.38
2.11 0.97 1.35 1.89 1.86 1.73
11.68 0.86
0.17 9.65
5 19 24
Fuel Oil 6.54 8.4 9.05 11.31 8.53 10.70
5.5 7.9 7.40
3.52 1.75 2.12
4.35 2.9 3.20
11.8
0.28 9.65
23 25 48
Gas Oil 3.36 5.67 3.4 5.79 3.38 5.73
4.09 3.59 3.83
0.69 2.04 1.39
3.08 1.9 2.47
6.09
4.81
1.29
2.37
Proposed Correlations The available set of liquid and vapor enthalpy data, which was obtained from Lenoir (1973), was divided into two groups. The first was used to develop the correlations and the second to test the developed correlations. The liquid fraction data cover a temperature range of 297-617 K and a pressure range of 0.21-9.65 MPa. The vapor fraction data cover a temperature range of 445617 K and a pressure range of 0-3.79 MPa. In studying the liquid enthalpy of the petroleum fractions, it has been observed that pressure has a negligible effect on the liquid enthalpy and that the relationships between liquid enthalpy and the characterization factor Kw and temperature T are of a power law form (Y ) aXb). From these observations, the following simple correlation for predicting the liquid enthalpy is proposed:
N
0.848
358
5.83
1.28 1.18 1.1 1.66 0.47 1.09 1.21 1.18
1.55 1.06 0.6 2.18 1.96 0.42 1.01 1.09
3.46 2.53 1.68 1.87 1.38 0.82 1.21 1.65
where T is in K and HL is in kJ/kg. The error limits for this equation were (2.48%.
4900 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 Table 4. Comparison of the Results of the Present Correlation with Previous Methods for Vapor Petroleum Fractions API gravity
Kw
P, MPa
N
PR
% av abs dev SRK PRSV API
present
50.5
11.63
0.00 0.41 0.69 2.07 2.76
Alaska Naphtha 12 1.25 0.33 9 0.7 0.94 9 0.47 1.11 3 0.69 2.3 2 0.15 1.73 35 0.80 0.94
44.4
11.48
0.00 0.21 0.41 1.03
Jet Naphtha 13 1.69 0.26 13 1.22 0.49 10 1.07 0.59 6 0.87 1.85 42 1.28 0.64
0.37 0.29 0.23 1.2 0.43
0.45 0.71 0.91 2.74 0.97
0.37 0.56 0.68 1.7 0.69
34.5
10.5
0.00 0.21 1.38 2.59
Aromatic Naphtha 7 23.84 26.38 9 24.44 27.01 7 23.35 25.71 4 27.29 29.77 27 24.42 26.92
21.67 22.18 20.98 24.5 22.08
3.08 2.58 0.7 3.63 2.38
2.12 2.12 0.13 2.76 1.70
59.9
12.1
0.21 0.69 1.38 2.07 2.76 3.10 3.45 3.79
Low-Boiling Naphtha 18 1.9 3.61 13 1.98 3.61 9 2.15 3.74 7 2.54 4.15 10 3.02 4.67 7 3.16 4.83 4 2.74 4.37 3 2.46 4.05 71 2.36 4.01
3.33 0.91 3.23 1.14 3.29 1.55 3.6 2.21 3.95 2.67 3.98 2.05 3.53 3.02 3.13 13.32 3.49 2.16
1.72 2.13 2.05 2.01 1.34 1.6 1.1 1.04 1.74
54.2
12.1
1.6 1.74 1.68 2.59 1.81
0.41 0.93 1.11 3.05 1.11
0.26 0.54 0.53 0.93 0.49
43.5
11.8
0.84 1.58 1.42 1.96 1.33
1.55 2.56 2.83 3.74 2.37
1.21 1.85 1.67 2.18 1.62
4.16
1.67
1.19
overall
High-Boiling Naphtha 0.21 12 0.43 2.17 0.48 8 0.71 2.39 0.69 7 0.72 2.35 1.38 5 1.83 3.47 32 0.78 2.47 0.00 0.21 0.48 0.69
9 10 6 2 27 234
Kerosine 0.29 1.3 0.59 2.13 0.52 2.06 1.15 2.68 0.52 1.88 4.05
5.13
0.12 0.57 0.71 1.61 0.63 0.54
1.52 0.75 0.33 1.67 0.46 0.97
0.69 0.34 0.37 0.64 1.57 0.56
For vapor petroleum fractions, the pressure has an appreciable effect on the predicted enthalpy. It has been found that the following simple correlation represents well the vapor enthalpy:
HV ) 2.4719T - 0.0253P + 56.221Kw - 899.232 (3) where T is in K, HV is in kJ/kg, and P in kPa. The error limits for this equation were (1.23%. The data given in Table 1 were used to develop eq 1, while the data given in Table 2 were used to develop eq 2. The error limits for these two equations were calculated from the data shown in Tables 3 and 4, respectively. It should be noted that the proposed correlations for liquid and vapor petroleum fractions require the characterization factor as the only input. The values of the constants included in these correlations were determined using the Solver method (Excel 5.0, Microsoft).
developing the correlations, comprises 155 liquid enthalpy data points and 109 vapor enthalpy data points. A comparison of the results of applying the developed correlations on the first group of data with other methods is shown in Tables 1 and 2. The second group, which was used to test the present correlations, comprises 358 data points for 8 liquid fractions and 234 data points for 6 vapor fractions. Hydrocarbons predominate in the makeup of petroleum fractions. On the basis of the comparison study of Manavis et al. (1994), it was expected that the API method, which is based on the LK EOS, will be the best for enthalpy prediction. For liquid fractions, as shown in Tables 1 and 3, the API method is better than all other methods, and the proposed method comes next to it. The PRSV EOS is slightly better than the other equations of state. Although the proposed correlation is simple and has a limited number of correlating constants (4 constants for the present correlation in comparison with more than 15 for the API method), its accuracy for predicting the liquid enthalpy is still reasonable. For vapor fractions, as shown in Tables 2 and 4, the proposed method is better than all other methods and the API method comes next to it. The PR and PRSV EOSs are equally better than the SRK equation of state. The present correlation (which has only four correlating constants) eliminates the need to solve complex mathematical relations that are required in equation of state models and the API model, which is based on the LK EOS. In addition, it gives a better prediction accuracy. It is to be noted that the equation of state models fail to predict the liquid and vapor enthalpies of aromatic naphtha. The higher the aromatics and the heavy component contents in the fraction, the higher the deviation in prediction. The API method works better than the proposed correlation in the liquid phase, but the reverse is true in the vapor phase. The present correlations work much better the other equations of state. They are not based on the data for naphthas, which contain polar or noncondensing compounds. Conclusion New correlations have been proposed for predicting the liquid and vapor enthalpies of the petroleum fractions. The proposed correlations give comparable results to more complex models found in the literature. The proposed correlation for the liquid phase gives an overall average deviation of (2.48% when tested on data points including eight petroleum fractions. The correlation for the vapor phase gives an overall average deviation of (1.23% when tested on data including six petroleum fractions. Nomenclature a, b ) EOS parameters H ) enthalpy Kw ) characterization factor N ) number of data points P ) pressure R ) gas constant T ) temperature Z ) compressibility factor
Results and Discussion
Greek Symbols
The present correlations have been applied to the two groups of data. The first group, which was used in
R ) temperature-dependent part of EOS γ ) specific gravity
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4901 ω ) acentric factor κ ) expression in PRSV EOS ν ) molar volume
3. Pressure Effect on Enthalpy
(
)
H0 - H H0 - H ) RTc RTc
Superscripts
(0)
ω
ID ) ideal (r) ) for reference fluid (0) ) for simple fluid
+
[(
ω(h)
c ) critical L ) liquid phase r ) reduced V ) vapor phase
Appendix I. API Method for Enthalpy Calculation of Petroleum Fractions 1. Liquid Phase
HL ) A1[T - 259.7] + A2[T2 - 259.72] + 3
3
A3[T - 259.7 ] (I.1)
( ) H0 - H RTc
{
(i)
A1 ) 10-3 -1171.26 + (23.722 + 24.907γ)Kw +
(
[
(
2Trνr
]
)]
( (
)]
(i)
d2 5Trνr5
)]
(i)
)
(I.2)
Z(0 or h) )
(0)
)
H0 - H RTc
Prνr B C D )1+ + 2+ 5+ Tr νr ν ν r
[
B1 ) 10-3 -356.44 + 29.72Kw +
(h)
c4 3
r
(
Tr νr3
β+
) ( )
π π exp - 2 (I.5) 2 νr νr
with
(
248.46 γ
)]
[
(
B ) b1 -
b2 b3 b4 - 2- 3 Tr T T r
B2 ) 10-6 -146.24 + (77.62 - 2.772K)Kw B4 301.42 -
253.87 γ
)]
C ) c1 -
r
c2 c3 + 3 Tr T r
-9
B3 ) 10 [-56.487 - 2.95B4] B4 ) 10.0 12.8 - 1.0 1.0 (γ - 0.885)(γ - 0.70)(104) Kw Kw
[(
)(
)
(I.4)
when the equation is applied to the heavy reference fluid, and Z(i) ) Z(0) or Z(h) is compressibility factor of the fluid, depending on whether the equation is applied to the simple fluid or the heavy reference fluid. νr, Z(0), and Z(h) are obtained from the LK equation
where HL is the liquid enthalpy at a reduced temperature of 0.8 calculated by eq I.1,
B4 295.02 -
}
+ 3E
)
H0 - H RTc
)
) (
H0 - H RTc
HV ) HL + B1[T - 0.8Tc] + B2[T2 - 0.64Tc2] + B3[T3 - 0.512Tc3] + RTc H0 - H 4.507 + 5.266ω MW RTc
+
) (
H0 - H RTc
2.3653 γ
(
(I.3)
when the equation is applied to the simple fluid,
T is in degrees Rankine. 2. Vapor Phase
[
(0)
where
13.817 γ
A3 ) -10-9 (1.0 + 0.82463Kw) 9.6757 -
c2 - 3c3/Tr2 2
1149.82 - 46535Kw γ
[
H0 - H RTc
-
b2 - 2b3/Tr + 3b4/Tr2 Trνr
) -Tr Z(i) - 1 -
where HL is the enthalpy of the liquid petroleum fraction with Tr e 0.8 and Pr e 1.0 in Btu/lb,
A2 ) 10-6 (1.0 + 0.82463Kw) 56.086 -
(h)
where (H0 - H)/RTc is the dimensionless effect of pressure on enthalpy of the fluid of interset, ((H0 - H)/ RTc)(0) is the effect of pressure on the enthalpy of the simple fluid, to be calculated from eq I.4, ((H0 - H)/ RTc)(h) is the effect of pressure on the enthalpy of the heavy reference fluid (n-octane), to be calculated from eq I.4, ω is the acentric factor of the fluid for which the pressure effect on enthalpy is being sought, and ω(h) is the acentric factor of the heavy reference fluid ) 0.3978. The dimensionless effects of pressure on the enthalpies of the simple fluid and the heavy reference fluid are to be calculated from the following equation:
Subscripts
[
)]
) (
H0 - H RTc
]
D ) d1 + 2
d2 Tr
where the parameters are given in Table 5.
4902 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
RTc b ) 0.0778 Pc
Table 5 b1 b2 b3 b4 c1 c2 c3 c4 d1 d1 β π
(0)
(h)
0.118 1193 0.265 728 0.154 79 0.030 323 0.023 6744 0.018 6984 0.0 0.042 724 0.155 488 0.623 689 0.653 92 0.060 167
0.202 6579 0.331 511 0.027 655 0.203 488 0.031 3385 0.050 3618 0.016 901 0.041 577 0.483 6 0.074 0336 1.226 0.037 54
where
R(Tr,ω) ) [1 + m(1 - Tr1/2)]2 and
m ) 0.37464 + 1.54226ω - 0.26992ω2 PRSV EOS The PRSV EOS is an extension of the Peng-Robinson EOS, utilizing an extension of the κ expression as shown below:
Appendix II. Equations of State Used in the Study RKS EOS
R ) [1 + κ(1 - Tr0.5)]2 Z)
ν a ν - b RT(ν + b)
(II.1)
κ ) κ0 + κ1(1 + Tr0.5)(0.7 - Tr) κ0 ) 0.378893 + 1.4897153ω - 0.17131848ω2 +
with
0.0196554ω3
R2Tc2 R(Tr,ω) a ) 0.42748 Pc
Literature Cited
RTc b ) 0.08664 Pc where
R(Tr,ω) ) [1 + m(1 - Tr1/2)]2 and
m ) 0.48 + 1.574ω - 0.176ω2 PR EOS
Z)
aν ν ν - b RT(ν(ν + b) + b(ν - b))
with 2
2
R Tc R(Tr,ω) a ) 0.45724 Pc
(II.2)
(1) Lee, B.-I.; Kesler, M. G. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 1975, 21, 510. (2) Lenoir, J. M. Measured enthalpies of eight hydrocarbon fractions. J. Chem. Eng. Data 1973, 18, 195-202. (3) Manavis, T.; Volotopoulos, M.; Stamatoudis, M. Comparison of fifteen generalized equations of state to predict gas-phase enthalpy. Chem. Eng. Commun. 1994, 130, 1-9. (4) Peng, D.-Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59. (5) Soave, G. Equilibrium constants from a modified RedlichKwong equation of state. Chem. Eng. Sci. 1972, 27, 1197. (6) Stryjek, R.; Vera, J. H. PRSV: An Improved Peng-Robinson equation of state for pure compounds and mixtures. Can. J. Chem. Eng. 1986, 64, 323-333. (7) Technical Data BooksPetroleum Refining; American Petroleum Institute: Washington, DC, 1976. (8) Toledo, P. G.; Reich, R. A comparison of enthalpy prediction methods for Nonpolar and polar fluids and their mixtures. Ind. Eng. Chem. Res. 1988, 27, 1004-1010.
Received for review February 17, 1998 Revised manuscript received September 14, 1998 Accepted September 15, 1998 IE9800997
