948
Ind. Eng. Chem. Res. 1987,26, 948-952
Prediction of Dew Points of Semicontinuous Natural Gas and Petroleum Mixtures. 1. Characterization by Use of an Effective Carbon Number and Ideal Solution Predictions Bert Willman and Amyn S. Teja* School of Chem.ica1Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100 The effective carbon number (ECN) is used as the distribution variable in a gamma distribution function to characterize natural gas and petroleum mixtures containing many components. The ECN offers the advantage over more conventional continuous distribution variables in that it can distinguish between isomers. The resulting function is combined with an ideal-solution theory to develop a simple method to predict dew points of natural gas condensates. The simple method is compared with the modified equation of state approach of Cotterman et al. and, although it is less accurate in its predictions, is shown to work surprisingly well. Since the method involves no density iterations, stability and convergence problems are minimized and the method can be used as a first approximation in many cases. Mixtures whose complete composition is not available are of considerable interest in oil and natural gas processing. Examples of these mixtures include reservoir fluids, coal liquids, and natural gas condensates. In order to apply conventional equilibrium thermodynamic relationships to these fluids, the undefined fractions are often represented by some equivalent pseudocomponents. The choice of pseudocomponents is, however, somewhat arbitrary and can sometimes lead to unreliable predictions. In particular, when the pseudocomponent method is combined with an equation of state to calculate phase equilibria, the arbitrary description of the (heavy) undefined fractions considerably diminishes the accuracy of the predictions (Whitson, 1983). This has also been demonstrated by Adler et al. (1977) who showed that the calculated dew points of a natural gas mixture differed by nearly 40 K, depending on the characterization adopted for the “heavy ends”. (Bubble points, on the other hand, remained unaffected by the Characterization.) An alternative to the pseudocomponent approach involves the representation of the composition of an undefined mixture by a continuous distribution function. While this approach is not new to chemical engineering (see, for example, the work of Bowman, 1949 and Edmister, 1955), it has recently been reformulated for use with modern equations of state and solution models by Ratzsch and Kehlen (1983), Cotterman and Prausnitz (1985), and Cotterman et al. (1985). We have combined this recent work on “continuous thermodynamics” with the effective carbon number (ECN) concept of Ambrose (1976) and Chase (1984) to calculate dew points in semicontinuous natural gas and petroleum mixtures. In the work of Cotterman and Prausnitz (1985) and Cotterman et al. (1985), the composition of an undefined mixture is described by a continuous distribution function whose argument is the molecular weight. The molecular weight cannot, however, distinguish between the various isomers of a compound. Distribution functions which employ this property as a characterization variable must therefore necessarily “lump” all isomers together. We show below that the effective carbon number (ECN) can be successfully employed as the single characterization variable in continuous thermodynamics studies and combine it with an ideal solution theory to predict dew points. In the second part of this work, we show how this simple ap0888-5885/87/2626-0948$01.50/0
proach may be modified via extensions of the virial equation and regular solution theory to continuous mixtures.
Effective Carbon Number Ambrose and Sprake (1970) have shown that if the boiling points of a homologous series of compounds are plotted against their carbon numbers, a smooth curve may be drawn through the points. This curve may then be used to calculate a nonintegral effective carbon number (ECN) of other (similar) compounds from their normal boiling points. The ECN, in turn, may be used to calculate the properties of these compounds from equations relating the properties of homologous series to the carbon number. Thus, for example, Ambrose and Sprake reported that the normal boiling points, Tb, of the straight-chain primary alkanols from C3 to CIS could be correlated with their carbon numbers, n, by
Tb = 306.83 + 21.732n - 0.1591n2 - 0.00406n3 (1) where T b is in degrees K. Their variation in vapor pressures Po with temperature T could be expressed as log
= 17.5832 + 0.96958n -
(3175.08 + 346.908n)Tl - (0.0199352 + 0.00124927n)T + (1.02395 + 0.0650235n)10-5p (2)
where Po is in kPa. If eq 1 is used to calculate the effective carbon number (ECN) of other alkanols from their boiling points, then it can be shown (Ambrose and Sprake, 1970; Ambrose 1976) that eq 2, with appropriate values of ECN, reproduces the vapor pressures of propan-2-01, butan-1-01, 2-methylpropan-1-01,and 2-methylpropan-2-01within 2% between 25 and 120 kPa. More recently, Willman and Teja (1985) showed that the ECN can be used to successfully predict the vapor pressures of a wide variety of fluids (including isomeric alkanes, alkenes, alkynes, cycloalkanes, alkadienes, aromatics, and sulfur-containing compounds) from a correlation of the vapor pressures of the alkanes. The approach can also be extended to other properties such as critical points, viscosities, and thermal conductivities, as demonstrated by Chase (1984). The ECN therefore appeared to be a singularly appropriate characterization variable to adopt in our continuous thermodynamics work. 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 949 Development of t h e Method Following the work of Cotterman and Prausnitz (1985) and Cotterman et al. (1985), the composition of a mixture with a continuous distribution of Components is described by a distribution function, F[A, whose argument, I , is the ECN in this work. (Square brackets are used to denote functional dependence.) The function, F[A, is normalized such that i F [ I ] dI = 1
(3)
Semicontinuous mixtures are those in which the mole fractions of some components have discrete values while the mole fractions of others are described by a distribution function. For m discrete components, the normalization is (4) where q = (1- Cyxi)is the total "mole fraction" of the continuous portion. Again, following Cotterman and Prausnitz (1985) and Ratzsch and Kehlen (1983), we may write the equilibrium relationship for continuous mixtures as follows:
PJ'Vl = FLIAACTII,TQlf*[I,Tl@*[I,Tl exp(W1) (5)
F"[qand FL[Oare the distributions of mole fractions in the vapor and liquid phases, respectively; f*[I,T] is the standard-state fugacity of the species, I , at the system temperature, T; ACT[I,T,P] is the liquid-phase activity coefficient of species I; and K [ A represents the correction due to and the pressure dependence of the liquid-phase fugacity of species I. As a first approximation for natural gas and petroleum mixtures at moderate pressures, we may assume ideal vapor and liquid phases and no pressure dependence of the liquid-phase fugacity. Hence, eq 5 reduces to
Pm4 = FL[4f*[I,T1
(6)
For subcritical components at temperatures well below their critical temperatures,
f*[I,T]= P*[I,T]
(7)
where PC is the vapor pressure. Note that as the critical point is approached, the correction @*[I,Tldue to the nonideal behavior of the saturated vapor of the pure species I must be included in eq 7. Thus, f*[I,TI = @ * [ I , T l m I , T l
(7')
For supercritical components, the correlation of Prausnitz and Shair (1961) was used to obtain f*. For a continuous fraction, eq 6 can be written as
vvpFvl= vLFL[Af*[I,T1
(8)
where qv and qL are the mole fractions of the continuous portions of the vapor and liquid phases, respectively. For discrete components, eq 6 reduces to Pyi = ~ i f i * [ T ]
(9)
Equations 8 and 9, together with the normalization equations for the vapor and liquid phases (eq 4), the distribution functions, F[A, and the standard-state fugacity relationships, f * [ I , T ]were , used to calculate dew points of several natural gas condensates. The results of our calculations and comparisons with the method of Cotter-
Table I. An A; A2
Constants of Equation 9 = 95.504 418 920 07 Aa = -22.662 298 239 25 = 3.742203001499 A; = -1660.893 846 582 As = 439.132 269 15 = 2295.530 315 13
A3 = -1042.572 560 80
man et al. (1985) are described below. Standard-State Fugacity Relationships. For subcritical components, the standard-state liquid-phase fugacity is assumed equal to the vapor pressure, eq 7. For reasons outlined elsewhere (Willman and Teja, 1985), we have chosen to represent the vapor pressure by the Wagner (1973) equation In PR= (A(1 - TR)+ B(1 - TR)"5 + c(1- TR)~" + D ( l - T R ) 6 . 0 ) / T R (10) where P R = PIP, is the reduced vapor pressure and TR = T I T , is the reduced temperature. In our earlier work (Willman and Teja, 1985), the boiling points of the n-alkane homologous series were correlated with the number of carbon atoms, I , by Tb (K) = A.
+ All +
+
+
A4 In (I)+ A S P s + A,P9 (11)
Values A,-& are given in Table I. In addition, the six constants in eq 10-T,, P,, A, B, C, D-were correlated with the number of carbon atoms for the n-alkane homologous series by using critical temperature, critical pressure, and vapor pressure data. The empirical formulas are given by
T, (K) = Tb(l.0 + (1.25127 + 0.1372420-1] (12) P, (MPa) = (2.33761 + 8.164480/(0.873159
+ 0.5428502 (13)
A = -6.90237 - 0,0415291- 0.006503P
(14)
B = 3.55130 - 0.534931 + 0.021867P
(15)
+ 0.4601981 - 0.029179P D = 5.54103 - 1.931881 + 0.029081P
(16)
C = -4.26807
(17)
Equation 11was used in our previous work to obtain the (nonintegral) effective carbon numbers of 92 isomeric alkanes, alkenes, alkynes, cycloalkanes, alkadienes, aromatics, and sulfur compounds from a knowledge of their boiling points. Then the ECN of each pure substance was used to predict the vapor pressure of that substance in the range, 1 (kPa) - P,, using eq 12-17. An overall absolute average deviation of 3.97% was obtained for the 92 substances studied (Willman and Teja, 1985). Thus, a single parameter (ECN) determined on the basis of the boiling point provides an accurate prediction of the vapor pressure of that substance over a broad range of conditions. Equations 10-17 were therefore used to obtain the vapor pressures required in the calculations performed in this work. The Prausnitz and Shair (1961) correlation for f* was used for supercritical components. A polynomial formula was fit to the original graphical correlation and is given by
f* = 3.59058 - 14.01535T~ + 15.76381T~'PC
5.22446TR3+ 0.54523T~~ (18)
This correlation gives results practically identical with the correlation derived recently by Antunes and Tassios (1983).
950 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 Table 11. Comparison of Compositional Data for Bergman et al. Natural Gas Condensate No. 48 at 1.03 MPa
Table 111. Comparison of Compositional Data for Bergman et al. Natural Gas Condensate No. 98 at 1.38 MPa
Xi
comDonent
NZ
co2 C1 CZ c3
i-C4 n-C4 i-C5 n-C, C6
c7 C8 C9 ClO
exptl v; 0.916 0.301 85.292 63 7.469 23 3.37061 1.202 90 0.8449 0.3304 0.1366 0.10301 0.025 66 0.00592 0.00105 0.000 06
c11
temp, K
265.9 0.51
VL
Xi
simple method Cotterman data discrete semicone t al. X; case tinuous method 0.008 0.044 0.0452 0.020 0.140 0.0830 0.0955 0.039 6.512 6.578 6.7366 3.358 4.391 3.316 3.497 3.652 8.734 6.912 7.419 7.292 8.944 7.065 7.685 7.789 9.613 7.504 8.206 9.588 11.071 8.779 9.739 10.161 6.492 5.004 5.575 7.336 17.521 15.076 14.354 14.781 12.560 13.394 9.394 5.382 2.149 0.931 0.030 274.9
272.5 0.46
component
N2 Cl C2 c3
i-C, n-C, i-C, n-C, c6 c7
C8 C9 ClO Cll Cl2 temp, K
262.2 0.44
Distribution Function Following the work of Whitson (1983), Cotterman and Prausnitz (1985), and Cotterman et al. (1985), we have chosen the gamma or Schultz distribution function (Hahn and Shapiro, 1967; Pearson, 1895) to represent the continuous portion of the mixtures studied. The distribution function is
(20)
The parameter, y, fixes the origin at which F = 0; in addition, F [m] = 0. The mean, M, and the variance, 2, of F[Zl can be expressed as = L m F [ QI
u2 =
dZ = ap + y
LmFII'j ( I - p)2 dZ = aP2
287.8 0.62
280.2 0.59
X,
component
N2 C1
c2
c3
i-C4 n-C, i-C5 n-C5 C8 C9 CIO Cll
with
294.5
Table IV. Comparison of Compositional Data for Bergman et al. Natural Gas Condensation No. 107 at 4.14 MPa
c6 c7
p
272.0 0.49
VL
The only supercritical components in the mixture studied by us were nitrogen, carbon dioxide, and methane. However, in principle, the correlation can be used for other (higher boiling) components via the relationships between T,,P,, and ECN.
L m F I I ]dZ = 1
exptl data yi Xi 0.019 0.977 98 6.781 85.31327 4.499 8.4344 7.171 2.4923 0.87108 6.397 0.72518 8.529 0.35809 10.372 0.1637 7.154 0.18568 19.515 0.08605 15.971 0.034 05 9.661 0.001 19 3.149 0.0001 0.705 0.073 0.004
simple method Cotterman discrete semiconet al. case tinuous method 0.0599 0.0618 0.0294 7.311 7.731 7.5471 3.292 3.760 4.682 3.955 5.370 6.023 3.583 4.393 4.777 4.334 5.382 5.825 5.748 7.387 7.637 3.507 4.55 4.753 13.756 21.571 28.593 3.917 0.9304
c 1 2
temp, K VL
exptl data yi
0.97 87.338 78 8.13098 2.38969 0.5659 0.4153 0.1158 0.0449 0.01834 0.00318 0.00107 0.00006
Xi
0.058 22.053 15.327 15.202 10.428 11.825 8.718 4.821 6.644 3.112 1.396 0.348 0.057 0.005 0.001 255.4 0.20
simple method discrete semicontinuous case 0.178 0.179 23.786 24.128 11.51 11.900 14.707 15.369 9.541 10.051 10.401 10.994 8.267 8.816 4.355 4.657 6.666 4.256 5.249 1.083
280.7
279.2 0.14
cotterman et al. method 0.111 27.94 15.31 16.23 9.514 10.135 7.156 3.766
256.1 0.10
in the continuous fraction, is obtained from (Whitson, 1983)
(21) (22)
Then, using the normalization condition, we may write for the k fractions
The calculation of the dew point of a mixture thus requires a knowledge of the three parameters, a,p, and y, of the vapor-phase distribution function, F"[q.These can be obtained from a nonlinear least-squares fit of the vaporphase discrete composition data. Following Whitson (1983), however, we have chosen to estimate y by using 7 = I,,,
- 4/7
(23)
where I,, is the ECN value at the last maximum in the experimental yi vs. I plot. If no maximum exists, it is assumed that I,, = 6.0. Once y has been determined, a number of techniques may be used to estimate a and p. In the first (method l), the frequency, fi,for each of the k discrete experimental mole fractions, zi, assumed to exist
Note that i refers to "component" 'i and I is the ECN in these equations. The limits of integration, I i and Zi-l, represent the effective carbon number (ECN) boundaries for Components having a cumulative frequency of occurrence, f i . Estimates of a and can now be obtained by a nonlinear least-squares fit of eq 25 and 26. The drawback of this method is that the ECN boundaries, Ii and Ii-l, must first be specified. (We assumed an ECN of 0.5 on either side of i.)
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 951 Table V. ComDarison of Calculated and Exoerimental Dew-Point TemDeratures and "Heavs Fractions ComDositions" exptl data system 1. Bergman 52 2. Bergman 48 3. Bergman 53 4. Bergman 105 5. Bergman 109 6. Bergman 98 7. Bergman 13 8. Bergman 15 9. Bergman 14 10. Bergman 112 11. Bergman 96 12. Bergman 76 13. Bergman 69 14. Bergman 45 15. Bergman 83 16. Bergman 60 17. Bergman 73 18. Bergman 33 19. Bergman 10 20. Bergman 4 21. Bergman 92 22. Bergman 51 23. Bergman 12 24. Bergman 19 25. Bergman 23 26. Bergman 95 27. Bergman 108 28. Bergman 65 29. Bergman 114 30. Bergman 42 31. Bergman 103 32. Bergman 77 33. Bergman 29 34. Bergman 71 35. Bergman 37 36. Bergman 68 37. Bergman 111 38. Bergman 91 39. Bergman 85 40. Bergman 22 41. Bergman 21 42. Bergman 107 43. Bergman 94 44. Bergman 25 45. Bergman 86 46. Bergman 72 47. Bergman 70 48. Bergman 90 49. Bergman 102 50. Bergman 84
pres, MPa 1.02 1.03 1.03 1.36 1.38 1.38 1.70 1.72 1.72 2.04 2.04 2.04 2.04 2.07 2.07 2.07 2.07 2.07 2.07 2.07 2.07 2.07 2.28 2.38 2.41 2.76 2.76 2.76 2.76 3.40 3.40 3.40 3.45 3.45 3.45 3.45 3.45 3.45 3.45 3.71 3.79 4.14 4.14 4.18 4.76 4.83 4.83 4.83 4.83 4.83
simple method
temp, K 260.9 265.9 260.9 263.7 255.4 272.0 255.4 260.4 250.9 270.9 288.7 254.8 254.8 270.9 244.8 272.0 253.7 255.4 249.8 257.6 256.5 260.9 260.7 250.9 261.5 288.7 255.4 246.5 274.8 283.2 263.7 248.2 254.3 252.0 260.9 259.3 264.8 255.9 2523 254.8 257.6 255.4 288.7 277.6 259.8 255.4 254.3 257.0 263.7 253.2
tL 0.49 0.51 0.38 0.40 0.32 0.49 0.67 0.68 0.54 0.33 0.58 0.77 0.82 0.35 0.49 0.33 0.70 0.83 0.79 0.83 0.78 0.17 0.62 0.50 0.62 0.50 0.19 0.73 0.27 0.28 0.21 0.74 0.74 0.62 0.72 0.76 0.14 0.73 0.43 0.44 0.46 0.12 0.38 0.69 0.48 0.61 0.62 0.65 0.14 0.40
A second method (method 2) for estimating a and p is to use the following empirical relationship proposed by Greenwood and Durand (1960): (27)
temp, K 266.6 272.5 272.5 273.5 272.3 287.8 266.5 267.1 263.8 288.8 294.5 270.9 270.4 282.2 275.1 285.2 276.3 263.1 263.2 302.6 274.7 276.4 274.9 269.6 275.4 306.5 273.0 268.4 290.8 304.9 291.6 288.8 263.9 272.5 273.4 288.7 290.0 300.5 285.2 280.3 281.4 279.2 312.3 286.9 298.9 292.2 291.5 309.1 299.5 294.4
VL 0.40 0.46 0.44 0.46 0.49 0.62 0.61 0.59 0.55 0.44 0.53 0.73 0.75 0.28 0.60 0.34 0.70 0.72 0.69 0.82 0.74 0.21 0.56 0.47 0.55 0.48 0.18 0.66 0.34 0.31 0.29 0.74 0.57 0.58 0.62 0.70 0.19 0.71 0.45 0.39 0.40 0.14 0.34 0.54 0.46 0.54 0.56 0.61 0.20 0.38
Cotter man et al. method temp, K 260.5 262.2 262.1 265.8 264.6 280.2 257.8 260.0 253.6 275.4 280.7 257.4 256.5 270.0 258.9 273.0 264.6 249.8 249.8 291.0 262.9 263.6 259.5 253.5 275.4 289.2 256.5 251.2 274.6 281.6 268.0 262.8 241.0 250.7 254.6 266.9 296.6 280.8 263.9 253.9 257.3 256.1 285.4 256.6 263.4 261.0 259.5 279.5 266.1 262.7
tL 0.38 0.44 0.42 0.43 0.47 0.59 0.60 0.59 0.55 0.40 0.48 0.73 0.75 0.26 0.55 0.32 0.73 0.74 0.71 0.85 0.77 0.18 0.54 0.44 0.55 0.47 0.16 0.68 0.32 0.25 0.23 0.75 0.59 0.62 0.65 0.82 0.14 0.73 0.42 0.28 0.32 0.10 0.27 0.54 0.24 0.52 0.49 0.66 0.21 0.18
A third method developed in this work for estimating a and /3 consists of matching the ECN value of the max-
imum in the gamma distribution to the maximum, I,,, in the experimental mole fraction vs. I relationship. By differentiation, we have a is thus obtained from eq 30 and ,f3 from eq 32.
CY
= (0.5000876
+ 0.1648852Y - O.O544174P)/Y
(30)
a 2 1.0
where I*and I G are the arithmetic and geometric averages of the ECNs for the continuous fraction. p may be estimated by using eq 21 as
0 = (IA - y ) / a
(31)
The three methods were used to predict the dew point of a natural gas mixture at 3.45 MPa. Method 1predicted a dew-point temperature of 355.5 K, method 2 yielded 315.0 K, and method 3 resulted in 275.8 K. The experimental dew-point temperature was 260.9 K. Similar trends were observed for other natural gas mixtures studied in this work. Method 3, which gave the best predictions, was therefore adopted throughout the rest of this work. It should be added that a minimum of vapor-phase information has been used to estimate the three parameters of the distribution function and that no liquid-phase data have been used in the calculations.
952 Ind. Eng, L'hem. Res., Vol. 26, No. 5, 1987
Results Some of the most complete compositional data on multicomponent mixtures have been presented by Bergman et al. (1975) on light gas condensates. These mixtures consist mostly of paraffinic hydrocarbons plus small amounts of nitrogen and carbon dioxide. Thus, as a first approximation, they may be treated as ideal mixtures even at moderate pressures, and the method developed above may be used for the calculation of their dew points. Dew points for natural gas condensates from the work of Bergman et al. (1975) were calculated, and the results are reported in Tables 11-V. Tables 11-IV contain detailed results of our calculations for three of these mixtures. Also shown are the results of the calculations for the discrete case (i.e., assuming that the mole fractions of all the components are known) and the results using the method of Cotterman et al. (1985). Cotterman et al. (1985) obtained the fugacities in the vapor and liquid phases from a continuous version of the Redlich-Kwong-Soave equation. Thus, they were able to account for the nonidealities in both the vapor and liquid phases. The average absolute deviation in dew-point temperatures was 8.22% using the method developed in this work, while the approach of Cotterman et al. (1985) had an average absolute deviation of 2.60%. As expected, the equation of state method proved to be better overall. However, the ideal mixture approximation worked surprisingly well even at moderately high pressures. Considering the ease with which it can be programmed and the absence of convergence and stability problems, the method is recommended for rapid calculations. A summary of our results for all 50 mixtures is given in Table V. Again, the equation of state method is superior but the ideal mixture approximation yields dew points that are quite close to the equation of state values. Conclusions A simple method involving the effective carbon number (ECN) has been described for the calculation of dew points of semicontinuous hydrocarbon mixtures. The semicontinuous treatment is shown to be an improvement over the conventional discrete thermodynamic treatment and avoids the arbitrary nature of the method of pseudocomponents. The new method can be used successfully to extend limited compositional data for one phase to compute the composition of another phase in equilibrium. An improved technique for fitting a continuous distribution to limited compositional data has also been proposed in this work. This technique can be used with other semicontinuous treatments which employ, for example,
equations of state in the calculations. Comparisons with the equation of state treatment of Cotterman et al. (1985) are also shown. It is obvious that the latter method, which allows for nonideal vapor and liquid phases, leads to better predictions of phase equilibria. Nevertheless, the simpler ideal mixture treatment yields surprisingly accurate dewpoint temperatures (average absolute errors of less than 9%) over a range of pressures. Moreover, since the method involves no density iterations, unlike the equation of state method, stability and convergence problems are minimized and the method can be used as a rapid fiist approximation in many calculations. In the second part of this paper, we extend the simple model to account for nonidealities of both the vapor and liquid phases and for the effect of pressure on the liquid-phase fugacity. More generally, the effective carbon number (ECN) concept and the techniques developed for fitting distribution functions to limited compositional data should prove useful in other treatments that employ continuous thermodynamics.
Literature Cited Adler, S. B.; Spencer, C. F.; Ozkardesh,H.; Kuo, C. M. In Equations of State Theory and Applications; ACS Symposium Series 60; Chao, K. C., Robinson, R. L., Eds; American Chemical Society: Washington, DC, 1977; p 150. Ambrose, D. J. Appl. Chem. Biotechnol. 1976,26, 712. Ambrose, D.; Sprake, C. H. S. J . Chem. Thermodyn. 1970,2,631. Antunes, C.; Tassios, D. Ind. Eng. Chem. Process Des. Dev. 1983,22, 457. Bergman, D. F.; Tek, M. R.; Katz, D. L. Retrograde Condensation in Natural Gas Pipelines;American Gas Association: Arlington, VA, 1975. Bowman, J. R. Znd. Eng. Chem. 1949, 41, 2004. Chase, J. D. Chem. Eng. Prog. 1984, 80(4), 63. Cotterman, R. L.; Bender, R.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 194. Cotterman, R. L.; Prausnitz, J. M. Znd. Eng. Chem. Process Des. Deu. 1985, 24, 433. Edmister, W. C. Znd. Eng. Chem. 1955,47, 1685. Greenwood, J. A.; Durand, D. Technometrics 1960, 2, 55. Hahn,G. J.; Shapiro, S. S. Statistical Models in Engineering;Wiley: New York, 1967. Pearson, K. Phil. Trans R. SOC.London, Ser. A 1985, 186A, 343. Prausnitz, J. M.; Shair, F. H. AZChE J. 1961, 7, 682. Ratzsch, M. T.; Kehlen, H. Fluid Phase Equilib. 1983, 14, 225. Wagner, W. Cryogenics 1973, 13, 470. Whitaon, C. H. SPE J. 1983,23, 683. Willman, B.; Teja, A. S. U Ind. Eng. Chem. Process Des. Dev. 1985, 24, 1033.
Received f o r review June 19, 1985 Revised manuscript received November 25, 1986 Accepted February 28, 1987
