Ind. Eng. Chem. Res. 1987,26, 953-957
953
Prediction of Dew Points of Semicontinuous Natural Gas and Petroleum Mixtures, 2. Nonideal Solution Calculations Bert Willman and Amyn S. Teja* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-01 00
A simple method was developed in part 1 of this work based on ideal solution theory for the calculation of dew points of semicontinuous natural gas condensates. The method employed a gamma distribution function with the effective carbon number as the continuous distribution variable. The simple method is extended in this paper to account for nonidealities of the vapor and liquid phases. The new modified approach employs extensions of the virial equation and regular solution theory to continuous mixtures. No adjustable parameters are required, and the predictions are comparable to those of an equation of state method. In addition, correlations are presented for the liquid density, the acentric factor, and the solubility parameters of nonpolar fluids as functions of the effective carbon number. In part 1of this work (Willman and Teja, 1987), a simple method was developed for the calculation of dew points of semicontinuous natural gas condensates. The method combined a gamma distribution function based on the effective carbon number as the distribution variable with ideal solution theory. I t yielded surprisingly accurate predictions of dew-point temperatures (average absolute deviations of 8.8% for 50 natural gas condensates). Although these predictions were not as good as those obtained using an equation of state method (Cotterman and Prausnitz, 1985; Cotterman et al., 1985),the simple method involves no iterations and is recommended as a first approximation in phase equilibrium calculations. In this work, we have modified the simple method to account for nonidealities of the vapor and liquid phases. The new method uses the virial equation for the vapor and Hildebrand's regular solution theory for the liquid, both of which we have reformulated for continuous mixtures in this work. In addition, liquid densities, acentric factors, and solubility parameters of hydrocarbons have been correlated with the carbon number, and the correlations are presented below. Development of the Method As discussed in part 1of this work, the composition of a mixture with a continuous distribution of components can be described by a distribution function, F[I],whose independent variable, I, is the effective carbon number (ECN). The function, F [ d , is normalized such that
Jy']dz = 1 "
i
+ g J F [ I ] dl = 1
= FL[~ACT[I,T91f*[I,TS1
(4)
where PC is the vapor pressure, 9*the fugacity coefficient of the pure saturated vapor, and the exponential term represents the (Poynting) correction for the pressure dependence of the liquid fugacity. For supercritical components, the correlation of Prausnitz and Shair (1961) can be used to obtain f*, as described in part 1 of this work. For a continuous fraction, eq 3 can be written as
77'J'"[IIP@[d sLFL[~ACTII,T,Plf*[I,T,Pl( 5 ) where gv and qL are the mole fractions of the continuous portions of the vapor and liquid phases, ACT[I,T,P] represents the activity coefficient of the continuous fraction, and @[Arepresents the fugacity coefficient of the continuous fraction. For discrete subcritical components, eq 3 reduces to YiP@i
(2)
where g = (1- I f x i ) is the "mole fraction" of the continuous portion. Following Cotterman et al. (1985), Cotterman and Prausnitz (1985) and Ratzsch and Kehlen (1983), we may write the equilibrium relationship for continuous mixtures as 0888-5885187 /2626-Q953$01.50/0 0
(3)
where J'"[Ijand P[II are the distribution functions in the vapor and liquid phases, respectively; f*[I,T,P] is the standard-state fugacity of the species, I, at the system temperature, T, and pressure P; ACT[I,TQ] is the activity coefficient of species I; and @[II represents the correction due to the real-gas behavior of the vapor phase. For subcritical components
=
(1)
Semicontinuous mixtures are mixtures in which the mole fractions of some components have discrete values, while the concentrations of others are described by a distribution function. For k discrete components, the normalization is $Xi
@[WJ'"[d
xiACTi[ T,P]Pi*[
[ TI exp
(
ViL(P - Pi*[TI) RT
1
(6)
The distribution function and the vapor pressure relationships were described in part 1of the work. The activity and fugacity coefficient expressions for continuous fractions are derived below. The fugacity coefficient, ai*[T], is approximately equal to one for pure components at temperatures well below their critical temperatures.
Activity Coefficient Expression To obtain the activity coefficient of the continuous fraction, we begin with the expression for the excess Gibbs energy of a multicomponent liquid mixture given by Hildebrand et al. (1970) 1987 American Chemical Society
954
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
Here, Vi is the molar volume of component i and volume fraction of component i, expressed as xi vi +=-
+ is the
necessary for the evaluation of the Poynting correction (eq 6). The solubility parameter may be calculated by using the definition
k
cxivi i where k is the total number of components in the mixture. Note that the second term in eq 7 is the Flory-Huggins term. The coefficient, A,, is related to the solubility parameters, 6i and 6j, by
A ?I, . = (6,1 - 6J. ) 2 + 21..6.6. v 1I
(9)
where lij is an empirical parameter which accounts for deviations from the geometric mean assumption implicit in regular solution theory. For semicontinuous (sc) mixtures, the analogous expression for the excess Gibbs energy is given by
GEBC=
where H a p is the vaporization enthalpy at temperature, T, R is the gas constant, and V Lis the saturated liquid molar volume. The vaporization enthalpy can be approximated by
The assumption'of ideal gas behavior of the pure saturated vapor used in writing down eq 16 and 17 is obviously only valid when the temperature is well below the critical temperature of the pure component. The standard-state fugacity relationships presented earlier may be differentiated with respect to temperature to obtain the heat of vaporization and, therefore, the solubility parameter as a function of temperature and carbon number. By use of this approach, solubility parameters were generated for ethane through n-eicosane a t temperatures between 248 and 368 K and fit to the relationship 6 = c[T] + d[T]I (18) with
(10)
+[I]represents the volume fraction for the continuous term, p[Ilrepresents the molar volume for the continuous terms, 7p,- represents the total number of moles in the liquid mixture, and VT represents the total volume of the liquid mixture. A[i,g represents the discrete-continuous solubility parameter interaction and A [I,I+]the continuouscontinuous solubility parameter interaction. The relationships for ViLand V [ I ]were obtained from a modification of the Spencer and Adler (1978) version of the Rackett equation. The Spencer-Adler form is given by =C
where ZRAis a specified constant for each compound. ZRA was correlated with the effective carbon number using the values supplied by Spencer and Adler for n-alkanes methane through eicosane. The correlation obtained in this work is given by 0.31421 + 21.992 ZRA= (3.12321 + 8.733)'
c[TI = 22.206 - 0.02987T
(19)
d [ T l = 0.284
(20)
where 6 is in (MPa)1/2and T in K. The resulting molar volumes and solubility parameters were checked against those of Barton (1983) and were in good agreement (less than 2% average absolute deviation for either property). It should be noted here that a t constant temperature, eq 18-20 for the solubility parameter predicts 6 to be linearly dependent on the carbon number. This is in agreement with the work of Rheineck and Lin (1968) and Sokolova and Pereverzev (1977). Equation 9 contains the binary interaction parameter, 1,. For the discrete-discrete interactions, the 1, values of King and AI-Najjar (1977) and Preston and Prausnitz (1970) were used in this work. King and Al-Najjar showed that it is possible to assign a constant value to the interaction parameter, l,,, for a given solute, i, dissolved in the entire range of alkane solvents, [ I ] . This method was followed here, using the values of I,, obtained from the work of King and Al-Najjar (1977). The binary-interaction parameter, L1p,for the continuous-continuous interaction was set equal to zero. Activity coefficients can now be obtained for the discrete components and continuous fraction through partial and functional differentiation, respectively, as shown by Salacuse and Stell (1982). For the discrete components
Saturated liquid molar volumes were then generated for effective carbon number 2-10 over a temperature range of 248-368 K and fitted to the simple expression VL[I] = u [ q + b [ I ] I
(13)
and, for the continuous fraction,
where
+ 0.1874T b [ q = 13.298 + 0.003334T u[U = -4.701
with
(14) (15)
in cm3/gmol and T in K. This relationship is also
Here, yi denotes the activity coefficient for discrete component i in the mixture and y[Il is the activity coefficient of the continuous fraction in the mixture.
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 955 Table 111. Comparison of Compositional Data for Natural Gas Condensate No. 98 at 1.38 MPa from Bergman et al.
Table I. Corrections to the Geometric Mean Critical TemDerature in the Virial Coefficient (EauRtion 40) i ai bi 2 ai bi Cz -0.03 0.01 -0.005 0.031 Nz 0.034 C3 -0.002 0.003 COZ 0.014 Cl -0.033 0.019
(1975) X;
component
Table 11. Comparison of Compositional Data for Natural Gas Condensate No. 48 at 1.03 MPa from Bergman et al. (1975)
N 2
Cl CZ c3
component N 2
COz C1 CZ c3
i-C4 n-C, i-C5 n-C5 c6 c7
cs C9 ClO
c11
exptl data y;
Xi
0.916 0.301 85.29263 7.46923 3.37601 1.20290 0.8449 0.3304 0.1366 0.10301 0.025 66 0.00592 0.00105 0.00006
0.008 0.039 3.358 3.652 7.292 7.789 9.588 10.161 7.336 17.521 14.354 12.560 5.382 0.931 0.030
temp, K
265.9 0.508
tL
this work Cotterman discrete semiconet al. case tinuous method 0.0088 0.0030 0.02 0.0203 0.01 0.14 3.002 4.09 6.51 3.332 3.36 4.39 7.055 7.037 8.73 7.304 7.276 8.94 7.777 9.61 7.730 9.249 9.104 11.071 5.194 5.283 6.49 15.685 15.319 13.711 9.39 2.068 268.1
268.2 0.57
c7
C8 C9 c 1 0 c11
ClZ temp, K TL
(24)
286.2
- P[TRl + wf[TRI
(25)
RTC where w is the acentric factor and and f1 are functions of the reduced temperature, TR,given by 0.33 0.1385 0.0121 0.000607 ~ [ T R= ]0.1445 - -- -- -TR TR2 TR3 TRS (26)
The cross virial coefficients, Bij,are determined from the combining rules
Tcij = (TciTCj)'/'(l- kij)
(28)
280.2 0.59
i-C4 n-C4 i-C5 n-C5 c6
c7
C8 C9 ClO C11 ClZ
nL
exptl data yi xi 0.97 87.338 78 8.136 98 2.389 69 0.56590 0.41530 0.11580 0.04490 0.01834 0.00318 0.00107 0.00006 0.000 06
0.058 22.053 15.327 15.202 10.428 11.825 8.715 4.821 6.644 3.112 1.396 0.348 0.057 0.005 0.001 255.4 0.20
this work Cotterman discrete semicone t al. case tinuous method 0.040 0.016 0.111 18.495 12.785 27.94 14.888 14.373 15.31 17.411 17.099 16.23 10.807 10.775 9.514 11.316 11.365 10.135 8.443 8.942 7.156 4.351 4.645 3.766 6.044 3.510 3.948 0.744
246.5
where B ~represents x the virial coefficient for a mixture of k components and yi is the vapor-phase mole fraction of substance i. Bij is the second virial cross-coefficient. Tsonopoulos (1974,1975,1979)employed a corresponding states correlation for B given by
BPC -
271.7 0.56
Xi
temp, K
k k
272.0 0.49
Table IV. Comparison of Compositional Data Natural Gas Condensate No. 107 at 4.14 MPa from B e r m a n et al. (1975)
c3
At moderate pressures, the effect of vapor-phase nonidealities may be conveniently represented by the truncated virial equation z = PV - = 1 + BRT V where P is the pressure, T , the temperature, R, the gas constant, Z, the compressibility factor, and B, the second virial coefficient. For mixtures i l
c6
component NZ C1 C2
264.2 0.44
Fugacity Coefficient Expressions
Bmix = CCyfijBij
i-C4 n-C4 i-C5 n-C5
this work Cotterman discrete semiconet al. yi xi case . tinuous method 0.97798 0.019 0.014 0.0045 0.0294 85.31327 6.781 5.100 3.638 7.5471 8.4344 4.499 3.320 4.309 4.682 2.4923 7.171 4.560 6.549 6.023 0.87108 6.397 3.65 5.660 4.777 0.72518 8.529 4.41 7.058 5.825 0.35809 10.372 5.88 10.317 7.637 0.1637 7.154 3.59 6.462 4.753 0.18568 19.515 14.13 0.08605 15.971 22.089 0.03405 9.661 29.00 0.001 19 3.149 3.32 0.0001 0.705 0.91 0.073 0.004 exptl data
w.. = Y
wi
248.3 0.20
256.1 0.10
+ wj
2
(30)
Vci is the critical volume of component i and kij is the binary-interaction coefficient. ki.'s were obtained from Chueh and Prausnitz (1967) and 7konopoulos (1979). For a semicontinuous mixture, the expression for Bmkmust be replaced with Bmix= BD-D + Bc-D + Bc-c (31) where BwD represents the discrete-discrete contribution to the virial coefficient, BC-D, the continuous-discrete contribution, and BC-c,the continuous-continuous contribution. These terms are given by k k
BD-D= CCYi.YjBij i l
(32)
956 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 Table V. Comparison of Experimental and Calculated Dew-Point Temperatures and “Heavy Fractions Compositions” Cotterman et al. exptl data this work method system 1. Beraman 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
temp, K 260.9 265.9 260.9 263.7 255.4 272.0 255.4 260.37 250.9 270.9 288.7 254.8 254.8 270.9 244.8 272.0 253.7 255.4 249.8 257.6 256.5 260.9 260.7 250.9 261.5 288.7 255.4 246.5 274.8 283.2 263.7 248.2 254.3 252.0 260.9 259.3 264.8 255.9 252.0 254.8 257.6 255.4 288.7 277.6 259.8 255.4 254.3 257.0 263.7 253.2
VL 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
temp, K 258.3 268.2 259.6 262.4 256.0 271.7 252.9 252.1 247.7 296.6 274.8 248.9 249.9 265.2 255.3 267.6 255.4 234.8 253.8 252.9 253.4 259.0 252.5 246.2 253.6 281.0 249.4 234.5 268.0 271.5 257.1 228.6 243.6 238.3 251.8 250.0 261.0 247.5 256.0 238.3 241.2 248.3 270.2 248.3 244.1 243.4 237.9 337.0 241.5 244.4
VL 0.39 0.57 0.43 0.45 0.38 0.56 0.65 0.64 0.57 0.39 0.50 0.77 0.85 0.25 0.55 0.31 0.74 0.81 0.87 0.79 0.83 0.17 0.58 0.46 0.59 0.48 0.13 0.80 0.31 0.22 0.19 0.80 0.71 0.71 0.85 0.80 0.14 0.73 0.42 0.28 0.32 0.20 0.27 0.54 0.24 0.52 0.49 0.66 0.21 0.18
o[Zl =
0.0092
temp, K 260.5 264.2 262.1 265.8 264.6 280.2 257.8 260.0 253.6 275.4 280.7 257.4 256.5 270.0 258.9 273.0 264.6 249.8 249.8 291.0 262.9 263.6 259.5 253.5 275.4 289.2 256.5 251.2 274.6 281.6 268.0 262.8 241.0 250.7 254.6 266.9 296.6 280.8 263.9 253.9 257.3 256.1 285.4 256.6 263.4 261.0 259.5 279.5 266.1 262.7
VL
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 0.38 0.10 0.32 0.49 0.24 0.52 0.49 0.66 0.21 0.18
+ 0.04771
(38)
and was obtained by a fit of the data of Ambrose (1980) for the n-alkanes through tridecane. Similarly, the relationship between the critical volume and the carbon number is given by Bi1 may be formulated through oil
=
oi
+ w[Z] 2
The relationship between the acentric factor, w [ A , and the carbon number is given by
Vc[I] = 37.08 + 56.041 cm3 mol-’
(39)
and was obtained by a fit of the data of Ambrose (1980) for n-alkanes methane through heptadecane. The binary-interaction parameters, kilt were also found to follow an approximate linear relationship with the effect carbon number, I k,i = ai + b,l (40) where ai and bi are constants for mixtures containing Nz, COz,C1, Cz,and C3 obtained by fitting the data of Chueh and Prausnitz (1967) and Tsonopoulos (1979). They are given in Table I. For alkanes heavier than propane, the k,;s were assumed to be zero as was also true for k p , which
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 957 is the correction to the continuous-continuous geometric mean critical temperature. The second virial coefficient, BII+,characteristic of the continuous-continuous interaction, may be obtained in a similar fashion. Fugacity coefficients for the discrete components and for the continuous fraction may now be obtained by using
R T In $411 =
Conclusions
A completely predictive phase equilibrium model has been developed for semicontinuousmixtures using a single characterization variable, the effective carbon number. The model contains no adjustable parameters and was used to calculate dew points of light natural gas condensates. Average absolute deviations between calculations and experiments were of the order of 3.35% in dew-point temperatures at pressures up to 40 bar. Good agreement was obtained between predicted and experimental liquid-phase compositions, while the method was found to be comparable with the semicontinuous equation of state approach of Cotterman et al.
s,-{(*~v~,))T,vp=IY} dV-RT1n
(42)
Results Dew-point calculations were performed for 50 natural gas mixtures by using the semicontinuous, nonideal solution theory. The results were compared with the data of Bergman et al. (1975). Due to the mathematical complexity of the equations, numerical integration is necessary, and in this work a 10-point Gauss-Legendre quadrature method (Carnahan et al., 1964) was used. These calculations are compared with the semicontinuous equation of state approach of Cotterman (1985) in Tables 11-V. The average absolute deviation in calculated dew-point temperatures for the methods developed in this work was 3.35% compared with 2.60% for the method of Cotterman et al. The two methods therefore are comparable. Tables 11-IV show a complete compositional analysis for 3 representative mixtures from the 50 studied. Presented in each case are the experimental phase equilibrium data, the computed dew points on a semicontinuous and discrete basis using the methods developed in this work, and the dew points calculated by using the method of Cotterman et al. Table V shows the calculated results for all 50 systems. In these calculations, the Flory-Huggins contribution to the excess Gibbs energy (eq 33) was assumed to be negligible since the molar volume ratios of the components in these natural gas mixtures were never large enough for the Flory-Huggins term to have a significant effect.
Literature Cited
Ambrose, D.National Physical Laboratory Report Chem 107,1980; National Physical Laboratory, Teddington, U. K. Barton, A. F., Ed. CRC Handbook of Solubility Parameters and Other Cohesion Parameters; CRC: Boca Raton, FL, 1983. Bergman, D.F.; Tek, M. R.; Katz, D.L. Retrograde Condensation in Natural Gas Pipelines; American Gas Association; Arlington, VA, 1975. Carnahan, B.; Luther, H. A.; Wilkes, J. 0. Applied Numerical Methods; Wiley: New York, 1964. Chueh, P. L.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1967, 6, 492. Cotterman, R. L.; Bender, R.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 194. Cotterman, R. L.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 433. Hildebrand, J. H.; Prausnitz,J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand: New York, 1970. King, M. B.; Al-Najjar, H. Chem. Eng. Sci. 1977, 32, 1241. Prausnitz, J. M.; Shair, F. H. AIChE J . 1961, 7, 687. Preston, G. T.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Deu. 1970, 9(2), 264. Ratzsch, M. T.; Kehlen, H. Fluid Phase Equilib. 1983, 14, 225. Rheineck, A. E.; Lin, K. F. J. Paint Technol. 1968, 40, 611. Salacuse, J. J.; Stell, G. J . Chem. Phys. 1982, 77, 3714. Sokolova, S.P.; Pereverzev, A. N. Zh. Fiz. Khim. 1977, 51, 1267. Spencer, L. F.; Adler, S.B. J . Chem. Eng. Data 1978,23, 82. Tsonopoulos, C. AIChE J . 1974,20, 263. Tsonopoulos, C. AIChE J. 1975,21, 827. Tsonopoulos, C. Adu. Chem. Ser. 1979, 182, 143. Willman, B.; Teja, A. S.Ind. Eng. Chem. Res. 1987, preceding paper in this issue. Received for review June 19, 1985 Revised manuscript received November 25, 1986 Accepted February 28, 1987
