Improved Modeling of the Phase Behavior of Asymmetric Hydrocarbon

Ind. Eng. Chem. Res. 1998, 37, 1651-1662

1651

Improved Modeling of the Phase Behavior of Asymmetric Hydrocarbon Mixtures with the Peng-Robinson Equation of State Using a Different Temperature Dependency of the Parameter a E. Flo1 ter,† Th. W. de Loos,* and J. de Swaan Arons Laboratory of Applied Thermodynamics and Phase Equilibria, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands

In this paper, for phase equilibrium calculations in asymmetric hydrocarbon mixtures the influence of the temperature dependency of the attractive parameter, a(T), in the Peng-Robinson equation of state is discussed. All systems studied contain methane at supercritical conditions and a heavy hydrocarbon, namely, decane, hexadecane, tetracosane, 1-phenyldodecane, phenanthrene, and 5-R-cholestane. The Peng-Robinson equation of state with the classical quadratic mixing rules is combined with eight different R functions. Next to the R functions taken from the literature also a correlation of R values optimized to IUPAC recommended fugacities of supercritical methane (T ) 250-500 K) is applied. The superior performance of the proposed correlation is most evident when all interaction coefficients are set to zero. This is partly compensated by the use of interaction coefficients. Nevertheless, it is shown that also when kij is optimized to vapor-liquid equilibrium (VLE) data or kij and lij are optimized to VLE and solid-liquid equilibrium (SLE) data, the proposed correlation still performs better than the other R functions. Introduction In the North Sea, gas-condensate reservoirs with very high pressures (p > 50 MPa) are found. Due to technological improvements these reservoirs can now be exploited. Except for the high pressures and high temperaturessup to 470 Ksalso the fluids found in these reservoirs are somehow peculiar. These reservoir fluids, also referred to as hyperbaric reservoir fluids, are characterized by their asymmetrical compositional distribution. This reflects the presence of large methane fractionssin excess of 60 mol %sand significant amounts of heavy hydrocarbons with up to 40 carbon atoms (Montel, 1993; Ungerer et al., 1995). A consequence of the composition of the hyperbaric reservoir fluids are elevated dew- and/or bubble-point pressures and the potential for the precipitation of solid-heavy hydrocarbon phases. In our laboratory we have conducted a systematic experimental study on the effect of the asymmetric composition on the phase behavior of hydrocarbon systems. Although reservoir fluids are complicated multicomponent systems, we have chosen to study the phase behavior of binary reservoir model fluids. The systems studied were all composed of methane and either tetracosane (Flo¨ter et al., 1997a), 5-R-cholestane (Flo¨ter et al., 1997b), 1-phenyldodecane, or phenanthrene (Flo¨ter et al., 1997c). For these systems almost the whole composition range was studied. In combination with experimental data gathered earlier in our laboratory (Rijkers, 1991, de Leeuw, 1995) and elsewhere (e.g., see Suleiman and Eckert (1995)), these data document the influence of the different properties of the * To whom correspondence should be addressed. † Current address: Unilever Research Laboratory, Olivier van Noortlaan 120, 3133 AC Vlaardingen, The Netherlands.

heavy component on the phase behavior of asymmetric systems incorporating methane. This compilation of experimental data can be used to evaluate the performance of models used for phase equilibrium calculations. The work presented here focuses on the influence of the temperature dependency of the attractive parameter a or, more precisely, the R function, in the Peng-Robinson equation of state (Peng and Robinson, 1976) on the phase equilibrium calculations in asymmetric hydrocarbon systems. In addition to various expressions given in the literature, also a direct optimization of R(T) for methane to IUPAC (1996) recommended pure component properties in the temperature range under consideration was employed. The effect of the different R functions for supercritical methane on phase equilibrium calculations is investigated for six different model systems. Equations of State Cubic equations of state are widely used in the petroleum industry. From this class of equations we selected the Peng-Robinson equation of state (Peng and Robinson, 1976) (eq 1) to allow discussion on the effects

p)

a(T) RT v - b v(v + b) + b(v - b) R2Tc2 pC

a(T) ) 0.45724R(T)

RTc b ) 0.0778 pC

(1)

of the temperature dependency of the attractive parameter a for supercritical components. For both the a and the b parameters the classical quadratic mixing rules

S0888-5885(97)00644-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/18/1998

1652 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998

were used (eq 2). If the binary interaction parameter N N

a)

∑i ∑j xixjaij

aij ) (aiaj)0.5(1 - kij)

ham et al. (1989) formulated a temperature-dependency model covering the sub- and the supercritical regions. For a better description of phase equilibrium calculations Danesh et al. (1995) introduced a constant factor for supercritical components into the R function of the original Peng-Robinson equation. Temperature Dependency of the a Parameter

N N

b)

∑i ∑j xixjbij

bij ) 0.5(bi + bj)(1 - lij)

(2)

lij is set to zero, the quadratic mixing rule for the b parameter becomes linear. A lot of work has been done to improve the description of the phase behavior of mixtures by the application of more advanced mixing rules (e.g., see Wong and Sandler (1992) and Dahl and Michelsen (1990)). The use of various mixing rules for binary reservoir model systems has been studied among others by de Leeuw et al. (1992). Other authors (e.g., Anastasiades et al. (1994), de Loos et al. (1984), Danesh et al. (1995), and Rijkers (1991)) changed the pure component properties of the constituents or one of the constituents in order to represent the experimental phase equilibrium data of mixtures better. de Loos et al. (1984) derived the a and b parameters at each temperature for the heavy component from pVT data to overcome the lack of a known critical temperature. Both Rijkers (1991), Danesh et al. (1995), and Twu et al. (1996) optimized the properties of a supercritical component on the basis of experimental data. Rijkers adjusted the critical properties of methane in order to match pVT data of methane in the temperature range of interest. Following this approach the experimental critical data have to be sacrified. Danesh et al. (1995) proposed determination of the parameters within the equation of state based on binary phase equilibrium data. Twu et al. optimized the R function of hydrogen for the Soave-Redlich-Kwong (SRK) equation on the basis of phase equilibrium data of mixtures. Anastasiades et al. (1994) evaluated the effect of the optimization of different parameters as Ωa, Ωb of the pure components and the binary interaction coefficient kij and combinations thereof on phase equilibrium calculations of synthetic gas-condensate systems. However, the quality of the performance of semiempirical equations of state depends to a large extent on the type of experimental data used to derive its parameters. As stated by Vidal (1983) and Soave (1972), the correct representation of the pure component behavior is a sound basis for the successful description of multicomponent systems. Many modifications of the temperature dependency of the attractive parameter a have been proposed in the past. Predominantly these modifications of the generalized R function concentrated on an improved description of the vapor pressure data in the subcritical region for high-boiling and polar components. As pointed out by, e.g., Boston and Mathias (1980), Mathias (1983), Vidal (1983), Melham et al. (1989), and de Leeuw et al. (1992) these different R functions might, due to the extrapolation to supercritical conditions, result in not justified attractive parameters a. To overcome this, Boston and Mathias (1980) proposed a dedicated extrapolation of the R function into the supercritical region. Also Mel-

The following different R functions for methane were used during this investigation. In the first four functions (eqs 3-6) only κ, the parameter relating R to the reduced temperature TR, is varied. Equation 3 is the refined version (Robinson and Peng, 1978) of the original proposal by Peng and Robinson (1976). Equation 4 is the R function for

R(T) ) [1 + κ(1 - TR1/2)]2 κ ) 0.3796 + 1.485ω - 0.1644ω2 + 0.01667ω3

(3)

R(T) ) [1 + κ′(1 - TR1/2)]2 κ′ ) 1.21κPR ) 1.21(0.3796 + 1.485ω - 0.1644ω2 + 0.01667ω3) (4) supercritical hydrocarbons proposed by Danesh et al. (1995). Another version (eq 5) of the original κ equation

κ ) 0.384401 + 1.52276ω - 0.21808ω2 + 0.03461ω3 - 0.001976ω4 (5) is given by Boukouvalas et al. (1994). Equation 6 is

κ ) κ0 + κ1(1 - TR1/2)(0.7 - TR) κ0 ) 0.378893 + 1.4897152ω - 0.1731848ω2 + 0.0196554ω3 TR > 0.7 f κ1 ) 0

(6)

proposed for κ by Stryjek and Vera (1986) to improve the representation of vapor pressures at low reduced temperatures. The generalized R function proposed by Melham et al. (1989) (eq 7) has been formulated to

ln(R(T)) ) m(1 - TR) + n(1 - TR1/2)2 mmethane ) 0.4045, nmethane ) 0.1799

(7)

overcome the deficiency of other R functions to cover both the subcritical and the supercritical regions without revealing unrealistic behavior. Twu et al. (1995) proposed another generalized R function dependent on temperature and the acentric factor. This generalized function is based on an R function (eq 8) proposed earlier NM

eL(1-TR R(T) ) TN(M-1) R

)

Nmethane ) 2.616 22, Mmethane ) 0.915 696, Lmethane ) 0.810 43 (8) (Twu et al. 1991). For the purposes of this work the parameters N, M, and L specifically derived for methane by fitting vapor pressure data from DIPPR (Daubert and Danner, 1990) were used (Twu et al., 1995). The R

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1653

Figure 1. Square of the relative deviation between experimental and calculated fugacities using the Peng-Robinson equation using eq 10 and the IUPAC recommended fugacities for methane. Dev ) [(fcalc(PRR) - fIUPAC)/fIUPAC]2 versus temperature and pressure.

function (eq 9) proposed by Boston and Mathias (1980)

R(T) ) [exp(ci(1 - TRdi)]2 ci ) 1 + di )

κ + 0.3pi 2 ci - 1 ci

κ ) 0.3796 + 1.485ω - 0.1664ω2 + 0.01667ω3

(9)

and Mathias (1983) for supercritical components splines the R function for subcritical conditions at the critical point. In this work the R function proposed by Peng and Robinson (eq 3) is used as the subcritical reference. The polar parameter pi, used to compensate for deficiencies of the R function at subcritical conditions, is set to zero for methane. Equation 10 is the correlation of R

R(T) ) 0.969617 + 0.20089TR - 0.3256987TR2 + 0.066653TR3 (10) values for methane that we propose. It is obtained by optimization of the Peng-Robinson equation of state to IUPAC (1996) recommended fugacity data. One might argue that the recommended fugacities are somehow artificial since they are only indirectly based on experimental data and derived by employing the IUPAC equation of state. However, the more desirable optimization of the R function to pVT data is difficult since these properties are not sensitive enough to the R function. Therefore, the proposed correlation being optimized to the IUPAC fugacities does yield a representation of experimental pVT data similar to other R functions. The applied approach seems to be justified since even the parallel incorporation of pVT data into the optimization routine does not yield significantly different results. The temperature range and pressure range covered by the correlation extends from 250 to 500 K and from 0.1 to 300 MPa, respectively. The optimization was performed straightforwardly to minimize the root mean square error. In Figure 1 the quality of the fit is documented. Here, the square of the relative deviation is plotted out as a function of pressure and temperature. As the figure indicates the Peng-Robinson equation of

Figure 2. Comparison of the different R functions for methane. [R(methane)(eqs 4-10) - R(methane)(Robinson and Peng, eq 3)] values versus the reduced temperature of methane. 1, Danesh et al. (1995) (eq 4); 0, Boukouvalas et al. (1994) (eq 5); +, Stryjek and Vera (1986) (eq 6); 4, Melham et al. (1989) (eq 7); 2, Twu et al. (1995) (eq 8); 3, Boston and Mathias (1980) (eq 9); b, this work (eq 10).

state is not able to reproduce the experimental data over the whole pressure range even though the R values were optimized. The description of the experimental data improves with increasing temperature. For the fit presented in this work (eq 10) the deviation decreases with decreasing pressure. However, this is caused by the fact that there are more low-pressure values available than high-pressure values. The eight different R functions used are shown in Figure 2. For reasons of clarity for this figure the different R functions relative to the R function proposed by Robinson and Peng (eq 3) are plotted versus the reduced temperature. Both extrapolations into the supercritical region proposed by Boston and Mathias (1983) (eq 9) and by Danesh et al. (1995) (eq 4) are only shown for TR > 1.0. The correlation of the optimized R values presented in this work is only depicted for temperatures between 1.3 and 2.6 (T ) 250 and 500 K). The various temperature dependencies almost perfectly coincide for subcritical temperatures. In the supercritical region the equation proposed by Boston and Mathias (1980) (eq 9) results in the highest R values. Also for supercritical temperatures the temperature dependencies proposed by Robinson and Peng (1978) (eq 3), by Boukouvalas et al.(1984) (eq 5), and by Stryjek and Vera (1986) (eq 6) almost coincide. Compared to these equations, the relationship given by Melham et al. (1989) (eq 7) results in slightly lower R values at increased temperatures. As a consequence of the increased κ value as proposed by Danesh et al. (1995) (eq 4), the R values decrease further at increased temperatures. The equation proposed by Twu et al. (1995) (eq 8) performs significantly different than the other equations. The decrease of the R values with increasing temperature is the most pronounced in this case. Finally the correlation given in this work yields, with exception of temperatures TR > 2.5 the lowest R values. Vapor-Liquid Phase Equilibrium Calculations The phase equilibrium calculations were performed according to the flash procedures proposed by Michelsen (1982a,b). To overcome significant convergence problems, a damping along the convergence path as proposed by Heideman and Michelsen (1995) was necessary. As


Solid-Fluid Phase Equilibrium Calculations

Table 1. Pure Component Properties component

formula

TC/K

pC/MPa

methane decane hexadecane tetracosane cholestane phenyldodecane phenanthrene

CH4 C10H22 C16H34 C24H50 C27H48 C18H30 C14H10

190.6 617.7 723.0 800.0 828.0 779.1 873.1

45.95 21.1 14.0 8.7 10.6 15.5 30.9

ω

κ1

0.01045 0.0 0.4905 0.0451 0.744 0.02665 1.033 0.1583 0.683 0.746 0.1168 0.5067 -0.209

already described above, quadratic mixing rules were used for both the a and the b parameters of the equation of states. As a deviation function the average sum of squares as given in eq 11 is used. However, the

dev )

[∑(

1 nx + ny

) ∑(

nx

xi,cal - xi,exp

i)1

xi,exp

2

)]

ny

yi,cal - yi,exp

j)1

yi,exp

+

2

(11)

evaluation of the performance of the different models is problematic in the vicinity of the critical point. Two general problems have to be taken into account. First, due to convergence problems in the critical region, it is not always possible to include all experimental points in the evaluation of the performance of a model. The second problem is related to the fact that the calculations are performed isobarically. Therefore, again it is difficult to account for the deviation between experimental data and calculations when the calculations yield a homogeneous phase at a pressure that experimantally results in a phase separation. As the R function for the heavy components, the version of Stryjek and Vera (1986) (eq 6) was employed. This was done in order to ensure that differences in the calculations are solely caused by the variation of the description of the supercritical methane. Only for the system containing 5-R-cholestane the R function given by Robinson and Peng (1978) (eq 3) was used since only very limited vapor pressure data were available. In Table 1 the pure component parameters are listed. The pure component data, see also Table 2, were gathered from various sources as DIPPR (Daubert and Danner, 1990), Dortmund Data Bank (1994), and Ambrose and Tsonopolous (1995a,b). For the optimization of the binary interaction parameters eq 11 was employed as the objective function.

The solid-fluid phase equilibrium calculations were performed in analogy with the vapor-liquid phase equilibrium calculations. Fugacities of solid phases were determined according to the following expression.

()

ln

fS

fL

(T,p) )

n

∑ i)1

[ ( (

TR ∆H TR i (p) T i (p)

RT TP i (p)

∆cP,i R

1-

T

T TR i (p) T

) ( )]

-1 +

+ ln

T TR i (p) T

Here, the superscript TR indicates both solid-solid and solid-liquid transitions, and n is the number of phase transitions which occur in the temperature range of interest. The derivation of this equation is among others described by Ungerer et al. (1995). Different than in this reference the pressure influence of the fugacity of the pure heavy component is in this work taken into account via a pressure-dependent phase transition temperature and heat of transition. This approach was also applied by van der Kooi (1981) for the system methane + eicosane. ∆HiTR is fitted to experimental phase transition data of the pure components using the Clapeyron equation. To be able to do this, information on ∆viTR, the difference in the molar volume of the coexisting phases, is necessary. In the case of 1-phenyldodecane and phenathrene this information is not available. Therefore, the ∆HiTR was assumed to be independent of pressure. However, this assumption in combination with pressure-dependent melting point temperatures results in a good approximation of the fugacities. If available, the differences of the heat capacities in the different phases are taken into account but assumed to be constant. The properties used per component are listed in Table 2. An equation similar to eq 11 was employed as the objective function for the optimization of the two binary interaction coefficients kij and lij. Results and Discussion The effect of the different temperature dependencies of the a parameter on vapor-liquid phase equilibrium calculations in asymmetric hydrocarbon systems was

Table 2. Pure Component Properties Used for Solid-Fluid Equilibrium Calculations component tetracosane

1-phenyldodecane

phenanthrene

(12)

properties per transition solid(hexagonal)-liquid transition: TSRL/K ) 323.63 + 0.2536(p/MPa) - 3.37 × 10-4(p/MPa)2 + 1.557 × 10-7(p/MPa)3 ∆hSRL/(kJ/mol) ) 0.2384(TSRL/K) - 22.4 ∆cP/(J/mol/K) ) 66 solid(hexagonal)-solid(tri-/monoclinic) transition: TSRSβ/K ) 319.57 + 0.2852(p/MPa) - 7.725 × 10-5(p/MPa)2 + 9.429 × 10-9(p/MPa)3 ∆hSRSβ/(kJ/mol) ) 9.952 × 10-2(TSRSβ/K) ∆cP/(J/mol/K) ) 0 solid(tri-/monoclinic)-liquid transition: TSβL/K ) 322.13 + 0.2655(p/MPa) - 2.426 × 10-4(p/MPa)2 + 7.59 × 10-8(p/MPa)3 ∆hSβL/(kJ/mol) ) 0.3393(TSβL/K) - 22.9 ∆cP/(J/mol/K) ) 66 solid-liquid transition: TSL/K ) 270.44 + 0.2158(p/MPa) ∆hSRSβ/(kJ/mol) ) 40.1(TSRSβ/K) ∆cP/(J/mol/K) ) 0 solid-liquid transition: TSL/K ) 372.44 + 0.3369(p/MPa) - 1.55 × 10-4(p/MPa)3 ∆hSRSβ/(kJ/mol) ) 16.474(TSRSβ/K) ∆cP/(J/mol/K) ) 16.3


Figure 3. Vapor-liquid equilibrium of (methane + decane) at 263.15 K: O, Experimental data (Rijkers et al., 1992). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ - ‚‚ -, eq 4; s s s, eq 7; - ‚- ‚-, eq 8; - - -, eq 9; - - -, eq 10.

Figure 5. Vapor-liquid equilibrium of (methane + hexadecane) at 293.15 K. O, Experimental data (Rijkers, 1991; Glaser et al., 1985). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚-, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10. (1.0e-7, for example, represents 1.0 × 10-7.)

Figure 4. Vapor-liquid equilibrium of (methane + decane) at 303.15 K. O, Experimental data (Rijkers et al., 1992). Calculations with Peng-Robinson EoS (kij ) 0) with different R functions: s, eq 3; - ‚‚ - ‚‚ -, eq 4; s s s, eq 7; - ‚- ‚-, eq 8; - - -, eq 9; - - -, eq 10.

Figure 6. Vapor-liquid equilibrium of (methane + hexadecane) at 350 K. O, Experimental data (Glaser et al., 1985). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

studied. The six systems studied were composed of methane and a heavy hydrocarbon. For these systems the experimental data cover almost the whole composition range so that an overall impression of the performance of the different models could be obtained. The calculations were performed for isothermal sections. In the first set of calculations all binary interaction parameters were set to zero to guarantee that differences in phase equilibrium calculation results are solely caused by the differences in the R functions. In Figures 3-8 the calculations are compared with experimental high-pressure vapor-liquid equilibrium data for (methane + n-alkane) systems. In Figures 3 and 4 the isothermal sections at respectively 263.15 and 303.15 K are shown for mixtures of methane and decane (Rijkers et al., 1992). Here the phase-boundary pressure is depicted versus the logarithm of the mole fraction of decane to show also the dew points at low concentrations. The relationship found for phaseboundary pressures calculated with the different R functions is such that lower R values result in higher phase-boundary pressures. See also Figure 2. The highest bubble- and dew-point pressures are calculated with the correlation proposed in this work. The lowest phase-separation pressures are obtained when the R function proposed by Boston and Mathias (1980) (eq 9)

Figure 7. Vapor-liquid equilibrium of (methane + tetracosane) at 325 K. O, Experimental data (Flo¨ter et al., 1997a). Calculations with Peng-Robinson EoS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

is employed. The choice of the R function does not significantly influence the vapor-liquid critical composition. At 263.15 K eq 10 yields the best predictions for the bubble-point pressures. As a result of the good prediction of the bubble points the critical pressure is overpredicted. The other equationssespecially the one


Figure 8. Vapor-liquid equilibrium of (methane + tetracosane) at 425 K. O, Experimental data (Flo¨ter et al., 1997a). Calculations with Peng-Robinson EoS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

proposed by Danesh et al. (1995)syield better predictions of the dew points at elevated pressures. At 303.15 K a similar picture is found; see Figure 4. Compared to Figure 3, the quality of the predictions with the R function proposed in the literature is slightly better, while the quality of the results of the proposed equation are worse. In Figures 5 and 6 the isothermal sections, at T ) 293.15 K and T ) 350 K, respectively, for the system (methane + hexadecane) are shown. The experimental data originate from Rijkers (1991) and Glaser et al. (1985). Again the sequence of the phase-separation pressures is opposite to the sequence found for the R values in Figure 2. At 293.15 K all equations underestimate the pressures in the critical region. Clearly the best predictions for both bubble- and dew-point pressures are obtained by employment of our correlation, eq 10. As Figure 6 reveals eq 10 still performs better than the other equations at 350 K. However, the other R functions yield somewhat better predictions at lower pressures (p < 20 MPa). The experimental data on the phase behavior of mixtures of methane and tetracosane shown in Figures 7 (T ) 325 K) and 8 (T ) 425 K) are taken from Flo¨ter et al. (1997a). The results obtained for this system resemble those for the system (methane + hexadecane). At lower temperatures (T ) 325 K, Figure 7) all R functions predict phase equilibrium pressures above 20 MPa too low. Again eq 10 performs better than the other R functions. At 425 K (see Figure 8) the use of eq 10 results in an excellent agreement between experiments and calculations at pressures above 40 MPa. This good prediction of both high-pressure bubble and dew points is accompanied by a less good performance at lower pressures. As could be expected from the values of R, see Figure 2, the R function proposed by Twu et al. (1995) (eq 8) yields for the high temperatures the highest but one phase separation pressures. The other three systems studied involve heavy components which significantly differ in their chemical nature from n-alkanes, 1-phenyldodecane, phenanthrene, and 5-R-cholestane. For each system two isotherms are shown (Figures 9-14). The experimental data are taken from Flo¨ter et al. (1997b,c). As found before, the clear relationship between the calculations employing the different R functions is found. For these kinds of systems the performance of the Peng-Robinson

Figure 9. Vapor-liquid equilibrium of (methane + 1-phenyldodecane) at 280 K. O, Experimental data (Flo¨ter et al., 1997c). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

Figure 10. Vapor-liquid equilibrium of (methane + 1-phenyldodecane) at 360 K. O, Experimental data (Flo¨ter et al., 1997c). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

Figure 11. Vapor-liquid equilibrium of (methane + phenanthrene) at 400 K. O, Experimental data (Flo¨ter et al., 1997c). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; - -, eq 5; - ‚‚- ‚‚ -, eq 6; s s s, eq 7; - ‚ - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

equation of state without binary interaction parameters is inferior to the performance for n-alkane systems. Only the system incorporating 1-phenyldodecane is described quite well at 360 K without a binary interaction parameter. The predictions of the phase behavior of the


Figure 12. Vapor-liquid equilibrium of (methane + phenanthrene) at 460 K. O, Experimental data (Flo¨ter et al., 1997c). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; - ‚ -, eq 8; - - -, eq 9; - - -, eq 10.

Figure 13. Vapor-liquid equilibrium of (methane + 5-R-cholestane) at 340 K. O, Experimental data (Flo¨ter et al., 1997b). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 10.

Figure 14. Vapor-liquid equilibrium of (methane + 5-R-cholestane) at 400 K. O, Experimental data (Flo¨ter et al., 1997b). Calculations with Peng-Robinson EOS (kij ) 0) with different R functions: s, eq 3; - ‚‚ -, eq 4; s s s, eq 7; - ‚ -, eq 8; - - -, eq 10.

system (methane + 1-phenyldodecane) are qualitatively very similar to those for the systems with hexadecane and tetracosane. At low temperatues (T ) 280 K) (Figure 9) all R functions yield too low phase-separation pressures. The correlation proposed in this work (eq

10) performs better than the other equations (eqs 3-9). The good prediction of high-pressure bubble and dew points at T ) 360 K by eq 10 is accompanied by a less good performance at pressures below 40 MPa. This is shown in Figure 10. Figures 11 and 12 show the phase behavior of the system (methane + phenathrene) at T ) 400 and 460 K, respectively. As shown in Figure 11 the PengRobinson equation state predicts too low phase-separation pressures at 400 K no matter which R function (eq 3-10) is used. Equation 10 again performs better than the other R function (eqs 3-9). Despite this, both the correlation proposed in this work (eq 10) and the R function given by Twu et al. (1995) (eq 8) yield at T ) 460 K too high pressures for lower concentrations. Finally, the phase behavior of the system (methane + 5-R-cholestane) at T ) 340 and 400 K is shown in Figures 13 and 14. The predictions of the phase boundary pressures for this system by the PengRobinson equation of state without using binary interaction parameters are in general of poor quality. This might be attributed to the uncertainty in the pure component properties of 5-R-cholestane. Only very few experimental data of pure component properties are available (Schiessler et al., 1959). Therefore, ω and the critical properties had to be determined on the basis of group contribution methods and correlations (Reid et al., 1977). Nevertheless, eq 10 performs better than the other R functions considered, which predict even lower phase-boundry pressures. Summarizing the information given in the 12 figures, it can be concluded that the different R functions result in qualitatively similar two-phase regions when the binary interaction coefficients are set to zero. The predicted vapor-liquid critical compositions seem not to differ significantly for the various R functions. However, it is found that smaller R values result in higher dew- and bubble-point pressures. On the basis of this relationship, the performance of the different R functions can directly be related to Figure 2. In general the Peng-Robinson equation tends to predict (kij ) 0) much too low vapor-liquid critical pressures. For more asymmetric systems this tendency is more pronounced. It becomes less pronounced with increasing temperature. The R function proposed in this work (eq 10) performs overall superiorly to the other R functions. Nevertheless, in cases of an excellent prediction of the high-pressure dew and bubble points the description of the bubble points at lower pressures deteriorates. Therefore, at lower pressures and high concentrations of the heavy component, the bubble points are sometimes somewhat better represented by the conventional R functions (eqs 3, 5, and 6). This mainly occurs at higher temperatures. The R functions dedicated to supercritical conditions as formulated by Boston and Mathias (1980) (eq 9), Melham (1983) (eq 7), and Danesh et al. (1995) (eq 4) do, with the exception of the latter one, not perform superiorly to the conventional R functions. To the contrary, the R function proposed by Boston and Mathias (1980) actually results in predictions of vapor-liquid equilibria inferior to the conventional R functions. The R function proposed by Twu et al. (1995) (eq 8) performs overall quite well due to the fact that the evolvement of the R values with temperature differs significantly from the other R functions as shown in Figure 2.

0.058 (1.7 × 0.053 (1.7 × 10-2) 0.057 (1.6 × 10-2) 0.058 (1.7 × 10-2) 0.057 (1.7 × 10-2) 0.054 (1.6 × 10-2) 0.060 (1.7 × 10-2) 0.032 (1.7 × 10-2) 0.070 (6.1 × 0.040 (6.1 × 10-1) 0.068 (6.1 × 10-1) 0.069 (6.1 × 10-1) 0.065 (6.1 × 10-1) 0.084 (1.1 × 10-1) 0.072 (6.1 × 10-1) 0.072 (9.6 × 10-2) 0.092 (2.2 × 0.064 (6.0 × 10-1) 0.071 (6.1 × 10-1) 0.092 (2.2 × 10-1) 0.092 (2.1 × 10-1) 0.087 (1.7 × 10-1) 0.092 (2.3 × 10-1) 0.077 (1.2 × 10-1) 0.067 (9.2 × 0.047 (8.0 × 10-2) 0.064 (8.7 × 10-2) 0.065 (8.7 × 10-2) 0.062 (8.8 × 10-2) 0.021 (7.7 × 10-2) 0.068 (7.8 × 10-2) 0.003 (7.3 × 10-2)

0.059 (7.9 × 0.054 (5.5 × 10-2) 0.057 (7.3 × 10-2) 0.057 (7.5 × 10-2) 0.056 (6.9 × 10-2) 0.051 (4.8 × 10-2) 0.058 (7.5 × 10-2) 0.040 (3.3 × 10-2) eq 3 (Peng and Robinson) eq 4 (Danesh et al.) eq 5 (Boukouvalas et al.) eq 6 (Stryjek and Vera) eq 7 (Melham et al.) eq 8 (Twu et al.) eq 9 (Boston and Mathias) eq 10 (this work)

T ) 460 K

0.075 (6.1 × 10-2) 0.045 (6.5 × 10-2) 0.074 (6.3 × 10-2) 0.075 (6.1 × 10-2) 0.092 (1.6 × 10-1) 0.013 (9.4 × 10-2) 0.077 (6.2 × 10-2) 0.006 (6.9 × 10-2) 0.084 (6.5 × 0.062 (6.0 × 10-2) 0.081 (6.4 × 10-2) 0.082 (6.5 × 10-2) 0.080 (6.5 × 10-2) 0.053 (6.1 × 10-2) 0.083 (6.5 × 10-2) 0.036 (6.5 × 10-2) 0.046 (1.3 × 0.027 (1.3 × 10-2) 0.046 (1.3 × 10-2) 0.047 (1.3 × 10-2) 0.044 (1.3 × 10-2) 0.024 (1.3 × 10-2) 0.048 (1.3 × 10-2) -0.0001 (1.4 × 10-2)

10-2) 10-2) 10-1) 10-1) 10-1) 10-2) 10-2)

T ) 400 K T ) 360 K

CH4 + C18H30

T ) 280 K T ) 430 K

CH4 + C27H48

T ) 340 K T ) 425 K

CH4 + C24H50

T ) 325 K

kij (dev)

Table 3. Results on the Optmization of the Interaction Parameter kij for Different r Functions (Equations 3-10) (dev As Defined by Equation 11)

In a second set of calculations the impact of the different R functions is studied when the optimization of the binary interaction parameter kij is allowed. In Table 3 the optimal values of the interaction coefficient kij and the respective average sum of squares are listed. As the table shows, the quality of the representation of the experimental data does not differ too much between the calculations involving the different R functions. However, it should be noted, as also could be expected from the results of the calculations with kij ) 0, that the calculations involving the R function proposed in this work result in the smallest optimal values for kij. Despite the suggested similar overall performance of the different models with the optimized interaction parameter kij, it should be stressed that the localsin terms of compositionsperformance varies. This is shown in Figures 15-18. In these figures only three R functions are considered. These are the functions proposed by Robinson and Peng (1978) (eq 3), the correlation proposed by us (eq 10), and one of the other R functions. This is so since most of the calculational results perfectly coincide. Figure 15 depicts the phase behavior of the (methane + tetracosane) system at T ) 325 K. The experimental data are compared with calculations with the Peng-Robinson equation using the R functions proposed by Robinson and Peng (1978) (eq 3) with kij set to 0.059, proposed by Danesh et al. (1995) (eq 4) with kij set to 0.054, and proposed by us (eq 10) with kij ) 0.042. The bubble points at lower pressures are best described by eq 3. However, the dew and bubble points at pressures above 30 MPa are best described by eq 10. As shown in the insert of this figure, the use of eq 10 also results in the smallest overprediction of the vapor-liquid critical pressure. The phase behavior of the system (methane + 1phenyldodecane) at T ) 360 K is depicted in Figure 16. Again for the sake of clarity only three different R functions are considered. The optimal values for the interaction parameter are +0.046 for eq 3, +0.024 for eq 8 proposed by Twu et al. (1995), and -0.001 for eq 10. For this system all three equations perform very similarly. However, in the intermediate pressure range eq 10 gives the highest and eq 3 the lowest bubble-point pressures, so that the same pattern as that found earlier is revealed. For the system composed of methane and phenanthrene the phase behavior at 400 K is shown in Figure 17. To elucidate the description of the dew points, the composition axes are logarithmical. In this figure the same R functions as those used in the previous figure are compared with the experimental results. The interaction parameters used are 0.081 for eq 3, 0.053 for eq 8, and 0.036 for eq 10. As found earlier eq 10 results in greater differences between the vapor and the liquid compositions than the other equations. This inherently means that eq 10 again yields the smallest vapor-liquid critical pressure. For this system the bubble-point pressures are described best with eq 10, while the dew-point pressures are best represented by eq 3. Figure 18 shows the comparison of the experimental vapor-liquid equilibrium data for the system (methane + 5-R-cholestane) at 400 K with again the same set of models. For this system the respective interaction parameters are 0.092 for eq 3, 0.084 for eq 8, and 0.072 for eq 10. Again it is found that eq 10 results in a broader, flatter two-phase region. Since for this system

CH4 + C14H10



Figure 15. Vapor-liquid equilibrium of (methane + tetracosane) at 325 K. O, Experimental data (Flo¨ter et al., 1997a). Calculations with Peng-Robinson EOS with different R functions: s, eq 3 (kij ) 0.059); - ‚‚ -, eq 4 (kij ) 0.054); - -, eq 10 (kij ) 0.042).

Figure 16. Vapor-liquid equilibrium of (methane + 1-phenyldodecane) at 360 K. O, Experimental data (Flo¨ter et al., 1997c). Calculations with Peng-Robinson EOS with different R functions: s, eq 3 (kij ) 0.046); - -, eq 8 (kij ) 0.024); - -, eq 10 (kij ) -0.001).

Figure 17. Vapor-liquid equilibrium of (methane + phenanthrene) at 400 K. O, Experimental data (Flo¨ter et al., 1997c). Calculations with Peng-Robinson EOS with different R functions: s, eq 3 (kij ) 0.081); - ‚ -, eq 8 (kij ) 0.053); - -, eq 10 (kij ) 0.036).

all three equations predict too low dew- and bubblepoint pressures, eq 10 performs superiorly to the other two equations. Prior to summarizing the above results, a general remark about the quality of the description of the experimental data presented should be made. The

Figure 18. Vapor-liquid equilibrium of (methane + 5-R-cholestane) at 400 K. O, Experimental data (Flo¨ter et al., 1997b). Calculations with Peng-Robinson EOS with different R functions: s, eq 3 (kij ) 0.092); - ‚ -, eq 8 (kij ) 0.084); - -, eq 10 (kij ) 0.072).

choice of the objective function used and the incorporation of the full experimental data set, covering also the critical region, have an obvious impact on the quality of the representation of the experimental data by the different models. Therefore, the optimized fit presented here might in some cases be disappointing compared to results of an optimization performed on only a limited data set, e.g., bubble points. However, due to the reasoning given above and the underlying objective to compare the performance of the different models, the procedure applied is justified. The description of vapor-liquid equilibria in asymmetric binary systems is very similar for all the R functions considered when the interaction coefficient kij is optimized. However, a slight superiority of the description of the experimental data by the calculations involving the correlation of R values proposed in this work (eq 10) is found. Apart from the slightly better description of the experimental points the use of eq 10 also yields lower optimal values of kij. Since with increasing kij the vapor-liquid critical pressure increases, the lower optimal kij values found for eq 10 are favorable. Above a certain value of the interaction coefficient the vapor and liquid equilibrium compositions do not approach a vapor-liquid critical composition anymore. Then the two-phase region becomes unrealistic and “hourglass” shaped. In a final set of calculations the performances of the Peng-Robinson equation with either the R function proposed by Robinson and Peng (1978) (eq 3) or eq 10 as proposed in this paper are compared for simultaneous computation of vapor-liquid and solid-fluid equilibria. Since the simultaneous description of the solid-fluid and vapor-liquid equilibria with only one interaction coefficient is not satisfying, both kij and lij had to be used. In Figures 19 and 20 the phase behavior of the system (methane + tetracosane) is shown. The vaporliquid two-phase region and solid solubility curve at 325 K as well as the solid solubility curve at 323 K are depicted. It should be noted that the temperature of 323 K is less than 0.5 K higher than the temperature at the second critical end point. Here, the isothermal solid solubility curve shows a horizontal point of inflection. kij and lij were simultaneously optimized to the isothermal data at 325 K and also used for the calculations of the solid solubilities at 323 K. For the two R functions this yields kij ) 0.060 and lij ) 0.017 for eq 3


Figure 19. Vapor-liquid of (methane + tetracosane) at 325 K and solid-fluid equilibria at 325 and 323 K. Experimental data (Flo¨ter et al., 1997a): 0, VLE 325 K; 3, SFLE 323 K; O, SFLE 325 K. Calculations with Peng-Robinson EOS in combination with eq 3: s, kij ) 0.060 and lij ) 0.017; - -, SLE 323 K with kij ) 0.059 and lij ) 0.0. The dotted line indicates a solid-solid transition of tetracosane.

Figure 20. Vapor-liquid of (methane + tetracosane) at 325 K and solid-fluid equilibria at 325 and 323 K. Experimental data (Flo¨ter et al., 1997a): 0, VLE 325 K; 3, SFLE 323 K; O, SFLE 325 K. Calculations with Peng-Robinson EOS in combination with eq 10: s, kij ) 0.042 and lij ) 0.015; - -, SLE 323 K with kij ) 0.042 and lij ) 0.0. The dotted line indicates a solid-solid transition of tetracosane.

and kij ) 0.042 and lij ) 0.015 for eq 10. The parameters determined correspond very well with interaction coefficients determined by Arnaud et al. (1996). They optimized the kij (0.054) to vapor-liquid equilibrium data and the lij (0.0225) to excess volumes at 374 K. The phase equilibrium calculations based on eq 3 are depicted in Figure 19. Compared to the calculations depicted in Figure 15 with kij ) 0.059, the results of the vapor-liquid equilibrium calculations are slightly worse. The dashed line indicates the calculated solid solubilities at 323 K calculated with kij ) 0.059. Figure 20 shows the same experimental data now compared to calculations involving eq 10. Here, the disimprovement of the description of the fluid phase equilibria is almost not detectable. The calculations with the single interaction coefficient optimized to vapor-liquid equilbrium kij ) 0.042 (lij ) 0) yield poor quality calculated solid solubilities at 323 K. As both figures reveal, both models describe the isothermal solid solubilities quite well when two interaction coefficients are used. Nevertheless, it should be pointed out that the evolvement of the solid solubilities at elevated pressures are better

Figure 21. Vapor-liquid of (methane + 1-phenyldodecane) at 280 K and solid-fluid equilibria at 280 and 285 K. Experimental data (Flo¨ter et al., 1997c): 0, VLE 280 K; 3, SFLE 285 K; O, SFLE 280 K. Calculations with Peng-Robinson EOS in combination with eq 3: s, kij ) 0.049 and lij ) -0.002; - -, SLE 280 K with kij ) 0.057 and lij ) 0.0.

Figure 22. Vapor-liquid of (methane + 1-phenyldodecane) at 280 K and solid-fluid equilibria at 280 and 285 K. Experimental data (Flo¨ter et al., 1997c): 0, VLE 280 K; 3, SFLE 285 K; O, SFLE 280 K. Calculations with Peng-Robinson EOS in combination with eq 10: s, kij ) 0.032 and lij ) -0.003; - -, SLE 280 K with kij ) 0.032 and lij ) 0.0.

described by eq 10, as the inserts show. One reason for this improved description is the lower vapor-liquid critical pressure calculated by employing eq 10. The horizontal dotted lines at low pressures and high tetracosane concentrations indicate a solid-solid transition from a hexagonal to a mono- or triclinic crystal polymorph. Similar results for the phase behavior of the system (methane + 1-phenyldodecane) are graphically represented in Figures 21 and 22. In both figures the experimental vapor-liquid and solid-fluid equilibrium data at 280 K and solid-fluid equilibrium data at 285 K are shown. The optimized interaction coefficients are kij ) 0.049 and lij ) -0.002 for eq 3 and kij ) 0.032 and lij ) -0.003 for eq 10. Also here the solid-fluid calculations at 285 K were performed with the interaction coefficient derived at 280 K. The results of eq 3 and eq 10 are shown in Figures 21 and 22, respectively. Again the description of the vapor-liquid equilibria is superior when eq 10 is used. Both equations represent the solid solubilties quite well using two interaction coefficients. A significant difference occurs when only the interaction coefficient optimized to vapor-liquid equilibrium data is used. Equation 10 (kij ) 0.032)


and 359.6 MPa. The three-phase equilibria calculated at 395 K reflect the experimental findings. However, the overall representation of the isothermal solid solubilities by the two models is good. The dashed curves in the two figures indicate that the employment of only kij optimized to vapor-liquid equilibria yields unsatisfactory results. Conclusions

Figure 23. Vapor-liquid of (methane + phenanthrene) at 400 K and solid-fluid equilibria at 395 and 405 K. Experimental data (Flo¨ter et al., 1997c): 0, VLE 400 K; 3, SFLE 395 K; O, SFLE 405 K. Calculations with Peng-Robinson EOS in combination with eq 3: s, kij ) 0.072 and lij ) -0.016; - -, SLE 405 K with kij ) 0.084 and lij ) 0.0.

Figure 24. Vapor-liquid of (methane + phenanthrene) at 400 K and solid-fluid equilibria at 395 and 405 K. Experimental data (Flo¨ter et al., 1997c): 0, VLE 400 K; 3, SFLE 395 K; O, SFLE 405 K. Calculations with Peng-Robinson EOS in combination with eq 10: s, kij ) 0.026 and lij ) -0.016; - -, sle 405 K with kij ) 0.036 and lij ) 0.0.

predicts the solid solubilities quite well for pressures up to 150 MPa. At higher pressures the solubilities are overpredicted. The combination of kij ) 0.057 and eq 3 results in very good predictions of the solid solubilities. But erroneously a solid-liquid-vapor three-phase equilibrium at a pressure of approximately 174 MPa is found. Figures 23 and 24 show experimental data and calculations of the phase behavior of the system (methane + phenanthrene). The vapor-liquid equilibrium at 400 K and the solid-fluid equilibria at 395 and 405 K are depicted. The interaction parameters are in this case optimized to the data at 400 and 405 K. For eq 3 the optimal interaction coefficients were found to be kij ) 0.072 and lij ) -0.016. For eq 10 these are kij ) 0.026 and lij ) -0.016. Again the description of vapor-liquid equilibria is less good with eq 3 than with eq 10. However, the dew points are very well represented by both models. Anyhow, both models dramatically overpredict the vapor-liquid critical pressure. This causes both models to give at T ) 405 K a solid-liquid-vapor three-phase equilibrium at approximately 375 MPa which has not been found experimentally. The experimental second critical end point is located at 403.3 K

The results shown in this work show that, even though being mentioned by several authors, the problem of the description of a supercritical component in phase equilibrium calculations has not been addressed sufficiently. The various temperature dependencies of the attractive parameter a in a cubic equation of state proposed in the past were mainly aiming at an improved description of the vapor pressure curve of high-boiling and polar compounds. Fortunately, most of them almost coincide in the R values generated for lighter components at supercritical conditions. However, these R values are not related to any experimental evidence. In this work a correlation of R values for supercritical methane optimized to IUPAC recommended fugacities is proposed. This correlation is compared to various R functions given in the literature. The employment of the optimized R function yields improved results of phase equilibrium calculation with the Peng-Robinson equation of state in asymmetric hydrocarbon mixtures. The Peng-Robinson equation tends to predict (kij ) 0) too low dew- and bubble-point pressures for these systems. In general decreasing R values cause increasing dew- and bubble-point pressures. From the set of R functions studied the proposed correlation yields the smallest R values. The superiority of the proposed R function for supercritical methane is the most evident in vapor-liquid equilibrium calculations with all binary interaction parameters set to zero. When the optimization of one interaction parameter kij is allowed, the influence of the different R functions is partly compensated. Nevertheless, using the proposed R function, the smallest values of kij are obtained. This is desirable since the smaller kij values result in a “flatter” vapor-liquid two-phase region. Consequently the vapor-liquid critical pressures tend less to increase dramatically and to cause unrealistic and “hourglass”shaped two-phase regions. Calculations of isothermal high-pressure solid solubilities made with the Peng-Robinson equation reveal that the employment of the interaction coefficient optimized to vapor-liquid equilibria does not yield satisfactory results. Nevertheless, due to the phenomenon described above, an “hourglass”-like vapor-liquid two-phase region, the conventional R functions, e.g., eq 3, might yield unrealistic solid-liquid-vapor threephase equilibria at elevated pressures. The equation proposed in this work (eq 10) suffers, due to the smaller values of kij, less from this deficiency. However, if the optimization of two interaction coefficients is allowed, both eqs 3 and 10 yield a good description of the isothermal solid solubility curves. It should be pointed out that the calculations were carried out close to the second critical end point. With the simultaneous fit of two interaction coefficients two main differences between the R functionsseq 3 and eq 10soccur. At first, the correlation of R values proposed here yields smaller kij values, while lij are very similar for both R functions. Second, the calculations incorporating eq 10 result in a


clearly better fit of the exprimental vapor-liquid data than with eq 3. Finally, it could be concluded that for the future development of a genaralized R function its performance for light components at supercritical conditions should be taken into account. Since for most of the lighter components of major interest IUPAC-recommended pure component properties tables exist, it is desirable and seems to be worthwhile to include this readily available information. The preliminary approach applied in this work to incorporate the available experimental information on fugacities of supercritical methane is premature in two ways. It has first not been tried yet to extend the temperature range covered to the subcritical region. This also means that the present form of the proposed correlation does not meet the criterion to equal unity for TR ) 1.0. Second, so far this study was limited to asymmetric systems incorporating methane as a supercritical component. Literature Cited Ambrose, D.; Tsonopoulos, C. Vapor-Liquid Critical Properties of Elements and Compounds. 2. Normal Alkanes. J. Chem. Eng. Data 1995a, 40, 531. Ambrose, D.; Tsonopoulos, C. Vapor-Liquid Critical Properties of Elements and Compounds. 2. Aromatic Hydrocarbons. J. Chem. Eng. Data 1995b, 40, 547. Anastasiades, K.; Stamataki, S.; Tassios, D. Prediction/correlation of vapor-liquid equilibria in synthetic gas and oil systems. Fluid Phase Equilib. 1994, 93, 23. Arnaud, J. F.; Ungerer, P.; Behar, E.; Moracchini, G.; Sanchez, J. Excess volumes and saturation pressures for the system methane + tetracosane at 374 K. Representation by improved EOS mixing rules. Fluid Phase Equilib. 1996, 124, 177. Boston, J. F.; Mathias, M. P. Phase Equilibria in a ThirdGeneration Process Simulator. Proceedings of the 2nd International Conference on Phase Equilibria and Fluid Properties in the Chemical Process Industries, Berlin, Mar 17-21, 1980; Dechema: Frankfurt/Main, FRG, 1980; p 823. Boukouvalas, C.; Spiliotis, N.; Coutsikos, P.; Tzouvaraz, N.; Tassios, D. Predictions of vapor-liquid equilibrium with the LCVM model: A linear combination of the Vidal and Michelsen mixing rules coupled with the original UNIFAC and the t-mPR equation of state. Fluid Phase Equilib. 1994, 92, 75. Dahl, S.; Michelsen, M. L. High-Pressure Vapor-Liquid Equilibrium with a UNIFAC Based Equation of State. AIChE J. 1990, 36, 1829. Danesh, A.; Xu, D.-H.; Tehrani, D. H.; Todd, A. C. Improving predictions of equation of state by modifying its parameters for super critical components of hydrocarbon reservoir fluids. Fluid Phase Equilib. 1995, 112, 45. Daubert, T. E.; Danner, R. P. Data Compilation. Physical and Thermodynamic Properties of Pure Compounds; Taylor and Francis: London, 1990. de Leeuw, V. V. Phase behaviour of selected nitrogen + hydrocarbon systems. Ph.D. thesis, TU Delft, Delft, 1995. de Leeuw, V. V.; de Loos, Th. W.; Kooijman, H. A.; de Swaan Arons, J. The experimental determination and modelling of vapourliquid equilibrium for binary subsystems of the quaternary system N2 + CH4 + C4H10 + C14H30 up to 100 MPa and 440 K. Fluid Phase Equilib. 1992, 73, 285. de Loos, Th. W.; Poot, W.; Lichtentaler, R. N. Fluid Phase Equilibria in Ethylene + n-Alkane Systems. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 855. Dortmund Data Bank, version 1994; DDBST Gubh: Oldenburg, FRG, 1994. Flo¨ter, E.; de Loos, Th. W.; de Swaan Arons, J. High-pressure solidfluid and vapour-liquid equilibria in the system (methane + tetracosane). Fluid Phase Equilib. 1997a, 127, 129. Flo¨ter, E.; Brumm, C.; de Loos, Th. W.; de Swaan Arons, J. HighPressure Phase Equilibria in the Binary System (Methane + 5-R-Cholestane). J. Chem. Eng. Data 1997b, 42, 1, 64.

Flo¨ter, E.; van der Pijl, P.; de Loos, Th. W.; de Swaan Arons, J. High-Pressure Phase Equilibria in the Systems (Methane + Phenanthrene) and (Methane + 1-Phenyldodecane). Fluid Phase Equilib. 1997c, 134, 1. Glaser, M.; Peters, C. J.; van der Kooi, H. J.; de Swaan Arons, J. Phase equilibria of (methane + n-hexadecane) and (p, VM, T) of n-hexadecane. J. Chem. Thermodyn. 1985, 17, 803. Heidemann, R. A.; Michelsen, M. L. Instability of successive substitution. Ind. Eng. Chem. Res. 1995, 34, 958. IUPAC. Wagner, W.; de Reuck, K. M. Methane; International Thermodynamics Tables of the Fluid State, 13; Blackwell Science: Oxford, U.K., 1996. Mathias, M. P. A Versatile Phase Equilibrium Equation of State. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 385. Melham, G. H.; Siani, R.; Goodwin, G. M. A Modified PengRobinson Equation of State. Fluid Phase Equilib. 1989, 47, 189. Michelsen, L. M. The isothermal flash problem. Part I. Stability. Fluid Phase Equilib. 1982a, 9, 1. Michelsen, L. M. The isothermal flash problem Part II. Phase split calculations. Fluid Phase Equilib. 1982b, 9, 21. Montel, F. High-Pressure High-Temperature Reservoir Fluid Properties. Presentation at the 13th European Seminar on Applied Thermodynamics, Jun 9-12, 1993, Carre-Le-Rouet, France. Peng, D.-Y.; Robinson, D. R. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Rijkers, M. P. W. M. Retrograde Condensation of Lean Natural Gas. Ph.D. thesis, TU Delft, Delft, 1991. Rijkers, M. P. W. M.; Malais, M.; Peters, C. J.; de Swaan Arons, J. Measurement of the phase behaviour of binary hydrocarbon mixtures for modelling the condensation behaviour of natural gas. Part I. The system methane + decane. Fluid Phase Equilib. 1992, 71, 143. Robinson, D. B.; Peng, D.-Y. The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs; GPA Research Report 28; Gas Processors Association: Tulsa, OK, 1978. Schiessler; et al. Am. Doc. Inst. Doc. 1959, 4597 (via Beilstein). Soave, G. Equilibrium constants for a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972, 27, 1196. 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. Suleiman, D.; Eckert, C. A. Phase equilibria of alkanes in natural gas systems. 1. Alkanes in methane. J. Chem. Eng. Data 1995, 40, 2. Twu, C. H.; Coon, J. E.; Cunningham, J. R. A new generalized alpha function for a cubic equation of state. Part I. PengRobinson equation. Fluid Phase Equilib. 1995, 105, 49. Twu, C. H.; Coon, J. E.; Harvey, A. H.; Cunningham, J. R. An approach for the application of cubic equation of state to hydrogen hydrocarbon systems. Ind. Eng. Chem. Res. 1996, 35, 905. Ungerer, P.; Faissat, B.; Leibovici, C.; Behar, E.; Moracchini, G.; Courcy, J. P. High pressure-high-temperature reservoir fluids: Investigation of synthetic condensate gases containing a solid hydrocarbon. Fluid Phase Equilib. 1995, 111, 287. van der Kooi, H. J. Measurements and calculations on the system methane + eicosane. Ph.D. thesis, TU Delft, Delft, 1981 (in Dutch). Vidal, J. Equations of state-Reworking the old forms. Fluid Phase Equilib. 1983, 13, 15. Wong, D. S.; Sandler, S. I. A theoretically correct new mixing rule for cubic equations of state for both highly and slightly nonideal mixtures. AIChE J. 1992, 38, 671.

Received for review September 10, 1997 Revised manuscript received December 30, 1997 Accepted January 6, 1998 IE970644O

Improved Modeling of the Phase Behavior of Asymmetric Hydrocarbon

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