Simultaneous Prediction of the Critical and Subcritical Phase Behavior

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Ind. Eng. Chem. Res. 2005, 44, 2292-2300

Simultaneous Prediction of the Critical and Subcritical Phase Behavior in Mixtures of Ethane-n-Alkanes Ilya Polishuk,*,† Jaime Wisniak,‡ and Hugo Segura§ Department of Chemical Engineering & Biotechnology, The College of Judea and Samaria, Ariel, Israel, Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel, and Department of Chemical Engineering, Universidad de Concepcio´ n, Concepcio´ n, Chile

The present study compares the ability of three semipredictive approaches, namely, the global phase diagram based semipredictive approach (GPDA), predictive Soave-Redlich-Kwong (PSRK), and linear combination of the Vidal and Michelsen mixing rules (LCVM), to describe the experimental data in the binary homologue series of ethane-n-alkanes. The applicability of GPDA has been extended by attachment of an additional empirical functionality. Now it can predict not only the data of a homologues series to which the reference system belongs but also other series that include solvents from the same chemical group. Yet, both GPDA and PSRK appear as entirely predictive models. However, LCVM has a correlative character rather than predictive. Although the models under consideration yield rather similar results for the light homologues of the series, starting from ethane-n-decane, GPDA shows a clear superiority over PSRK and LCVM. Moreover, it is demonstrated that the GE mixing rules of both PSRK and LCVM generate the nonphysical behavior of liquid-liquid equilibrium. This result strongly affects the reliability of these models in predicting the data of heavy homologues. In contrast, GPDA predicts robust results even for very asymmetric systems. These facts confirm the clear advantage of an approach based on consideration of the complete phase diagram and not of its separate parts. 1. Introduction The phase behavior of the ethane-n-alkanes systems is very important from a practical point of view. Substantial amounts of ethane are usually present in crude oil under high-pressure conditions, and it is partially released when pumping the oil up. In addition, the initial transformation of the crude oil at a refinery is done through the topper process, where the crude oil is heated at atmospheric pressure to accomplish the first rough distillation “cut”, which consists mostly of ethane and other light hydrocarbons. This vaporized fraction passes further refining in the catalytic cracking unit or the reforming unit. The liquid fractions, which cannot be vaporized in this process, are separated afterward in the vacuum distillation unit, a process that removes the remaining amount of ethane from the gasoline products almost completely.1 Natural gas usually also includes about 3% of ethane and about 0.6% of heavier hydrocarbons.2 Although the content of these compounds in the gas phase can be considered as insignificant, they represent the mean constituent of the condensate that appears when depressurizing the natural gas. Supercritical ethane finds also very important implementations as a solvent. In particular, it is used to convert the heavy molecules into the fluid phase, which is a requirement for compounds analyzed by mass * To whom correspondence should be addressed. E-mail: [email protected]. † The College of Judea and Samaria. ‡ Ben-Gurion University of the Negev. § Universidad de Concepcio´n.

spectrometry.1 Moreover, supercritical ethane is sometimes preferred over carbon dioxide as an extraction agent.3 An appropriate design of these and other processes requires development models capable of describing the phase behavior of ethane-n-alkane mixtures.4-7 So far, an important contribution to modeling of these mixtures was made,8,9 using the Soave-Redlich-Kwong equation of state (EOS) and classical mixing rules with two binary adjustable parameters. It was found that the performance of this model is not satisfactory because the values of the binary parameters showed significant and unpredictable temperature dependence. This result was explained by strong nonideality characteristic for the mixtures under consideration. A significant improvement in the description of the bubble-point data for these mixtures was achieved,10 using the EOS based on the perturbed-hard-chain theory. However, this model was not able to predict reliable results in the near-critical region. Recently, it has been demonstrated11,12 that another theoretically based model, statistical associating fluid theory, overestimates the critical data under consideration in a similar manner. Such results do not permit this model to yield an accurate description of the pertinent dew-point data.12 The above observation confirms the fact that a successful thermodynamic model should be reliable in describing all parts of the thermodynamic phase space and recognize that all of them are always closely interrelated. In addition, it should be realized that, because the available experimental data cannot provide all of the necessary information, the engineering models

10.1021/ie048982u CCC: $30.25 © 2005 American Chemical Society Published on Web 03/04/2005

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should be attached with a predictive character. The present work continues the previous studies13-17 that have examined the ability of the novel global phase diagram based semipredictive approach (GPDA) to predict experimental data, in comparison with the two most successful approaches, namely, the predictive Soave-Redlich-Kwong (PSRK) and the linear combination of the Vidal and Michelsen mixing rules (LCVM). PSRK was proposed by Holderbaum and Gmehling18 and then adjusted for a wide variety of systems.19-21 LCVM was proposed by Boukouvalas et al.,22 and its parameter matrix was extended.23-26 The main advantage of GPDA over PSRK and LCVM finds expression in its better predictive ability. In contrast to the latter models, which require a large basis of experimental data for evaluation of their parameters, GPDA needs no more than two to three key experimental points in order to predict the complete phase diagram in a whole homologue series of binary systems. The accuracy with which GPDA predicts the particular data that have been included for fitting of the parameters of PSRK and LCVM is comparable with the correlative results of these models. However, GPDA has a clear superiority over PSRK and LCVM in describing the overall picture of phase behavior, which includes not only the vapor-liquid equilibria (VLE) but also the liquid-liquid equilibria (LLE), the liquid-liquid-vapor equilibria (LLVE), and the critical curves. At the same time, it should be pointed out that the applicability of GPDA is restricted by the fact that the key experimental points needed for evaluation of its parameters, namely, the critical pressure maximum (CPM) and some LLE critical point for at least one of the homologues, are sometimes unavailable for a whole series. In the present study, we propose to solve the problem of predicting data in such series by implementing the binary parameters evaluated for the similar series, with the correction factor universal for all series. In other words, here we present the final development of GPDA, which requires two to three key points of one homologue in order to predict the complete picture of phase behavior not only of a given series but also of similar series. In particular, the data of ethane-nalkane systems will be predicted using the parameters that have been obtained so far17 for the system methanen-pentane, as shown below.

Table 1. Values of K11, K22, and L12 homologue series

K11

K22

L12

CO2-n-alkanes n-alkanes-n-alkanes

0.1 -0.03

0.35 0.21

0.02 -0.025

The cross-interaction parameters are obtained using the following combination rules:

a21 ) a12 ) (1 - k12)xa11a22 b11 + b22 2

b12 ) b21 ) (1 - I12) c12 ) c21 )

c11 + c22 2

d12 ) d21 )

d11 + d22 2

where k12 and l12 are binary adjustable parameters whose values are estimated at the intersection of the loci that present the experimental values of CPM and upper critical end point (UCEP) [or, alternatively, tricritical point (TCP)] on the global phase diagram in the k12-l12 projection (klGPD) of the reference homologue. According to GPDA, the values of the binary parameter l12 change proportionally to the values of the pure compound parameters b. In addition, investigation of the klGPDs of different systems has clearly demonstrated that the desired intersection of loci is dependent on the critical temperature of the lighter compound. As this temperature increases, the value of l12 increases as well (for example, as shown by the values obtained so far for the series methane-n-alkanes and carbon dioxide-n-alkanes; Table 1). At the same time, our experience shows that different series, which include lighter compounds that belong to the same chemical family, can be described using similar values of binary parameter k12. In other words, it appears that the dependency of attractive forces of the molecular size of the solvent can be neglected so that such forces are mainly defined by the chemical properties of the solvent. However, the repulsive forces are dependent on both the molecular size and the chemical properties of the solvent. Investigation of the klGPDs of two families of homologue series, namely, n-alkanes-n-alkanes and n-alkanes-n-alkanols, has led to the following empirical correction factor:

2. Theory

δ ) b11

The EOS implemented by GPDA is

P)

RT(Vm + 0.125b) Vm(Vm - 0.875b)

m2

-

aTrm1Tr

(Vm + c)(Vm + d)

(1)

The way of evaluating the parameters of eq 1 was presented in previous studies.13-17 The values of these parameters for mixtures are obtained using the following classical van der Waals mixing rules:

z) where z ) a, b, c, and d.

∑ij xixjzij

(2)

(3)

[

]

Tc1 - Tc1* Tc1

1/3

(4)

where Tc1 is a critical temperature of the solvent in the given homologue series and Tc1* is a critical temperature of the solvent in the reference series, for which the values of binary parameters have been evaluated. Thus, here Tc1* is a critical temperature of methane (190.564 K). Thus, for the reference series (methane-n-alkanes), the value of correction factor δ is equal to zero and positive for solvents heavier than methane. Obviously, if the molecule of the solvent in the reference series is branched, then δ becomes negative for the series, which includes lighter solvents from the same chemical family. Because experience shows that there are clear regularities in the behavior of global phase diagrams of different systems, we estimate that eq 10 has a universal

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character and is not restricted only to the families of systems considered in our research. This corrects the value of l12 as follows:

l12 )

b22 - b11 L +δ b22 + b11 12

(5)

where L12 is a value characteristic for a given homologue series. At the same time, as pointed out, we neglect the dependence of the binary parameter k12 on the molecular size of the solvent. Therefore, the contribution that the correction factor δ makes for l12 may be removed here as follows:

[

]

Tc2 (1 - t) + K22t k12 ) K11 - (l12 - δ) Tc1

(6)

where K11 and K22 are characteristic values for a given homologue series and t is given by the following dimensionless functionality:

t ) tanh

[(

T - Tc1 Tc2* - Tc1

)] 2

(7)

For homologues that include solutes heavier than the solute of the reference homologue (CH4-n-pentane), Tc2* ) Tc2, and for lighter ones, Tc2* is taken as equal to Tc2 of the reference homologue (e.g., n-pentane, 469.7 K). The values of the characteristic parameters for the series investigated so far are listed in Table 1. We will now compare the ability of GPDA, LCVM, and PSRK to predict the phase equilibria data in the homologue series under consideration. 3. Results 3.1. General Procedures. The basic criterion that defines the predictive character of thermodynamic models is the ratio between the amount of data input and the predicted ones. The LCVM model considers ethane as a separate UNIFAC group. The LCVM parameters for the homologue series under consideration were evaluated considering the data of all homologues starting from ethane-propane and up to ethanen-octacosane.22 In other words, LCVM appears here as a correlative model and not as a predictive model. In contrast, PSRK does not define ethane as a separate UNIFAC group. Thus, both PSRK and GPDA estimate the data of homologue series under consideration in an entirely predictive manner. In what follows, we will discuss the accuracy with which these models describe the data of the homologues for which their phase behavior has been studied experimentally in the most detailed manner. 3.2. Ethane-Propane. Figure 1 presents the results predicted by the models for the system ethane-propane. This system is quite symmetric; hence, the quality of the description of the VLE binary data in such a system is dependent mostly on the accuracy with which the models represent the pure compound vapor-pressure lines. Because all equations correlate the latter data in a satisfactory manner, they are accurate in representing the binary ones as well. The figure demonstrates that both GE-based models yield almost identical results and that both tend to slightly overestimate the data. Contrary to this, GPDA tends to underestimate the data. Thus, it can be seen that GPDA is superior in predicting

Figure 1. VLE of ethane (1)-propane (2). Solid lines: VLE predicted at 260, 270, and 280 K. Experimental VLE data:27 (b) 260 K; (3) 270 K; (1) 280 K.

the dew points, while both GPDA and LCVM yield a somewhat better description of the bubble-point data. However, it should be noted that the differences between the results of models are insignificant, and they do not really exceed the possible experimental uncertainty. 3.3. Ethane-n-Pentane. Figure 2 presents the results predicted by the models for the system ethanen-pentane. In a comparison with the previous system, this one is more asymmetric. Therefore, the accuracy of the results is determined not only by the quality of the correlation of the vapor-pressure lines but mostly by the ability of the model to treat the binary data. The figure demonstrates that again all models yield rather similar results. Nevertheless, it can be seen that GPDA is slightly less accurate than the GE-based models predicting the data at low temperatures and the dew points at high temperatures. At the same time, GPDA has a clear superiority in predicting the bubble-point data at high temperatures. 3.4. Ethane-n-Hexane. The results for the system ethane-n-hexane are presented in Figure 3. The experimental data30 seem questionable in the near-critical pressures because they present phase envelopes wider


Figure 3. VLE of ethane (1)-n-hexane (2). Solid lines: VLE predicted at 338.71, 394.26, and 449.82 K. Experimental VLE data: 30 (1) 338.71 K; (4) 394.26 K; (2) 449.82 K.

Figure 2. VLE of ethane (1)-n-pentane (2). Solid lines: VLE predicted at 310.93, 344.26, 377.59, 410.93, and 444.26 K. Experimental VLE data:29 (1) 310.93 K; (3) 344.26 K; (b) 377.59 K; (2) 410.93 K; (4) 444.26 K.

than those observed in other members of this homologue series. Nevertheless, the figure shows that the pattern of behavior described for the previous homologues is also valid in the present case. In particular, it can be seen that GPDA is the most accurate model for predicting the composition of the liquid phase and that PSRK has a clear superiority in predicting the vapor compositions. Although LCVM appears as a correlative model, in the present case it is less accurate that the two other predictive models. 3.5. Ethane-n-Heptane. Figure 4 presents the data of the system ethane-n-heptane. It can be seen that at low temperatures all of the models yield very similar results. Again, at high temperatures, GPDA is the most accurate model for predicting the bubble-point data and less accurate for predicting the dew-point ones. At these temperatures, PSRK predicts more accurate overall results than both LCVM and GPDA.

3.6. Ethane-n-Decane. This system has been studied experimentally in a very detailed manner, and the pertinent results are shown in Figure 5. Again, it can be seen that both the predictive GPDA and PSRK are more accurate than the correlative LCVM. Although GPDA once again slightly underpredicts the liquidphase compositions, it is clearly superior to PSRK in predicting the critical data. The latter model tends to overestimate the critical pressures. LCVM correlates the high-temperature data inaccurately because of the overestimation of both the composition and pressure of the critical points. In other words, the close interrelation between the ability of the model to estimate the critical data and its accuracy in predicting the subcritical VLE is clearly seen. 3.7. Ethane-n-Hexadecane. The next system in the series for which experimental data are available in a wide temperature range is ethane-n-hexadecane (see Figure 6). The difference between the critical temperatures of its components exceeds 420 K; therefore, this system should already be considered as very asymmetric. Nevertheless, it still exhibits type I phase behavior according to the classification of van Konynen-

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Figure 4. VLE of ethane (1)-n-heptane (2). Solid lines: VLE predicted at 338.71, 366.48, 394.26, 422.04, and 449.82 K. Experimental VLE data:31 (3) 338.71 K; (1) 366.48 K; (4) 394.26 K; (2) 422.04 K; ([) 449.82 K.

Figure 5. VLE of ethane (1)-n-decane (2). Solid lines: VLE predicted at 310.93, 344.26, 377.59, 410.93, 444.26, 477.59, and 510.93 K. Experimental VLE data:32 (1) 310.93 K; (3) 344.26 K; (2) 377.59 K; (4) 410.93 K; (b) 444.26 K; (O) 477.59 K; ([) 510.93 K. Experimental VLE data:33 (1) 310.93 K; (3) 344.26 K; (2) 377.59 K; (4) 410.93 K. Experimental VLE data:34 (4) 410.95 K.

burg and Scott.35 The figure shows that all of the models under consideration yield the qualitatively correct description of the phase behavior in this system. However, their quantitative results are now substantially different. GPDA is the only model that predicts the experimental data accurately and only slightly overestimates the near-critical data. In contrast, PSRK overestimates the data in the entire pressure-temperature range. LCVM yields an accurate correlation of the data only below 80 bar, and afterward it presents a significant deviation from the experimental data. The accuracy of the prediction of data is strongly dependent on the ability of the model to represent the global phase behavior characteristic for the homologue series. Because GPDA was developed to describe this behavior in the most appropriate manner, it has an obvious superiority over the GE-based models, whose parameters are developed considering only certain kinds of data, namely, VLE. Thus, in contrast to GPDA, an increasing asymmetry of the system usually strongly affects the robustness and reliability of these models and may even lead to the prediction of nonphysical results. This is

exactly what happens in the case of the homologue series under consideration. 3.8. Ethane-Heavy n-Alkanes. The most asymmetric system in the series for which a comprehensive set of data are available is ethane-n-tetracosane. Therefore, it has been selected for a detailed analysis of the results yielded by the models under consideration. First, let us consider the VLE phase diagram, as presented in Figure 7. It can be seen that, although GPDA yields an overall robust prediction, it tends to slightly overestimate the data. Therefore, the results predicted by GPDA seem less accurate than the correlative results of LCVM. However, Figure 7 demonstrates an unusual behavior of this equation. While in the case of the less asymmetric system ethane-nhexadecane LCVM has overestimated the near-critical data, here it underestimates them. The results of PSRK seem even more surprising. Usually this model tends to overestimate the LLE range in asymmetric systems, and indeed here it substantially overestimates the VLE data at low pressures. However, at the same time, it significantly underestimates the near-critical data.


Figure 7. VLE of ethane (1)-n-tetracosane (2). Solid lines: isopleths predicted by GPDA. Dashed lines: isopleths predicted by LCVM. Dot-dot-dashed lines: isopleths predicted by PSRK. Experimental isopleths:9 (3) x(1) ) 0.1192; (9) x(1) ) 0.2358; (0) x(1) ) 0.3924; (b) x(1) ) 0.6079; (O) x(1) ) 0.7839; (2) x(1) ) 0.9484.

Figure 8. Critical lines of ethane (1)-n-tetracosane (2). Solid lines: critical lines predicted by GPDA. Dashed lines: critical lines predicted by LCVM. Dot-dot-dashed lines: critical lines predicted by PSRK. Dotted line: vapor pressure of ethane. 0: critical points of pure compounds. 2: critical points approximated from the data.41

Figure 6. VLE of ethane (1)-n-hexadecane (2). Solid lines: predicted isopleths. Experimental isopleths:36 (b) x(1) ) 0.199; (O) x(1) ) 0.439; (2) x(1) ) 0.606; (4) x(1) ) 0.718; (9) x(1) ) 0.792; (0) x(1) ) 0.875; (1) x(1) ) 0.946.

An explanation of this unusual behavior is provided by Figure 8, which presents the critical curves of the system ethane-n-tetracosane in the P-T projection, predicted by the models under consideration. The figure demonstrates that, although GPDA slightly overestimates the available data points, it predicts the qualitatively correct and robust picture of phase behavior in the system, namely, type IV. In contrast, it can be seen that both PSRK and LCVM predict the nonphysical behavior of the LLE. This behavior resembles the closed loops of LLE (type VI), which may be occasionally reproduced by semiempirical equations because of numerical pitfalls.37 However, this type VI behavior has a basic difference when compared to the one presented in Figure 8. In particular, the phase behavior of type VI can be characterized by two LLE closed-loop parts at low and high pressures, which meet at the hypercritical point. In contrast, the nonphysical phase be-

havior predicted by the GE-based models under consideration does not include any low-pressure LLE closedloop part. The high-pressure LLE closed-loop part meets at the transitional point the crucial locus of VLE. Because the phase diagram generated by PSRK for the system under consideration is very close to such a point, the CPM of VLE has an enormously high value. This nonphysical LLE shape leads to the prediction of nonphysical global phase behavior. In particular, PSRK predicts a small range of limited LLE in the vicinity of the critical point of pure ethane for the system ethane-n-eicosane, which is in qualitative agreement with the experimental data. However, it predicts a continuous shape of the VLE critical curve for the heavier homologues. Such predictions are characteristic also for the recent modifications of PSRK.14 These results can be originated by the numerical pitfalls generated either by the empirical temperature functionalities incorporated by the PSRK and LCVM models.37-39 To detect the part of the model responsible for the prediction of nonphysical results in this particular case, we have replaced the original temperature functionalities of PSRK and LCVM by the temperature functionality of eq 1, which does not generate any numerical pitfalls. The results yielded by the equations

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Figure 9. Solubility of n-tetracosane (2) in supercritical ethane (1). Solid lines: data predicted by GPDA. Dashed lines: data predicted by LCVM. 2: experimental data.40

Figure 10. Solubility of n-hexatriacontane (2) in supercritical ethane (1). Solid lines: data predicted by GPDA. Dashed lines: data predicted by LCVM. Experimental data:42 (0) 308.2 K; (4) 318.2 K; (9) 328.2 K; (2) 333.2K; (b) 348.2 K.

are almost identical with those presented in Figure 8. In other words, this time the GE mixing rules are the ones responsible for the present erroneous predictions. Thus, it can be seen that the nonphysical way in which both PSRK and LCVM predict the LLE in the series under consideration is responsible for underestimation of the near-critical bubble-point data at low temperatures. Thus, the accuracy of the latter model should be recognized as just occasional and local. This conclusion may be verified by considering the dew-point plot (Figure 9), which represents the very important data of the solubility of n-tetracosane in supercritical ethane. It should be realized that that data on supercritical solubility are very difficult to measure, and several pertinent data sets probably present incorrect results.12 In contrast, the data of Gregorowicz40 are consistent with the reliable data sets of Peters et al.9 and du Rand and Nieuwoudt.41 Figure 9 clearly indicates the robust character of the results predicted by GPDA in comparison with LCVM and PSRK. Similar conclusions can be reached considering the solubility data of n-hexatriacontane in supercritical ethane (Figure 10). LCVM continues to predict nonphysical results, namely, the continuous critical line between pure ethane and n-hexatriacontane with another high-pressure immiscibility loop. As a result, LCVM fails to predict the absolute immiscibility that takes place in the system under consideration at low and moderate temperatures. Although the predictions of GPDA are not exact, this model correctly describes

Figure 11. LLVE of ethane (1)-n-tetracosane (2). Solid lines: LLVE lines predicted by GPDA. Dashed lines: LLVE lines predicted by LCVM. 2: experimental data44 for the n-tetracosanerich phase. 4: experimental data44 for the carbon dioxide rich phase.

the patterns of the solubility of n-hexatriacontane in supercritical ethane. Important information may also be learned from the three-phase LLVE diagram (Figure 11). It should be realized that the major part of three-phase equilibria in the homologue series under consideration cannot be determined experimentally because the pertinent UCEPs are located below the melting points. Therefore, only very small parts of LLVE near the critical point of pure ethane have been observed experimentally. Obviously, it is very difficult to expect predictive models to yield precise results in a very small temperature range of just 3-4 K, especially when considering that the LLVE prediction is extremely sensitive to the parameter values. As mentioned above, PSRK predicts a continuous critical line for the system under consideration; hence, the pertinent nonphysical phase diagram does not include any LLVE. In addition, it can be seen that while GPDA underestimates the compositions of the n-tetracosane-rich phase, LCVM overestimates them. However, the most important topological information can be learned from Figure 11b. In particular, it can be seen that the experimental data involve an inversion of the phase densities (the so-called barotropic effect). This means that topologically the experimental system is already located above the mathematical double point and its critical line has a shape of type III.43 However, parts of this critical line are located below the vaporpressure curve of ethane and, therefore, become metastable. As a result, the stable part of the phase diagram exhibits type IV or V behavior. The figure demonstrates that GPDA yields a qualitatively correct prediction of


this behavior, including inversion of the phase densities. Because LCVM describes the LLE in an erroneous manner, it cannot generate the barotropic effect and its results for LLVE are qualitatively incorrect. 4. Conclusions Modern thermodynamic models allow a precise correlation of the experimental data. However, the problem of their reliable prediction without resource to experimental results for particular systems is still open. This is because the properties of real fluids are influenced by many factors, which still cannot be properly treated by entirely theoretical approaches. A possible solution to this problem seems to be extrapolation of the fit of certain systems to others using a semipredictive procedure. It is generally agreed that cubic EOSs combined with Huron-Vidal-type mixing rules, such as PSRK and LCVM, have good predictive ability. These models have many adjustable parameters, which have attached to them an excellent flexibility and correlative ability. In addition, the activities of the UNIFAC model incorporated by the Huron-Vidal-type mixing rules have large parameter matrices, which are believed to contribute to the predictive ability. However, the multidimensional nature of the GE-based models substantially hinders consideration of the complete thermodynamic phase space, which includes VLE in the entire temperature range and LLE, LLV, and critical lines as well. The present study compares the ability of three semipredictive approaches, namely, GPDA, PSRK, and LCVM, for describing the experimental data in the binary homologue series of ethane-n-alkanes. GPDA predicts the data assuming that a basic property such as the balance between VLE and LLE (the relation between the UCEP and CPM) remains approximately constant over the homologue series. Therefore, in contrast to PSRK and LCVM, GPDA requires no more that two to three experimental critical points of one homologue in order to predict the data of the whole series. In the present study, the applicability of GPDA has been extended by the addition of an empirical functionality. It can now predict not only the data of a homologue series to which the reference system belongs but also of other series that include solvents from the same chemical group. Thus, the data of the homologue series of ethane-n-alkanes are predicted here using the values of binary parameters obtained thus far17 for the system methane-n-pentane. In other words, GPDA appears as an entirely predictive model. The results of the present and previous studies13-17 clearly demonstrate a robust predictive character of GPDA. Because GPDA is free of numerical pitfalls, it is reliable in the entire temperature-pressure range and for all types of phase equilibria. The results of this and our previous studies13-17 also show that the way in which the parameters of LCVM and GPDA are evaluated tends to ignore the fact that all regions of the thermodynamic phase space are closely inter-related. In other words, the accuracy of correlating some data, such as VLE, is achieved by the improper description of other data, such as LLE or critical lines. As a result, those models may predict a nonrealistic global phase behavior over the entire homologue series and fail to describe LLE and LLV. Although in the particular case of the series ethanen-alkanes all models under consideration yield rather similar results for the light homologues, both PSRK and

LCVM models generate the nonphysical behavior of LLE for asymmetric systems. In contrast, GPDA predicts robust results even for very asymmetric systems. These facts confirm a doubtless advantage of the approach based on consideration of the compete phase diagram and not just of its separate parts. Hence, GPDA is highly recommended for the prediction of data not available experimentally. Acknowledgment This work was financed by the Israel Science Foundation (Grant 340/00) and by FONDECYT, Santiago, Chile (Project 1020340). Nomenclature a ) cohesion parameter b ) covolume c, d ) attraction density dependence parameters in eq 1 GE ) excess Gibbs energy P ) pressure R ) universal gas constant T ) temperature x ) mole fraction of the lighter compound in the liquid phase y ) mole fraction of the lighter compound in the vapor phase V ) volume Greek Letters δ ) correction factor of the binary parameter l12 ζ ) correction factor of the critical volume ω ) acentric factor Subscripts c ) critical state m ) molar property TP ) triple points Abbreviations CPM ) critical pressure maximum DCEP ) double critical end point EOS ) equation of state GPDA ) global phase diagram approach klGPD ) global phase diagram in the k12-l12 projection LCEP ) lower critical end point LCVM ) linear combination of the Vidal and Michelsen mixing rules LLE ) liquid-liquid equilibria LLVE ) liquid-liquid-vapor equilibria MDP ) mathematical double point PSRK ) predictive Soave-Redlich-Kwong group contribution EOS TCP ) tricritical point UCEP ) upper critical end point VLE ) vapor-liquid equilibria

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Received for review October 19, 2004 Revised manuscript received January 18, 2005 Accepted February 2, 2005 IE048982U

Simultaneous Prediction of the Critical and Subcritical Phase Behavior

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