Ind. Eng. Chem. Res. 1998, 37, 2957-2965
2957
1. Comparison of 12 Equations of State with Respect to Gas-Extraction Processes: Reproduction of Pure-Component Properties When Enforcing the Correct Critical Temperature and Pressure Oliver Pfohl,*,† Tim Giese, Ralf Dohrn,† and Gerd Brunner TU Hamburg-Harburg, Arbeitsbereich Thermische Verfahrenstechnik, 21071 Hamburg, Germany
Modeling gas-extraction processes carried out at temperatures and pressures slightly above the critical temperature and pressure of the supercritical solvent requires equations of state (EOS) that correctly reproduce the critical temperature and pressure of the solvent. EOS that are based on statistical mechanics instead of on simple van der Waals theory are known to be superior at high densities, as encountered in gas-extraction processes. But most of these EOS overpredict the critical temperature and pressure of pure compounds significantly, if conventional purecomponent parameter determination methods are applied. Here, a universal pure-component parameter determination method is proposed that enforces the correct reproduction of a pure compound’s critical temperature and pressure. This guarantees qualitatively correct calculation results for mixtures in the region important for gas-extraction processes. This method reduces the number of adjustable pure-component parameters by two, compared to conventional methods, and enables the determination of unique parameter sets that describe a component best. Based on this method, the abilities of 12 EOS to reproduce experimentally determined densities and vapor pressures of 6 nonpolar compounds are compared. 1. Introduction Gas extraction is a separation process which is increasingly used in industry. Modeling phase equilibria in gas-extraction processes is the key to reduce expensive and time-consuming high-pressure experiments carried out before designing plants for gasextraction processes. However, calculating phase equilibria requires a suitable equation of state. In this study, a comparison of different EOS gives information about their principal suitabilities to model phase equilibria related to gas-extraction processes. These are not investigated by most authors presenting new EOS because this requires different testing methods than for conventional phase equilibria. On one hand, there are cubic EOS that are very often based on the treatment of intermolecular forces as proposed by van der Waals (1873), although van der Waals already pointed out that his treatment is not appropriate at high densities (Randzio and Deiters, 1995). Cubic EOS can be solved for the volume analytically. Further, equations that allow the determination of the pure-component EOS parameters from the critical temperature and critical pressure to be calculated, named Tc′ and pc′ in this study, are easily derived from the critical isotherm’s point of inflection at the critical point for cubic EOS. The commonly used way to determine the pure-component EOS parameters is to set Tc′ and pc′ equal to the experimentally determined values Tc and pc which are tabulated for most lowboiling compounds. This enforces the correct calculation of the high-temperature limit of the vapor-pressure curve. * Corresponding author. Fax: +49/40/7718-4072. Telephone: +49/40/7718-3040. E-mail:
[email protected]. † Present address: Bayer AG, Zentrale Technik-Technische Entwicklung, Geb. B310, 51368 Leverkusen, Germany.
On the other hand, many EOS are based on the treatment of hard spheres according to Carnahan and Starling (1969), which was derived from basic concepts of statistical mechanics with the aim to model thermodynamic properties with physically meaningful parameters describing the structure and energetic interactions of the real molecules. These EOS should allow safer extrapolations to conditions not investigated experimentally due to their theoretically founded background. Especially at higher densities, the advantageous use of EOS incorporating the hard-sphere term by Carnahan and Starling is evident: the repulsive term RT/(v - b) according to van der Waals becomes negative and leads to meaningless results. Since most of these EOS are complex, it is not always possible to obtain analytical expressions for the determination of the pure-component parameters based on Tc′ and pc′. This is not necessarily a shortcoming, because many EOS are developed for calculations with polymers or other high-boiling compounds, where no critical data are available so that Tc′ and pc′ could not be derived from Tc and pc anyway. Second, calculations with these EOS are often performed far away from the pure compounds’ critical temperatures, not enforcing the necessity of a correct reproduction from the engineering point of view. Purecomponent EOS parameters are determined in a way that deviations between the calculated and experimentally determined saturation pressures and liquid densities at saturation conditions are then minimized. A good reproduction of liquid densities at saturation conditions in general also leads to a good prediction of densities at higher pressures. One disadvantage of this procedure is that it often leads to a significant overestimation of Tc and pc, i.e., Tc′ . Tc and pc′ . pc, because the slope of the saturation pressure curve vs temperature is correct but not its length (Figure 1).
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2958 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 1. Adjustable Parameters for Different EOS parameter
Figure 1. P-F diagram and vapor-pressure curve of carbon dioxide. Points: experimental data. Splines: calculations with SAFT and the parameter set by Huang and Radosz (1990). This set does not enforce the correct reproduction of the experimentally determined critical temperature, Tc, and pressure, pc: Tc′ and pc′ are the calculated propertiesssignificantly overestimating Tc and pc.
EOS
no. 1
no. 2
no. 3
PHCT Deiters Pfennig BACK SAFT PHSCT all others
q/k, K a, K T*, K u0/k, K u0/k, K /k, K Tc′, K
rv0, cm3/mol b, cm3/mol v0, cm3/mol v00, cm3/mol v00, cm3/mol NAπσ3/6, cm3/mol pc′, MPa
c c c R r r ω′
Consequently, a pure-component EOS parameter determination method enforcing at least the reproduction of the experimentally determined critical temperature of the supercritical solvent has to be applied, if calculations of phase equilibria for gas extractions near the critical point of the solvent are to be carried out. In this paper, such a method of pure-component EOS parameter determination is proposed. Further, the description of pure-component phase behavior of 6 nonpolar compounds by 12 different EOS is compared using this parameter determination method for all EOS, guaranteeing a common basis for an objective comparison. 2. Equations of State Investigated
Figure 2. Pxy diagram for the system carbon dioxide (1) + n-butane (2) at 319.3 K, a temperature where pure carbon dioxide is supercritical (Tc ) 304.1 K). Points: experimental data from Hsu et al. (1985). Splines: calculations with SAFT and parameters from Huang and Radosz ((1990) vdW mixing rule with k12 ) 0.13) and the PR EOS (quadratic mixing rule with k12 ) 0.11). The overprediction of the critical temperature using SAFT leads to a calculated vapor pressure of 9.0 MPa and a qualitatively wrong phase diagram.
The overestimation of Tc and pc of the supercritical solvent also leads to erroneous phase splitting in mixtures at temperatures and pressures slightly above the critical point of the solvent. Conventional separation processes are carried out far away from this region so that this overestimation does not affect the calculation results. However, gas-extraction processes are preferably carried out in this region, because the compressibility of the fluid is very high there, allowing a significant change of solvent density and solvating power when the pressure is slightly changed. Figure 2 shows an example for a systematic error caused by such overestimation: using SAFT with pure-component parameters determined not enforcing Tc ) 304.1 K (Huang and Radosz, 1990) results in Tc′ ) 320.7 K so that, according to SAFT, carbon dioxide still shows a vaporliquid transition at T ) 319.3 K, where Hsu et al. (1985) investigated the binary system carbon dioxide + nbutane.
This section very briefly lists all EOS which are compared in this work. Each of the EOS needs exactly three numerical inputs for each nonpolar pure component, except BACK and SAFT which need four. This common denominator for the selection of the EOS to be compared tries to do all EOS justicesalthough their ages and preferred fields of application are quite different. No approaches have been tested where additional pure-component parameters are used to finetune the performance of the EOS as often applied to cubic EOS where the so-called R functions are modified in order to improve vapor-pressure calculations and adjustable volume translations are added in order to improve the liquid density calculations. Well-known approaches are those by Mathias (1983), Stryjek and Vera (1986), Mathias et al. (1989), Melhem et al. (1989), and Twu et al. (1992), enforcing up to five additional parameters. Also, no EOS with terms for near-critical corrections are tested, because these terms also enforce additional EOS parameters (compare Kiselev, 1997). Table 1 lists the required pure-component input information for the different EOS. Cubic EOSs are as follows: (a) SRK EOS (SoaveRedlich-Kwong EOS (Soave, 1972)), (b) SRK-VT EOS (volume-translated SRK EOS (Peneloux et al., 1982)), (c) PR EOS (Peng-Robinson EOS (Peng and Robinson, 1976)), (d) 3P1T EOS (Yu and Lu, 1987), and (e) SPHCT (simplified perturbed hard-chain theory (Elliot et al., 1990)). Peneloux et al. shifted the volume of the well-known SRK EOS by a constant value in order to calculate the liquid volumes more accurately. Their “RKSc3” model needs no fourth parameter in addition to Tc′, pc′ and ω′, in contrast to other proposed volume translations, which are therefore not included in this study. The 3P1T EOS is a result of an intensive evaluation of the capabilities of 14 different cubic EOS to represent 8 pure-component properties of the first 10 members of n-alkanes (Yu et al., 1986). Anderko (1989a) used the 3P1T EOS as a physical part for an EOS incorporating chemical association. Incorporation of association in-
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2959
creases the interest in an EOS here, because associating modifiers such as ethanol are often added to nonpolar supercritical solvents in order to enhance their capability to dissolve polar compounds. SPHCT is a simple semiempirical EOS accounting for nonspherical shape by introducing a shape factor c according to the ideas of Prigogine (1957). Prigogine’s theory treats densitydependent degrees of freedom as equivalent translational ones. Elliot et al. also proposed an extension to the associating components. The authors of all the above EOS supply equations to calculate the purecomponent EOS parameters from Tc′ and pc′ and the acentric factor ω′ (Pitzer et al., 1955). They can be set equal to the experimental values Tc, pc, and ωsor they can be used as adjustable pure-component parameters, like in this study. The EOS with reference term based on Carnahan and Starling (1969) are as follows: (a) DP EOS (DohrnPrausnitz EOS (Dohrn and Prausnitz, 1990a)), (b) PHCT (perturbed hard-chain theory (Beret and Prausnitz, 1975)), (c) Deiters’ EOS (Deiters, 1982), (d) Pfennig’s EOS (Pfennig, 1988), (e) BACK EOS (Boublik-Alder-Chen-Kreglewski EOS (Chen and Kreglewski, 1977)), (f) SAFT (statistical associating fluid theory (Chapman et al., 1989, 1990; Huang and Radosz, 1990, 1991, 1993), and (g) PHSCT (perturbed hardsphere chain theory (Song et al., 1994)). The DP EOS incorporates a modified van der Waals perturbation term, designed to predict the critical isotherms of the nonpolar pure compounds accurately. PHCT is a combination of the perturbed hard-sphere theory for small molecules according to Carnahan and Starling and Prigogine’s (1957) theory for chain molecules. Pfennig and Deiters modified this EOS. Deiters paid special attention to an accurate reproduction of the critical density, which is of interest for gas-extraction processes. The BACK EOS uses the reference term for convex bodies by Boublik (1975), which reduces to the Carnahan-Starling reference term if the shape parameter R is unity. It uses the temperature dependence of the core size according to Barker and Henderson (1967) and the power series dispersion term by Alder et al. (1972) with refitted constants. In the present study, the fourth parameter of BACK, e/k, is set equal to the noncentral energy for inert liquids, 0.6ωTc, as proposed by Chen and Kreglewski. Compared to the other EOS with only three parameters, the fourth parameter is an advantage, but when setting it equal 0.6ωTc, the advantage is smaller than fine-tuning this parameter for each compound separately. SAFT has been developed to predict phase equilibria incorporating associating compounds and systems, where molecules with totally different sizes are present like in polymer + solvent systems. Such EOS can become necessary when working with gas extractions of high-boiling macromolecules from natural sources or phase equilibria incorporating supercritical fluids and polymers (Bungert et al., 1997). The physical term for the nonassociating reference fluid is identical to the BACK EOS, except that SAFT accounts for the nonspherical shape by modeling molecules as chains of r covalently bonded spheres instead of allowing the spheres to become convex bodies through R. Huang and Radosz proposed setting the fourth purecomponent parameter e/k to 10 K for most components except CO2 (40 K), CH4 (1 K), and Ar (0 K). PHSCT was developed for systems with polymers and copolymers and was only included in the test after obtaining
strange test results for SAFT (section 6) in order to compare the physical term of SAFT with PHSCT, because both theories model molecules as chains of covalently bonded spheres. 3. Compounds Investigated in This Study The different EOS have been tested to reproduce the pure-component pvT behavior of six nonpolar compounds. Testing with argon (data by Angus and Armstrong (1971), data range: 0.56 e T/Tc e 0.97) and methane (Setzmann and Wagner (1991), 0.51 e T/Tc e 0.98) shows how well the properties of simple spherical molecules are reproduced. Carbon dioxide (Span and Wagner (1996), 0.71 e T/Tc e 0.97), ethane (Sychev et al. (1987), 0.49 e T/Tc e 0.98), and propane (Sychev et al. (1990), 0.51 e T/Tc e 0.97) are commonly used supercritical solvents. n-Hexane (Smith and Srivastava (1986), 0.48 e T/Tc e 0.97) has been tested in order to verify the ability to model molecules with shapes more different from spheres. Carbon dioxide has a dipole moment of zero, like all other compounds investigated, but a significant quadrupole moment (Kay and Kreglewski, 1983) and is weakly self-associating (Anderko, 1989b). 4. Parameter Determination Based on pSAT and vLIQ Only If no analytical equations exist to calculate EOS purecomponent parameters based on Tc′ and pc′, then these parameters are normally determined by an optimization procedure, minimizing the deviations between the calculated and experimentally determined vapor pressures and densities. This procedure often leads to an overestimation of Tc and pc, with the consequences described above. This section shows the extent of overestimation using the different EOS and the pure-component parameter determination method. The function to be minimized (eq 1) is the same as used in section 6.
deviation )
x ( x ( n
1
1
2
n i)1
∑
)
SAT pSAT exp (Ti) - pEOS(Ti)
pSAT exp (Ti)
n
1
1
2
n i)1
∑
2
+
)
LIQ vLIQ exp (Ti) - vEOS(Ti)
vLIQ exp (Ti)
2
(1)
Table 2 lists the best parameter sets, the values of the first and second roots of eq 1, eq 1, and the overestimations Tc′ - Tc and pc′ - pc for carbon dioxide and propane. The PR and 3P1T EOS manage to predict Tc within 0.5-1 K, while the SRK EOS and SPHCT overestimate Tc by ∼6 K. The SRK-VT EOS is only slightly better. Using the DP, Deiters, Pfennig, or BACK EOS with the Carnahan-Starling term, the overestimations are ∼3 K. With PHCT, SAFT, and PHSCT, huge overestimations result: Tc′ - Tc ) 1020 K, pc′ - pc > 1 MPa. Extremely low deviations for carbon dioxide using the DP or Deiters EOS are not representative: both model carbon dioxide behavior far better than the others’ behavior, which is to some extent due to the incorporation of carbon dioxide data in the development of these two EOS.
2960 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 2. Optimization Results Not Enforcing Tc and pc parameter EOS
no. 1
no. 2
no. 3
Tc′ - Tc, K
SRK SRK-VT PR 3P1T DP SPHCT PHCT Deiters Pfennig BACK SAFT PHSCT
310.11 307.55 304.78 305.21 303.94 310.03 322.65 325.15 256.22 286.58 150.03 119.27
8.282 7.846 7.463 7.593 7.424 8.215 17.705 20.408 12.714 19.706 6.9104 3.290
0.194 0.204 0.215 0.210 0.234 0.195 1.516 1.195 1.721 1.066 2.495 4.445
5.98 3.42 0.65 1.08 -0.19 5.90 13.2 -0.08 4.66 0.58 13.3 10.0
SRK SRK-VT PR 3P1T DP SPHCT PHCT Deiters Pfennig BACK SAFT PHSCT
376.98 375.02 369.55 370.43 376.79 375.85 371.15 324.44 326.23 352.34 193.00 187.85
4.669 4.492 4.193 4.327 4.438 4.589 38.080 43.919 28.657 42.824 13.457 13.429
0.121 0.123 0.153 0.152 0.073 0.132 1.410 1.106 1.481 1.035 2.696 2.712
7.13 5.14 -0.3 0.58 6.94 6.00 14.8 4.35 4.96 2.28 18.2 16.8
pc′ - pc, MPa
dev, %
dev(pSAT), %
dev(vLIQ), %
0.91 0.47 0.09 0.22 0.05 0.84 2.52 0.02 0.51 -0.14 1.91 1.34
0.94 1.96 2.36 2.00 1.56 1.91 1.51 0.26 2.48 0.87 0.93 1.36
0.11 0.13 0.32 0.69 0.14 0.16 0.89 0.26 0.46 1.57 0.84 0.73
1.76 3.80 4.40 3.31 2.99 3.66 2.12 0.27 4.50 0.18 1.02 1.99
0.42 0.24 -0.06 0.08 0.19 0.34 1.48 0.18 0.23 0.12 0.98 0.86
1.62 2.41 3.23 2.28 2.91 3.09 2.64 2.22 2.65 1.22 2.32 2.01
0.93 0.96 1.07 1.12 5.21 1.10 2.85 3.43 1.14 2.14 2.50 2.31
2.31 3.86 5.40 3.44 0.62 5.08 2.43 1.01 4.16 0.30 2.15 1.72
CO2
C3H8
5. Parameter Determination, Enforcing Tc and pc A simple iterative pure-component EOS parameter determination method that enforces the correct reproduction of Tc and pc and at the same time minimizes the deviations of the calculated vapor pressures and liquid densities from experimental values is proposed in this section. For a realistic initial guess, c0, for the shape parameter (third parameter in Table 1), one pair of size and energy parameters, (σ,)crit(c0), exists, allowing the exact reproduction of Tc and pc. (σ,)crit(c0) is easily found with an initial guess of (σ,) and subsequently modifying and σ in such a way that the differences |Tc′-Tc| and |pc′-pc| decrease. After obtaining (σ,)crit(c0) for c0, the parameter c is changed by subsequently adding small increments, ∆c, and finding (σ,)crit,c(i+1) for the new value, ci+1 ) ci + ∆c. (σ,)crit,c(i) is used as the initial guess. For each triple (σ,,c)crit, the deviation of experimentally determined vapor pressures and liquid densities (eq 1) is calculated and plotted against c, yielding the optimum of (σ,,c)crit where the calculated deviation is minimal. The calculation of Tc′ and pc′ for EOS with no analytical equations that allow the EOS parameter determination based on Tc′ and pc′ is an indirect method: starting with a vapor-pressure calculation at low temperature, the temperature is subsequently increased by 1 K and the vapor pressures are calculated. Initial guesses for the vapor-pressure calculations are obtained as linear extrapolations based on the last two calculations (ln(Psat) vs 1/T). The increase in temperature is canceled and the step width is reduced if a vapor-pressure calculation did not converge, if the routine to determine v(T,p,x) by Topliss (1985) sets the error flag, or if the vapor-pressure calculation results were wrong: vLIQ(Ti) < vLIQ(Ti-1), vVAP(Ti) > vVAP(Ti-1), 1/vLIQ - 1/vVAP < 10-6 kmol/m3. Tests with EOS where
Tc′ and pc′ can be calculated with analytical equations alternatively resulted in determinations of Tc′ within 0.01 K. 6. EOS Comparison, Enforcing Tc and pc Based on the parameter determination method described in section 5, the reproduction of the purecomponent behavior of the six compounds from section 3 with the EOS from section 2 is compared. The deviation plots (Figure 3) show the deviations of the calculated vapor pressures and liquid volumes from the experimentally observed values dependent on the shape parameter, when enforcing the correct reproduction of Tc and pc. An EOS performs well for a given compound if the minima for saturation pressure deviation and liquid density deviation occur at the same value of the shape parameter. In this case, one set of purecomponent parameters allows the optimum reproduction of both properties and leads to a small value of eq 1. Using the same EOS for the six different components, all plots look very similar. Propane gives a typical plot for each EOS. Plots for the SRK EOS show a minimum for the saturation pressure deviations at ∼1% near ω′ ) ω, while the liquid volumes there are off by ∼7% (argon, methane) to 17% (n-hexane). This is caused by the kind of incorporation of the acentric factor in the EOS, which intends to allow a good prediction of vapor pressures but not of densities. The best reproduction of the liquid volumes is at extremely high ω′ and confirms that Soave’s modification of the RK EOS is an engineeringlike method to obtain correct saturation pressures. De Santis et al. (1976) already pointed out that a correct calculation of vapor pressures, i.e., phase equilibria, with an EOS which yields wrong volumes is only due to a balancing of errors because the calculation of fugacities requires an integration over the volume. The SRK-VT EOS gives identical vapor pressures as the SRK EOS but shifts the minima of the liquid density
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2961
Figure 3. Deviation plots for propane (if not stated differently). Using the 3P1T EOS with Anderko’s association part for methanol results in a better deviation plot, than using the 3P1T EOS alone. The best pure-component reproduction using SAFT is always achieved with r ∼ 1 if the correct reproduction of Tc and pc is enforced.
deviations very close to the vapor-pressure minima. The liquid density deviations remain high, because the liquid densities are only described accurately at T < 0.7Tc but not at T > 0.7Tc. Plots for the PR EOS show saturation pressure deviation minima below 1% at ω′ ) ω for the same reasons as discussed above, but the liquid volumes here are only off between 5% (carbon dioxide) and 10% (argon). The best density reproduction with ω′ ∼ ω is achieved for carbon dioxide only. Saturation pressures are also described best with the 3P1T EOS using ω′ ) ω: the minima are lower than 1.5% (propane). In contrast to the PR and SRK EOS, the best density reproductions (3-6%) are also close to ω′ ) ω. For n-alkanes, this was not unexpected because Yu and Lu (1987) used the reproduction of n-alkane behavior to develop their EOS. But tests carried out with carbon dioxide, oxygen (Wagner and de Reuck, 1987), nitrogen (Angus et al., 1979), and propylene (Angus et al., 1980) give even better results for eq 1. This improvement is attributed to the thorough investigation of cubic EOS by Yu et al. (1986) resulting in
three parameters (a, b, c) instead of only two (PR, SRK: a, b). Deviation plots for SPHCT have vapor-pressure deviation minima of ∼2% near ω′ ) ω but with the liquid volumes off by ∼10%. Plots for the DP EOS show that the best liquid density reproduction (∼3-5%) is very close to ω′ ) ω, while the saturation pressures there are off by 10-15% for argon, methane, ethane, and propane. The best saturation pressure reproductions for these compounds are at ω′ , ωsand high by 6-12%. This confirms Dohrn and Prausnitz’s conclusions that their EOS represents liquid densities better than the PR EOS but that it is not superior for vapor pressures. The diagrams for carbon dioxide and n-hexane are better: at ω′ ) ω, saturation pressures are off by only ∼1% and liquid volumes by ∼3%. Using PHCT from 1975, the least useful results are obtained when enforcing Tc and pc. The minima for saturation pressure and liquid volume are at totally different values of c for all compounds. This leads to errors in the calculation of liquid densities of 30% if the
2962 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
saturation pressure is accurate within 10%. For no compound does a parameter set exist that enforces Tc and pc and simultaneously allows a sufficient reproduction of liquid volumes and saturation pressures. The minimum saturation pressure and liquid density deviations for Pfennig’s EOS occur at identical values of c for all fluids yielding one parameter set, allowing both the best possible vapor pressure and liquid density representation. The deviation in saturation pressures (volumes) rises with increasing molecular weight and nonspherical shape from 0(5)% for argon and methane to 3(11)% for hexane. Deiters’ EOS allows an extremely accurate description of argon and carbon dioxide: the minima for vapor pressure and liquid density appear at the same value of c and ∼0.5%. For n-alkanes, the optimum for the saturation pressures rises from 2% for methane to 10% for hexane. The reproductions of the liquid volumes are off by 3% for methane and 5% for the others at these points. Deviation plots with SAFT are surprising. To understand them, a deviation plot for methane is also shown: the saturation pressure deviation curve has two minima, with the left one near the optimum density reproduction and at a chain length of r ∼ 1 (sphere)sit therefore is the physically reasonable one. The deviation plots for the other compounds look similar: one saturation pressure minimum for propane on the right side is at r ) 2.6 (if e/k ) 0.6ωTc and r ) 3.8 if e/k ) 10 K), but from the methane plot, it has to be concluded that the left minimum belongs to the desired EOS parameter setsat a value of r ∼ 0.9. Less than one sphere making up molecules such as ethane, propane, n-hexane, and carbon dioxide is not realistic, and theory enforces values of at least r ) 1. The plots stop at r ∼ 0.9, because iterations did not converge with too low values of rsbut the result is obvious: SAFT does not allow the correct reproduction of Tc and pc with physically meaningful parameters. The EOS finds the way back to a chain length of r ∼ 1 when Tc and pc are enforced, i.e., back to the BACK equation. Setting e/k ) 0.6ωTc heresas recommended by Chen and Kreglewski (1977) for the BACK EOSsreduces the vaporpressure deviation significantly compared to e/k ) 10 K used by Huang and Radosz (1990). PHSCT had been investigated in order to find if the assumption of freely joined tangent spheres, covalently bonded forming molecules, leads to the results obtained for SAFT. The plots for PHSCT look completely different from those for SAFT, disproving this assumption. The vapor-pressure deviations have minima between 3% (argon) and 8% (n-hexane), with liquid volume reproductions off by 15-20%. The minima for the liquid volume deviations are at much higher r, not allowing the selection of one parameter set accurately reproducing both vapor pressure and liquid volumes. The Lennard-Jones SAFT recently proposed by Kraska and Gubbins (1996) gives deviation plots (not shown here) which look similar to those of PHSCT. A comparison of BACK and SAFT, which are very similar equations, reveals that R used for convex shape in the BACK EOS yields much better results than the chain length r in SAFT when Tc and pc are enforced: the minima of the vapor pressures and liquid density deviations occur at similar values of R. In cases where the minima occur at slightly different values of R, one deviation is ∼1% and the other 3-4%. In addition to
Table 3. Optimization Results Enforcing Tc and pc parameter no. 2 no. 3
dev, %
dev(pSAT), dev(vLIQ), % %
EOS
no. 1
SRK SRK-VT PR 3P1T DP SPHCT PHCT Deiters Pfennig BACK SAFT PHSCT
304.13 304.13 304.13 304.13 304.13 304.13 328.02 324.12 244.32 283.76 311.49 101.71
7.377 7.377 7.377 7.377 7.377 7.377 22.083 20.512 12.869 19.645 26.036 2.526
CO2 0.227 7.31 0.227 4.31 0.220 2.48 0.213 2.73 0.227 1.68 0.232 7.33 1.643 17.9 1.194 0.50 1.914 5.61 1.058 2.31 0.8426 4.08 5.942 10.1
0.52 0.53 0.29 0.89 0.59 0.87 0.91 0.41 1.56 4.28 2.85 1.96
14.1 8.08 4.67 4.57 2.77 13.8 35.0 0.60 9.67 0.33 5.32 18.3
SRK SRK-VT PR 3P1T DP SPHCT PHCT Deiters Pfennig BACK SAFTa SAFTb PHSCT
369.85 369.85 369.85 369.85 369.85 369.85 371.96 334.90 315.84 354.75 444.31 386.47 149.14
4.248 4.248 4.248 4.248 4.248 4.248 48.371 44.888 29.219 42.460 57.012 53.816 9.470
C3H8 0.149 6.47 0.149 4.57 0.154 3.37 0.152 2.60 0.131 4.51 0.155 6.06 1.493 18.5 1.125 3.92 1.571 4.84 1.048 1.64 0.7903 9.24 0.8576 3.08 3.907 11.3
1.64 1.64 0.98 1.41 5.76 1.51 3.14 4.51 1.78 2.58 15.9 2.25 5.50
11.3 7.50 5.76 3.78 3.26 10.6 33.9 3.33 7.90 0.70 2.61 3.91 17.1
a
e/k ) 10 K. b e/k ) 0.6ωTc ) 33.95 K.
this good result, the pure-component property reproduction can be further improved by fine-tuning the fourth parameter, e/k, so that the minima of both curves appear at identical values of R and both deviations decrease to ∼1% (Pfohl and Brunner, 1998). n-Hexane is an exception: the minimum of eq 1 occurs at a liquid volume deviation of ∼1% and vapor pressures are off by ∼8%, indicating that the convex body model is a bad reference system for large molecules. Table 3 lists the pure-component parameter sets that enable the smallest values of eq 1 and the values themselves for carbon dioxide and propane when enforcing the correct reproduction of Tc and pc. Using PHCT, PHSCT, SPHCT, and the SRK EOS, liquid volume deviations > 10% occur. With the SRK-VT EOS and Pfennig’s EOS, the representations of liquid volumes are better than 10%, but the volumes are still extremely off at 1/3 < p/pc < 3. The EOS by Deiters and DP manage to model carbon dioxide far better (eq 1: ∼1%) than the other compounds so that the values achieved for propane (∼4%) are more representative. The best representations (2-3%) are achieved using the 3P1T, PR, and BACK EOS. 7. Conclusions The newest EOS with van der Waals reference term (3P1T) and the BACK EOS have been shown to reproduce the pure-component pvT behavior of commonly used nonpolar supercritical fluids best, when using parameters which enforce the correct reproduction of Tc and pc. Both EOS offer parameter sets that allow the optimum reproduction of both vapor pressures and liquid volumes. Deiters’, Pfennig’s, and the DP EOS do similarly well as the PR and SRK-VT EOSsalthough the relatively low pressure (
