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Ind. Eng. Chem. Res. 1996, 35, 829-836

829

Cubic Equation of State for Pure Compound Vapor Pressures from the Triple Point to the Critical Point Marcelo S. Zabaloy† and Juan H. Vera* Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7

A new procedure is proposed for obtaining the temperature dependence of the attractive energy parameter of equations of state (EOS’s) . The procedure can be applied to two-parameter cubic and noncubic EOS’s, and it is used for the Peng-Robinson (PR) EOS. The modified form of the PR-EOS, the ZVPR-EOS, can accurately reproduce the vapor pressure for a wide variety of pure compounds from the triple point to the critical point. The experimental information required for each pure compound is the acentric factor, the normal boiling point temperature, and the temperature and pressure coordinates at the critical and triple points. Tables of parameters and complete vapor-pressure relative-error information are provided for more than 100 compounds of industrial interest. Additionally, for the PR-EOS, Soave’s noniterative calculation procedures have been extended to the low-temperature range. Introduction Cubic equations of state (EOS’s) are widely used in process simulation (Mathias and Klotz, 1994). They have long been applied to the representation of properties for pure compounds and mixtures. Well-known EOS’s are the SRK-EOS (Soave, 1972) and the PR-EOS (Peng and Robinson, 1976). Even when the theoretical basis of cubic EOS’s is not strong, they can, in principle, accurately represent the relation among temperature, pressure, and phase compositions in binary and multicomponent systems. This is only possible when the pure compound vapor pressure curves are precisely reproduced. The temperature dependence of the attractive energy parameter of most cubic EOS’s follows the empirical form proposed by Soave (1972). In this work, we take advantage of a universal relation between the saturation temperature and pressure, obtained from cubic EOS’s. This universal relation was first found and correlated by Soave (1986). Adachi (1987) extended Soave’s work to the general cubic EOS, and Sugie et al. (1989) found an analytical relation between the attractive energy parameter of cubic EOS’s and the reduced vapor pressure. Among the cubic EOS’s available in the literature, a modification of the PR-EOS, the PRSV2-EOS, has been found to describe well the pure compound vapor pressure curve (Stryjek and Vera, 1986b). Recently, a new temperature dependence for the Redlich-Kwong EOS (Redlich and Kwong, 1949) has been proposed by Twu et al. (1991). Both of the above modifications of cubic EOS’s require three adjustable parameters per pure compound. As discussed by Mathias and Klotz (1994), this may lead to the need of multiproperty parameter fitting. In this work, we have reduced the number of adjustable parameters by constraining the EOS to exactly reproduce the experimental vapor pressure at some selected temperatures. This does not introduce any additional complexity if the direct calculation procedures proposed by Soave (1986) are applied. One of the fixed temperatures is that of the triple point. Thus, the EOS is forced to properly reproduce the low* To whom correspondence should be addressed. FAX: (514) 398-6678. E-mail: [email protected]. † Permanent address: PLAPIQUI/UNS/CONICET cc 717, 8000 Bahı´a Blanca, Argentina.

0888-5885/96/2635-0829$12.00/0

temperature region. This constraint should be met if the dew points of mixtures containing heavy components are to be successfully calculated. The Peng-Robinson Equation of State According to the PR-EOS (Peng and Robinson, 1976) the pressure of a pure fluid is related to the temperature and molar volume by

P)

a RT v - b v(v + b) + b(v - b)

(1)

In eq 1, R is the universal gas constant and P, T, and v are the absolute pressure, the absolute temperature and the molar (liquid or vapor phase) volume, respectively. The parameters a and b are specific for each compound. It has been conventional to consider the covolume parameter b as independent of temperature while a depends on temperature so as to adequately reproduce the vapor pressure curve of the pure compound. The parameter a is given by

a ) Rac

(2)

where ac is the value of a at the critical isotherm and R is a function of reduced temperature. Throughout the following discussion, all EOS’s satisfying eqs 1 and 2, but differing in the way that the R parameter depends on the reduced temperature, are generically called EOS’s of PR type. Examples of such models are the original Peng-Robinson (1976), the modification proposed by Mathias and Copeman (1983), the PRSV-EOS (Stryjek and Vera, 1986a,c) and the PRSV2-EOS (Stryjek and Vera, 1986b). Equation 1 can be expressed as a relation among dimensionless variables, as follows:

B)

A 1 V - 1 V2 + 2V - 1

(3)

where A, B, and V are dimensionless groups defined in the list of symbols . After introducing the van der Waals constraints at the critical point, we obtain Ac ) 5.877 359, Bc ) 0.077 796 07, and Vc ) 3.951 373. The subscript “c ” denotes the values of the functions evaluated at the © 1996 American Chemical Society

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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

to the huge value of the vapor phase molar volume at saturation. However, there are compounds, such as propane, propylene, and some alcohols and ethers, for which the triple point coordinates correspond to a value of A higher than the maximum shown in Figure 1. Therefore, if the vapor pressure has to be calculated in all the temperature range, it is necessary to have an alternative procedure to relate A to B, in the range of high A values. Relation between A and B in the Range of High A Values As outlined in Appendix A, the judicious simplification of the PR pure compound equilibrium equations, for values of A higher than or equal to the maximum value of Figure 1, results in the following equations for the relation between A and B, at saturation.

Figure 1. Universal relation between A and B, for pure compounds at saturation, generated by the Peng-Robinson EOS.

VL ≈

A - 2 - (A2 - 8A + 8)1/2 2 when

A g 37.393 36 (6)

and

critical point. Thus,

ln B ) RAc A) T/Tc B ) Bc

P/Pc T/Tc

(4)

(5)

where Pc and Tc are the experimental values of the critical pressure and temperature, respectively. Universal Relation between A and B The condition of equilibrium at a fixed temperature requires the equality of the pressure and of the pure compound fugacities of the coexisting phases. For cubic EOS’s, Soave (1986) demonstrated that, for pure saturated compounds, the equilibrium conditions result in a universal relation between the variables A and B. Such a universal relation is shown in Figure 1 for the PR-EOS. On the basis of this universal relation, Soave (1986) proposed a non-iterative method of calculation of pure compound vapor pressures from the values of the R parameter. He considered three cubic EOS’s, namely, the van der Waals, the Redlich-Kwong, and the PR-EOS’s. Soave (1986) also provided equations to perform the reverse calculation, i.e., the computation of R from experimental vapor pressure data. Basically, the direct procedures of Soave arise from either the correlation of B as a function of A, or the correlation of A as a function of B, both performed in such a way that the curve of Figure 1 is accurately reproduced. The range covered in Figure 1 is slightly larger than that considered by Soave (1986). The existence of universal relations is general for any EOS which can be written in a dimensionless form similar to that of eq 3. Thus, direct calculation procedures can be used even with a noncubic EOS, such as the MCSV-EOS (Aly and Ashour, 1994). It can be shown that, for values of A larger than the maximum value shown in Figure 1, the pure component equilibrium problem cannot be quantitatively solved using the PR-EOS. Even with a double precision compiler, it is not possible to fulfill a reasonable relativeerror convergence criterion (Appendix A). This is due

[

(-1) 1 +

VL2 + 2VL - 1 (VL - 1)(2x2)

(

ln

VL + 1 + x2

VL + 1 - x2

)

+

]

ln(VL - 1) (7) where VL is the dimensionless saturated liquid phase volume. From eqs 6 and 7, it is clear that B is an explicit function of A. Therefore, eqs 6 and 7 represent the extension of Soave’s direct calculation of vapor pressures to the low-temperature range, for the PR-type EOS’s. Eqs 6 and 7 can be applied as long as B does not exceed the minimum limit of the compiler used in the computations. The advantage of this procedure is that all approximations are clearly established (Appendix A). We have tested the procedure by using it to calculate the value of B for the highest value of A of Figure 1. Within the precision of the calculation, the result was identical to that coming from the solution of the original problem without simplifications. At larger values of A the calculation results of eqs 6 and 7 become closer to the actual PR-EOS values. In contrast, the conventional iterative procedure of solving the equilibrium system of equations does not converge, when a double precision compiler is used, if the convergence criterion is based on the relative difference between the pressures of the phases. Expressions analogous to eqs 6 and 7 can be obtained for other cubic EOS’s following the procedure explained in Appendix A. Calculation of A as a Function of B in the Range of High A Values The calculation of A as a function of B, at saturation, in the high A range, is important for the regression of values of R from experimental information. This calculation can be done iteratively by adjusting the value of A in successive evaluations of B, through eqs 6 and 7, until the value of B, calculated from eq 5 using experimental information, is recovered. The calculation can also be done by a direct method, as described below. Following Soave (1986), we calculated the values of B as a function of A in a suitable range, using eqs 6

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 831 Table 1. Constants for Eq 8

Table 2. Options for the ZVPR Equation of State

j

Kj

j

Kj

1 2 3 4 5

9.488773 × 10-3 9.101083 × 10-1 -2.782375 × 10-1 4.200238 × 10-1 -2.386644 × 10-1

6 7 8 9 10

-7.021087 × 10-2 -1.077287 × 10-1 4.269334 × 10-1 -2.061501 × 10-1 -3.879711 × 10-2

parameters in eq 14

and 7. Then, we correlated the values of A as a function of B by

KjY ∑ j)1

when

κ1

κ2

a b c

0 *0 (adjustable) *0 (adjustable)

0 0 *0 (adjustable)

we write

[ ]

RAc PrBc ) c1 ln + c2 Tr Tr

10

A ) 200

option

j-1

(11)

or

2 × 10-31 e B e 1.337 843 × 10-9 (8)

where

Y)

(-1) ln B 100

(9)

The values of the constants Kj are reported in Table 1. Equation 8 reproduces the values of A corresponding to the A-B relation as given by eqs 6 and 7, with an average error of 3 × 10-4% (maximum error 1 × 10-3%). The 10 constants of Table 1 are general for the PR-EOS, and they are not related to any experimental datum. The use of these constants avoids iterative calculations. Since the adjustment of parameters is done once for all, we have attempted to provide here the best possible reproduction of the relation stated by eqs 6 and 7. New Expression for the Parameter r When making multicomponent equilibrium calculations, the use in eq 2 of generalized semipredictive expressions of R, like that of the PRSV-EOS (Stryjek and Vera, 1986a,c), is probably the best choice for compounds for which the experimental vapor pressure curve is only partially known. However, for components for which accurate data are available, it is desirable to have better correlations. One of such correlations corresponds to the PRSV2-EOS (Stryjek and Vera, 1986b). In this section we propose a new temperature dependence for the parameter R, with some theoretical basis which makes the new function reliable from the triple point to the critical point. The equation that follows relates the reduced pressure to the reduced temperature of saturated pure compounds. It was obtained by extending an expression given in standard texts (Denbigh, 1971). We assumed a polynomial expansion in temperature for the difference in molar constant-pressure heat capacity, between the saturated vapor and liquid phases. The vaporization enthalpy was then obtained by integration and used with the Clapeyron equation to derive eq 10.

ln Pr ) c1 + c2 ln Tr +

c3 + c4Tr + c5Tr2 + c6Trc7 Tr (10)

In eq 10 all or part of the constants c1 to c7 are compound-specific. As an example, the equation used in the DIPPR compilation (Daubert and Danner, 1989) is a particular case of eq 10. On the other hand, Figure 1 suggests that as a first approximation the relation between ln(B) and A can be considered to be roughly linear (see eqs 4 and 5). Thus,

[

]

RAc Bc ) c1 ln Pr + ln + c2 Tr Tr

(12)

where c1 and c2 are universal constants, and Tr ) T/Tc and Pr ) P/Pc are the reduced temperature and pressure, respectively. Combining eqs 10 and 12, we obtain

R ) C1 + C2Tr ln Tr + C3Tr + C4Tr2 + C5Tr3 + C6Tr(C7+1) (13) where the Ci’s are constants obtained from the combination of the constants in eqs 10 and 12. The functional form of R, eq 13, arises from clear assumptions regarding the temperature dependence of the heat capacity difference of the saturated phases, and from the observation of the form of the relation between A and B in cubic equations of state. This is a new approach, and it provides a semi-theoretical temperature dependence for R. The number of adjustable parameters in eq 13 can be reduced by making the function R exactly reproduce a number of experimental vapor pressure points. In this work we have chosen the critical, boiling, and triple points and the point corresponding to a reduced temperature of 0.7. Additionally, after some preliminary calculations, we set the parameter C7 at a universal value of 11. Thus, eq 13 has at most two adjustable parameters, C5 and C6, which following the more conventional nomenclature, we denote by κ1 and κ2, from here on. With the above choice of fixed points, the following expression for R is obtained:

R ) [F1(Trbp,Trtp)][F2(Tr,Trbp,Trtp,Rtp,Rbp,Rω) + κ1F3(2,Tr,Trbp,Trtp) + κ2F3(11,Tr,Trbp,Trtp)] (14) The functions F1 , F2, and F3 are given in Appendix B. The subscripts bp and tp stand for boiling and triple points, respectively, while subscript ω corresponds to Tr ) 0.7. The symbols Rtp , Rbp, and Rω represent the values of the R parameter which make the PengRobinson EOS exactly reproduce the experimental vapor pressure at the corresponding temperatures. Equation 14 has, at most, two adjustable parameters (κ1 and κ2). Equation 14 is built so that R(Tr ) 1) ) 1, R(Tr ) 0.7) ) Rω, R(Trbp) ) Rbp, and R(Trtp) ) Rtp. The R function defined by eq 14 is not as complicated as it looks. This function is evaluated explicitly at any temperature following the procedure given in Appendix B. The values of R at the fixed points need to be calculated only once, for a given component. Similarly the expressions of dR/dT (required for enthalpy depar-

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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

Table 3. Vapor Pressure Correlation with the ZVPR Equation of Statea option a

option b

option c

AMXD%b

AAD%c

κ1

AMXD%

AAD%

κ1

κ2

AMXD%

AAD%

hydrogen nitrogen oxygen neon chlorine argon bromine iodine ammonia water hydrogen sulfide hydrogen chloride hydrogen iodide carbon monoxide sulfur dioxide methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane cyclohexane neopentane bicyclohexyl ethylene propylene

0.09 0.14 0.60 0.04 0.22 0.31 0.42 0.66 0.12 0.40 0.43 0.04 0.24 0.07 0.32 0.34 0.73 3.50 1.00 1.57 0.42 1.81 1.08 1.23 1.20 1.11 1.11 1.76 1.92 2.70 2.34 2.97 3.44 0.10 0.29 0.64 0.34 2.18

0.04 0.05 0.13 0.01 0.08 0.13 0.14 0.26 0.06 0.12 0.16 0.02 0.09 0.03 0.10 0.13 0.23 0.81 0.29 0.40 0.19 0.49 0.29 0.34 0.32 0.32 0.30 0.48 0.52 0.76 0.66 0.85 0.99 0.04 0.12 0.17 0.13 0.50

-0.1404 -0.2591 0.2331 -0.0684 -0.0955 -0.6386 -0.5870 -1.0006 0.1820 0.3102 -0.7052 -0.0306 -0.4187 -0.0515 -0.1009 -0.5601 0.1593 0.3454 0.2760 0.3377 -0.1732 0.6077 0.5495 0.5683 0.7097 0.6082 0.6975 1.0573 1.2196 1.6376 1.4486 1.7582 2.0528 -0.1924 -0.8672 0.3272 -0.0462 0.2283

0.03 0.03 0.27 0.01 0.27 0.03 0.12 0.05 0.10 0.27 0.03 0.04 0.02 0.07 0.35 0.03 0.46 0.56 0.47 0.51 0.32 0.49 0.41 0.39 0.38 0.35 0.31 0.36 0.39 0.40 0.35 0.33 0.35 0.06 0.03 0.24 0.31 0.53

0.01 0.01 0.12 0.00 0.07 0.01 0.03 0.02 0.03 0.08 0.01 0.01 0.01 0.03 0.09 0.01 0.16 0.20 0.16 0.19 0.11 0.16 0.14 0.12 0.12 0.11 0.10 0.11 0.11 0.11 0.10 0.10 0.10 0.02 0.01 0.08 0.13 0.19

-0.2350 -0.2685 0.6350 -0.0669 0.6293 -0.6522 -0.5761 -1.0154 0.2098 1.0789 -0.7030 -0.0515 -0.4212 -0.2567 -0.1009 -0.5650 0.4853 0.3732 0.7425 0.6833 0.2078 1.1124 1.2135 1.0911 1.3769 1.0696 1.1528 1.4034 1.6037 1.8587 1.5357 1.8278 2.1222 -0.1882 -0.8822 0.6565 0.4993 0.3085

0.001 176 0.000 086 -0.008 786 -0.000 026 -0.009 397 0.000 121 -0.000 239 0.000 161 -0.000 271 -0.008 070 -0.000 044 0.000 194 0.000 025 0.001 551 -0.000 904 0.000 050 -0.010 884 -0.001 139 -0.012 094 -0.008 595 -0.009 242 -0.010 658 -0.011 688 -0.009 385 -0.010 643 -0.007 291 -0.006 704 -0.004 997 -0.005 357 -0.003 126 -0.001 459 -0.001 162 -0.001 134 -0.000 074 0.000 108 -0.005 653 -0.010 760 -0.002 682

0.01 0.03 0.10 0.01 0.09 0.03 0.12 0.05 0.10 0.14 0.03 0.04 0.02 0.07 0.36 0.03 0.11 0.51 0.21 0.30 0.10 0.28 0.22 0.23 0.23 0.20 0.18 0.26 0.28 0.34 0.32 0.31 0.33 0.06 0.03 0.15 0.11 0.49

0.00 0.01 0.04 0.00 0.04 0.01 0.02 0.02 0.02 0.05 0.01 0.01 0.01 0.02 0.08 0.01 0.04 0.18 0.07 0.13 0.04 0.10 0.07 0.07 0.07 0.07 0.07 0.08 0.09 0.10 0.09 0.09 0.10 0.02 0.01 0.05 0.04 0.16

benzene toluene ethylbenzene n-propylbenzene o-xylene m-xylene p-xylene 1,2,3-trimethylbenzene indan styrene biphenyl diphenylmethane naphthalene 1-methylnaphthalene 2-methylnaphthalene phenanthrene

0.43 1.76 2.28 2.83 0.42 0.93 0.32 0.95 1.33 1.13 0.46 0.62 0.30 4.21 0.53 0.30

0.15 0.44 0.55 0.70 0.15 0.25 0.08 0.29 0.36 0.26 0.15 0.18 0.11 1.07 0.27 0.07

-0.8452 0.3726 0.4047 0.3977 0.1557 0.4179 0.3471 -0.0876 0.3833 0.5880 -0.4379 0.3349 -0.4539 1.1277 -0.5790 0.2595

0.09 0.53 0.50 0.51 0.35 0.41 0.19 0.90 0.47 0.44 0.28 0.30 0.24 0.65 0.24 0.12

0.02 0.19 0.16 0.14 0.14 0.14 0.06 0.30 0.16 0.17 0.12 0.10 0.07 0.19 0.10 0.04

-0.8357 0.7822 0.5085 0.4212 0.8210 0.9994 1.1113 1.2269 0.8386 1.2739 0.4953 0.8105 0.6988 1.6211 -0.0711 0.5050

-0.000 171 -0.010 743 -0.002 687 -0.000 753 -0.012 347 -0.011 490 -0.006 925 -0.028 241 -0.010 817 -0.014 318 -0.012 632 -0.008 015 -0.011 451 -0.011 641 -0.008 873 -0.003 184

0.09 0.33 0.59 0.53 0.10 0.20 0.11 0.13 0.28 0.12 0.11 0.17 0.09 0.37 0.10 0.08

0.02 0.11 0.14 0.14 0.03 0.06 0.04 0.05 0.09 0.04 0.04 0.05 0.03 0.14 0.04 0.03

dimethyl ether methyl ethyl ether diethyl ether methyl n-propyl ether methyl isopropyl ether methyl n-butyl ether methyl tert-butyl ether ethyl propyl ether di-n-propyl ether diisopropyl ether

1.73 0.72 0.47 4.65 1.57 0.47 1.60 4.36 1.14 2.71

0.41 0.38 0.17 1.18 0.70 0.14 0.76 1.36 0.28 0.72

0.6168 -0.5308 -0.0832 0.8052 -0.2152 0.0457 -0.5638 -0.8938 0.1494 1.4934

0.56 0.58 0.42 1.09 1.41 0.43 1.24 0.37 0.50 0.57

0.19 0.18 0.15 0.34 0.49 0.12 0.43 0.11 0.12 0.19

1.2093 -0.0079 0.3614 1.1562 0.8054 0.3050 0.7182 -0.8762 0.1719 2.4318

-0.014 630 -0.011 605 -0.011 692 -0.009 300 -0.034 393 -0.006 670 -0.034 425 -0.000 547 -0.000 675 -0.017 277

0.24 0.43 0.16 0.76 0.54 0.51 0.20 0.37 0.52 0.30

0.07 0.12 0.04 0.27 0.08 0.07 0.07 0.10 0.11 0.08

acetone methyl ethyl ketone diethyl ketone 2-pentanone 2-hexanone 3-hexanone 3-heptanone 5-nonanone 3,3-dimethyl-2-butanone

1.41 2.39 0.66 3.56 1.29 0.82 1.68 2.34 1.73

0.35 0.67 0.16 0.98 0.43 0.24 0.51 1.16 0.78

0.6157 1.0525 0.4569 1.5322 -0.3772 0.3841 0.9197 -2.6518 -1.3630

0.46 0.33 0.33 0.57 1.09 0.41 0.88 1.05 1.02

0.15 0.11 0.14 0.18 0.36 0.14 0.29 0.37 0.38

1.1734 1.4169 1.3605 2.1622 1.1926 1.0297 2.7440 -0.0871 0.5293

-0.011 823 -0.007 633 -0.013 996 -0.012 953 -0.034 185 -0.012 234 -0.032 403 -0.042 175 -0.035 852

0.23 0.21 0.09 0.35 0.15 0.20 0.38 0.17 0.18

0.06 0.05 0.03 0.09 0.05 0.06 0.09 0.05 0.06

chemical

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 833 Table 3 (continued) option a

option b

option c

AMXD%b

AAD%c

κ1

AMXD%

AAD%

κ1

κ2

AMXD%

AAD%

4.76 9.43 14.78 8.21 2.91 3.69 7.22 0.67 2.43 2.37 5.22 9.25 9.60 0.60

1.47 2.78 4.33 2.25 0.75 0.95 3.03 0.29 1.19 0.74 1.51 2.53 2.71 0.22

1.9996 2.3562 2.4460 2.4359 0.8133 1.8270 -5.1578 -0.3819 -2.2157 1.5791 2.4561 2.0049 2.0014 -2.1841

0.22 1.05 0.92 0.91 0.68 0.93 2.27 0.53 1.66 1.33 0.60 0.86 0.93 0.07

0.08 0.24 0.41 0.31 0.21 0.28 0.81 0.19 0.55 0.39 0.21 0.26 0.34 0.03

2.1968 2.3592 2.4465 2.4868 1.0174 3.3242 -0.6888 0.8006 2.7677 4.9017 2.7329 2.0336 2.0244 -2.1651

-0.005 049 -0.000172 -0.000 030 -0.002 551 -0.004 186 -0.027 217 -0.084 220 -0.020 986 -0.077 545 -0.043 059 -0.005 064 -0.001 587 -0.001 301 -0.000 247

0.17 1.04 0.92 0.79 0.68 0.50 0.53 0.29 0.31 0.66 0.67 0.79 0.87 0.07

0.05 0.24 0.41 0.31 0.18 0.14 0.17 0.06 0.09 0.27 0.17 0.26 0.34 0.03

acetic acid propionic acid

0.30 2.19

0.13 0.63

0.5894 2.0262

0.16 0.60

0.04 0.23

0.8735 3.6720

-0.002 126 -0.023 359

0.12 0.30

0.04 0.07

n-propylamine isopropylamine n-butylamine tert-butylamine sec-butylamine isobutylamine diethylamine

1.40 1.36 1.02 0.92 1.42 2.32 0.60

0.33 0.31 0.23 0.20 0.51 1.20 0.13

0.8108 0.7589 0.7579 0.7282 -0.2700 -1.6387 0.5076

0.57 0.49 0.46 0.42 1.24 1.30 0.36

0.22 0.19 0.21 0.18 0.42 0.45 0.13

1.7805 1.6235 2.2221 2.1243 0.9557 0.1667 1.9815

-0.019 028 -0.015 746 -0.021 465 -0.019378 -0.033 627 -0.040 102 -0.017 136

0.11 0.16 0.12 0.11 0.19 0.21 0.10

0.04 0.07 0.04 0.04 0.06 0.06 0.04

chloroform methyl chloride 1,1-dichloroethane 1,2-dichloroethane

0.65 0.39 0.33 0.63

0.15 0.13 0.15 0.17

0.4227 -0.4207 0.0760 0.5683

0.20 0.26 0.38 0.27

0.09 0.07 0.13 0.10

0.8343 0.3193 0.4564 1.4451

-0.007 438 -0.009 368 -0.010 084 -0.010 469

0.09 0.10 0.10 0.12

0.03 0.04 0.04 0.04

nitromethane m-nitrotoluene o-nitrotoluene p-nitrotoluene

0.26 2.75 2.16 1.64

0.07 0.88 0.56 0.62

0.0965 2.0368 1.1558 2.2908

0.19 0.26 0.60 0.17

0.08 0.10 0.19 0.07

0.6170 2.3503 2.0263 2.8658

-0.007 541 -0.004 887 -0.015 771 -0.006 608

0.08 0.19 0.34 0.10

0.03 0.06 0.09 0.03

2.36 0.94 1.78 1.83 1.11 0.65 0.45 1.49 5.78 26.29 1.41 1.81

1.20 0.26 0.83 0.54 0.29 0.18 0.18 0.61 1.67 8.91 0.52 0.74

-1.6488 0.1600 -0.9352 0.9441 0.8638 0.6130 0.1048 -0.4628 1.8437 -8.4901 -3.9804 -1.6659

1.29 0.53 1.25 0.28 0.32 0.20 0.43 1.18 0.44 3.60 0.19 0.90

0.45 0.20 0.43 0.10 0.13 0.09 0.17 0.40 0.13 1.39 0.05 0.38

0.1475 0.6034 0.7275 1.1882 1.5952 1.3028 0.8077 0.7115 2.2658 -5.7032 -3.9261 0.4688

-0.039 856 -0.012 382 -0.037 952 -0.004 664 -0.011 143 -0.008 738 -0.014 282 -0.031 687 -0.011 009 -0.060 719 -0.000 812 -0.035 716

0.20 0.26 0.19 0.18 0.15 0.09 0.12 0.19 0.18 1.81 0.19 0.19

0.06 0.10 0.06 0.07 0.05 0.04 0.04 0.06 0.06 0.92 0.05 0.07

chemical methanol ethanol n-propanol n-butanol 1-pentanol 1-hexanol 1-octanol 1-decanol 1-heptadecanol 1-octadecanol isopropyl alcohol isobutanol sec-butanol tert-butyl alcohol

anisole tetrahydrofuran ethyl acetate furfural m-cresol acetonitrile pyridine N-methylpyrrolidine 3-methoxypropionitrile N,N-dimethylformamide hexafluorobenzene benzothiophene

a Temperature range: triple to critical point. b AMXD% ) maximum absolute relative deviation, percent. c AAD% ) average absolute relative deviation, percent.

tures) and of d2R/dT2 (required for heat capacities), need to be obtained only once. The complete set of equations for the evaluation of R and its derivatives can be written in a small program block. These simple steps are justified if the vapor pressure curve, in all the temperature range, needs to be accurately reproduced. The R parameter becomes undefined if Trbp is exactly equal to 0.7 or if Trbp ) Trtp. When R becomes undefined, the ordered pairs (Trbp ,Rbp ), or (Trtp,Rtp), or both, can be exchanged by any other pair corresponding to a known point of the pure compound vapor pressure curve. This exchange may also be used if a particular problem has to be solved in a restricted temperature range. Correlation Results for the Vapor Pressure of Pure Compounds From here on the modification of the Peng-Robinson EOS for which R is given by eq 14 will be denoted as the ZVPR-EOS. The procedure for the calculation of the vapor pressure is divided into two main steps. In step I, the values of R at the fixed points (Rω, Rbp, or Rtp) are evaluated

from the experimental values of the vapor pressure at the corresponding temperatures. This is not done for the critical point since the function R defined by eq 14 meets the condition R ) 1 at Tr ) 1. Step I is carried out only once for each compound. Although the user may use any procedure to obtain Rω, Rbp, and Rtp, we recommend the direct procedures of Soave (1986) coupled with eq 8. This was the option chosen in this work. In step II, the information obtained in step I is used for the calculation of R at the desired temperature T, using eq 14. Then, the vapor pressure can be obtained, in the lower A range, either in the normal iterative way or using Soave’s (1986) direct procedure, and in the higher A range using eqs 6 and 7. In order to assess the performance of the ZVPR-EOS, for a large number of compounds, we have taken as “experimental data” the values of the vapor pressure obtained from the DIPPR compilation (Daubert and Danner, 1989). We have used the information in a standardized way. For each chemical, we have selected as our primary DIPPR information the saturation vapor pressure correlation, the critical temperature, and the triple point temperature. From this information we

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Figure 2. ZVPR-EOS vapor pressure correlation results for n-propylamine from the triple point to the critical point, using one adjustable parameter in eq 14: κ1 ) 0.8108 and κ2 ) 0.

have calculated the “experimental” properties at the fixed points, i.e., the critical pressure, the normal boiling point temperature and the vapor pressure at the triple point and at a reduced temperature of 0.7. We have studied the correlation of vapor pressures of pure compounds in the complete subcritical range using two, one, or no adjustable parameters per pure compound. Table 2 summarizes the different options studied. Table 3 shows the results obtained in the correlation of vapor pressures for more than 100 compounds, covering a wide range of molecular size, shape, and chemical nature. For all the compounds listed in Table 3 the temperature range is from the triple point to the critical point. The order of magnitude for the pressure at the triple point can be as low as 10-4 Pa. Critical pressures can be of the order of 107 Pa (water). For most of the compounds studied, with the exception of alcohols, the average error in vapor pressure, in the temperature range from the triple point to the critical point, is of the order of 1%, when no adjustable parameters are used. Excluding the alcohols, maximum errors are normally less than 5% . If one adjustable parameter (κ1) is used, the average error for most compounds is about 1% or less, while the maximum error for almost every compound decreases to values below 2%. If two adjustable parameters are considered, it is possible to reduce the maximum error to less than 2% for all chemicals, and for most compounds to values smaller than 1% . The parameter κ2 has influence at high reduced temperatures only. Figure 2 shows a sample correlation error curve, for the vapor pressure of an associating compound, by the ZVPR-EOS with one adjustable parameter. Note that the relative error is zero at the four fixed points. Since we have used “experimental data” obtained from the sets of constants of the DIPPR compilation, the resulting vapor pressure values correspond to smoothed data. Thus, larger errors than those reported in Table 3 should be expected when using raw experimental data.

The correlation accuracy for all compounds in Table 3 lies within the accuracy stated by the quality codes in the DIPPR data base even when the one-adjustableparameter option is used. However, in order to demonstrate that the EOS is able to properly reproduce any smooth vapor-pressure-like curve, we also show results for the two-parameter option in Table 3. This option may be used when the accuracy and availability of experimental data justify its application. Certain problems, as the simulation of difficult separation units, where the dominant components have a very similar molecular nature, may require such an accurate description of the pure compound vapor pressure curve. For most practical applications, the use of a single adjustable parameter produces satisfactory results. There is no evident correlation with molecular structure for the ZVPR-EOS adjustable parameters. The cost of an EOS can be measured, as a first approximation, by adding the number of adjustable parameters to the number of vapor pressure data points required as input information. The number of adjustable parameters in the PRSV2-EOS (Stryjek and Vera, 1986b) is 3. This equation requires also the introduction of the critical point coordinates and the acentric factor. Thus, a cost of 5 can be assigned to this equation. The ZVPR-EOS requires the introduction of 4 vapor pressure data points, and the number of adjustable parameters ranges from 0 to 2, depending on the nature of the compound represented. Thus, the cost of the ZVPR-EOS ranges from 4 to 6. However, a cost of 5 is enough for most needs. Therefore, the ZVPR-EOS has a cost similar to the PRSV2-EOS but using more experimental information and a smaller number of adjustable parameters. This feature reduces the probability of having multiple solutions to the problem of parameter fitting, as clearly illustrated by Mathias and Klotz (1994). The technique used here for the generation of eq 14 for the R parameter may be applied to any EOS having the direct-calculation property. If necessary, nonlinear relations can be used instead of the linear form chosen in eq 11. Eq 14 should not be used to obtain values of R at supercritical temperatures. A suitable extrapolation, like that proposed by Mathias (1983), may be applied instead. Although not reported here, predictions of pure compound heats of vaporization, obtained with the ZVPR-EOS, give good results from the triple point to reduced temperatures of about 0.9. When two adjustable parameters are used in the ZVPR-EOS the curve R(T) is very close to the optimum curve corresponding to the Peng-Robinson EOS. The optimum curve is that which exactly reproduces the vapor pressure from the triple point to the critical point. Thus, the ability of the two-parameter ZVPR-EOS to predict the heat of vaporization is limited not by eq 14 but by the form of the Peng-Robinson EOS itself. Conclusions In the present work the range of applicability of Soave’s (1986) direct calculation procedures has been extended to cover the low-pressure range, for any compound, i.e., for the case of a very low value of the triple point pressure, for the PR-EOS. The universal relations associated to the PR-EOS have been used to obtain a new temperature dependence of the parameter R for the PR-EOS. The new expression for the PR- EOS attractive energy parameter, the ZVPREOS, can accurately represent the vapor pressure curve

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 835

of pure compounds, in the complete subcritical temperature range, regardless the polarity, molecular size, or shape of the particular chemical species under study. For a large number of compounds, including some amines and carboxylic acids, the vapor pressure curve can be properly represented, without using any adjustable parameter. If 3% is considered to be an acceptable limit for the maximum error, only the alcohols and no more than 10 other compounds in Table 3 require the use of at least one adjustable parameter. The method used in this work can be easily extended to other cubic and noncubic EOS’s having the direct calculation property. Acknowledgment This work has been accomplished thanks to a scholarship awarded by the “Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas de la Repu´blica Argentina” to one of us (M.Z.). The authors are also grateful to NSERC, Canada, for financial support. Nomenclature a ) Peng-Robinson EOS attractive energy parameter A ) Rac /bRT b ) Peng-Robinson EOS covolume parameter B ) Pb/RT ci ) constants in eq 10 ci ) constants in eqs 11 and 12 Ci ) constants in eq 13 Fi ) functions for the ZVPR-EOS R parameter Kj ) constants in the analytical expression of a PR-EOS universal relation κ1, κ2 ) adjustable parameters P ) absolute pressure q ) dummy variable used in the definition of functions for the R parameter R ) universal gas constant T ) absolute temperature v ) molar volume V ) v/b Greek Letters R ) temperature dependent parameter in the PR-EOS ω ) acentric factor Subscripts

t ) t(V) ) [V2 + 2V - 1]-1

(A-4)

f ) f(V) ) h + ln h

(A-5)

g ) g(V) ) Vt + (1/4)x2 ln [(V + 1 + x2)/(V + 1 - x2)] (A-6) and subscripts L and V correspond to the saturated liquid and vapor, respectively. The variable B is related to A and V (liquid or vapor) by eq 3. The residual value of eq A-1 is a direct measure of the relative difference between the values of B for the two saturated phases. Equation A-1 can be solved for VV if VL is considered to be the independent variable. The residual value of eq A-1 must be much lower than unity. The value of VL varies between 1 and Vc. The value of A is determined by eq A-2 while B comes from eq 3, either with V ) VL or V ) VV. Figure 1 can be generated either in this way or in the way suggested by Soave (1986). For VL lower than 1.03, for example, the solution value VV does not have a satisfactory representation in a double precision compiler. This problem can be overcome by simplifying the above equations. The solution corresponding to the maximum value of A of Figure 1 has an acceptable residual value for eq A-1. This solution is VL ) 1.06, VV ) 7.474 714 × 108, A ) 37.393 36, and B ) 1.337 843 × 10-9. Thus, when VL e 1.06 the value of VV becomes larger than 7.474 714 × 108. These large values make it possible to neglect a number of terms in eqs A-1 and A-2. The resulting equations can be then combined giving the following equivalent system:

ln VV ≈ hLgLtL-1 - fL when VL e 1.06 (A-7) A ≈ hLtL-1 when VL e 1.06

(A-8)

If eq 3 is written for the vapor phase, combined with eq (A-8), and the proper terms are neglected in the result, we obtain

(A-9)

Appendix B. Definition of Functions in Eq 14

F1(Trbp,Trtp) ) [F4(Trbp)F7(1,Trtp,Trbp)]-1

The equations of equality of fugacity coefficients and pressures for the phase equilibrium of a Peng-Robinson pure fluid can be combined to give the following equations:

hV(gL - gV) - tV(fL - fV)

(A-3)

The result of the combination of eqs A-7 and A-9 is eq 7 of the text. Equation A-8 is of second order in VL. The solution is eq 6 of the text.

Appendix A: Derivation of Eqs 6 and 7

-1)0

h ) h(V) ) [V - 1]-1

B ) BV ≈ VV-1 when VL e 1.06

bp ) normal boiling point c ) critical L ) saturated liquid r ) reduced tp ) triple point V ) saturated vapor ω ) corresponding to a reduced temperature of 0.7

hL(gL - gV) - tL(fL - fV)

where

(A-1)

(B-1)

F2(Tr,Trbp,Trtp,Rtp,Rbp,Rω) ) F7(1,Tr,Trbp) [RtpF4(Trbp) F8(Trtp,Trbp,Rbp,Rω)] + F8(Tr,Trbp,Rbp,Rω) F7(1,Trtp,Trbp) (B-2) F3(q,Tr,Trbp,Trtp) ) F7(q,Tr,Trbp) F7(1,Trtp,Trbp) F7(q,Trtp,Trbp) F7(1,Tr,Trbp) (B-3)

and

where q is a dummy variable.

A ) (fL - fV)/(gL - gV)

(A-2)

F4(Tr) ) Tr ln Tr + (7/3)(Tr - 1) ln 0.7

(B-4)

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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

F5(q,Tr) ) (10/3)(0.7q+1 - 1)(Tr - 1) + Trq+1 - 1 (B-5) F6(Tr,Rω) ) (10/3)(1 - Rω)(Tr - 1) + 1

(B-6)

F7(q,Tr,Trbp) ) F5(q,Tr) F4(Trbp) - F5(q,Trbp) F4(Tr) (B-7) F8(Tr,Trbp,Rbp,Rω) ) F6(Tr,Rω) F4(Trbp) + F4(Tr)[Rbp F6(Trbp,Rω)] (B-8) Calculations follow the order F4 to F8 and then F1 to F3. Literature Cited Adachi, Y. Generalization of Soave’s Direct Method to Calculate Pure-Compound Vapor Pressures through Cubic Equations of State. Fluid Phase Equilib. 1987, 35, 309-312. Aly, G.; Ashour, I. A Modified Perturbed Hard-Sphere Equation of State. Fluid Phase Equilib. 1994, 101, 137-156. Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Compounds; Hemisphere Publishing Co.: New York, 1989. Denbigh, K. F. R. S. The Principles of Chemical Equilibrium; Cambridge University Press: Cambridge, UK, 1971; p 202. Mathias, P. M. A Versatile Phase Equilibrium Equation of State. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 385-391. Mathias, P. M. ; Copeman, T. W. Extension of the Peng-Robinson Equation of State to Complex Mixtures: Evaluation of the Various Forms of the Local Composition Concept. Fluid Phase Equilib. 1983, 13, 91-108. Mathias, P. M.; Klotz H. C. Take a Closer Look at Thermodynamic Property Models. Chem. Eng. Prog. 1994, June, 67-75. Peng, D.; Robinson D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59-64.

Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V: An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev. 1949, 44, 233-244. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197-1203. Soave, G. Direct Calculation of Pure-Compound Vapour Pressures through Cubic Equations of State. Fluid Phase Equilib. 1986, 31, 203-207. Stryjek, R.; Vera, J. H. PRSV: An Improved Peng-Robinson Equation of State for Pure Compounds and Mixtures. Can. J. Chem. Eng. 1986a, 64, 323-333. Stryjek, R.; Vera, J. H. PRSV2: A Cubic Equation of State for Accurate Vapor-Liquid Equilibria Calculations. Can. J. Chem. Eng. 1986b, 64, 820-826. Stryjek, R.; Vera, J. H. An Improved Cubic Equation of State. Paper presented at the 189th National Meeting of the American Chemical Society, Miami Beach, Florida, April 28-May 3, 1985. Volume Equations of StatesTheories and Applications; Chao, K. C., Robinson, Robert L., Jr., Eds.; ACS Symposium Series 300; American Chemical Society: Washington, DC, 1986c; pp 560-570. Sugie, H.; Iwahori, Y. ; Lu, B. C.-Y. On the Application of Cubic Equations of State: Analytical Expression for R/Tr and Improved Liquid Density Calculations. Fluid Phase Equilib. 1989, 50, 1-20. Twu, Ch. H.; Bluck, D.; Cunningham, J. R.; Coon, J. E. A Cubic Equation of State with a New Alpha Function and a New Mixing Rule. Fluid Phase Equilib. 1991, 69, 33-50.

Received for review May 19, 1995 Revised manuscript received October 17, 1995 Accepted November 6, 1995X IE950306S

X Abstract published in Advance ACS Abstracts, January 15, 1996.