Ind. Eng. Chem. Res. 2002, 41, 931-937
931
An Extension of CEOS/AE Zero-Pressure Mixing Rules for an Optimum Two-Parameter Cubic Equation of State Chorng H. Twu,*,† Wayne D. Sim,‡ and Vince Tassone‡ Hyprotech Ltd., 2811 Loganberry Court, Fullerton, California 92835, and 707 8th Avenue SW, Suite 800, Calgary AB T2P 1H5, Canada
Cubic equations of state are widely used in refinery and petroleum reservoir industries for the prediction of fluid-phase behavior. Two of the most well-known cubic equations of state broadly accepted in industry are the SRK and PR equations. However, neither the SRK nor the PR equation of state yields the best results for the prediction of liquid densities of polar components and heavy hydrocarbons. An approach is proposed to locate an optimum two-parameter cubic equation of state. A methodology is also proposed to modify Twu’s CEOS/AE zero-pressure mixing rules to extend the range of application of these mixing rules. The mixing rule developed in this work is incorporated into our new cubic equation of state for the prediction of phase equilibria of highly nonideal chemical mixtures. Introduction Since van der Waals cubic equation of state (CEOS) appeared more than a century ago,1 many different cubic equations of state have been proposed to improve the van der Waals (vdW) equation for the computation of the liquid volumes of hydrocarbons, reservoir-fluid systems, or polar components. A well-known improvement of the van der Waals equation is the RedlichKwong (RK) equation.2 Since its development, there have been numerous attempts to improve RK equation by changing the alpha function formulation, by modifying the volume function or by improving the mixing rules. Although numerous equations of state have been published and promising new equations keep appearing, the Soave-Redlich-Kwong (SRK) and Peng- Robinson (PR) equations proposed by Soave3 and Peng and Robinson,4 respectively, are still two of the most recognized modifications of the RK equation. Because of the simplicity and accuracy in predicting K values, the SRK and PR equations of state are broadly used in the refinery and petroleum reservoir industries for the successful prediction of phase behavior. Because the SRK and PR cubic equations of state predict similar K values, the significant difference between them lies in the precision of predicting the liquid density. The SRK equation represents methane PVT behavior closely. The PR equation improves the prediction of the liquid density for C6 hydrocarbons. As a consequence, the predictions of the liquid enthalpy from the SRK and PR equations differ from one hydrocarbon to another. It is worth mentioning that an equation of state representing PVT behavior well results in better accuracy of the predicted enthalpy departure. Although the SRK equation gives good results for the prediction of liquid density for light hydrocarbons and the PR equation better predicts liquid density for midrange hydrocarbons both * Correspondingauthor.E-mail:
[email protected]. Telephone: (403) 520-6000. Fax: (403) 520-0606. † Fullerton, CA. ‡ Calgary, Canada.
equations predict liquid densities poorly for heavy hydrocarbons and polar systems. Currently, cubic equations of state have been applied for the prediction of phase equilibria of polymer solutions. The equation-of-state parameters for polymers are generally obtained by fitting polymer density data. Because of the poor prediction of liquid density for heavy hydrocarbons from the SRK and PR equations, neither equation of state is quite appropriate for use as a polymer equation of state. In addition, the recent development of combining cubic equation of states with excess Helmholtz or Gibbs energy models has advanced the cubic equation of state as an accurate method for correlating and predicting the phase-equilibrium behavior of highly polar systems. Despite these modern advances in CEOS/AE mixing rules for polar systems, the prediction of liquid volumes from the SRK or the PR cubic equation of state still remains a weak point, especially for polar components. In this paper, we propose a methodology for finding an optimum twoparameter cubic equation of state that provides better liquid density predictions than the SRK and PR equations for heavier hydrocarbons and polar components. Our new cubic equation of state, along with the PR and SRK equations, eventually covers the entire range of hydrocarbons and polar systems. The CEOS/AE zero-pressure mixing rules recently developed by Twu et al.5-8 have been shown to reproduce accurately the incorporated excess Gibbs free energy, as well as the liquid activity coefficients of any activity model without requiring any additional binary interaction parameters. The approach of the zeropressure mixing rule requires the value of the zeropressure liquid volume at the system temperature for each of the pure components and for the mixture. However, at high temperatures, it might become impossible to find a zero-pressure liquid volume from an equation of state for one or more of the pure components. We present a methodology to overcome this obstacle and to extend the usefulness and application range of a zero-pressure mixing rule. The mixing rule developed in this work is incorporated into our new
10.1021/ie0101588 CCC: $22.00 © 2002 American Chemical Society Published on Web 07/14/2001
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
cubic equation of state for the prediction of phase equilibria of highly nonideal polar mixtures. An Optimum Two-Parameter Cubic Equation of State The SRK and PR equations of state predict constant values for the critical compressibility factor, Zc, equal to 0.3333 and 0.3074, respectively, for all substances. However, for most real substances, the value of Zc is below 0.3 and in the range of 0.2-0.3. For example, the Zc value for methane is 0.289, and that for water is 0.23. The Zc values predicted by the PR equation is apparently closer to those of real fluids than are the values predicted by the SRK equation. For this reason, when a volume estimation is required, the PR equation is often preferred to the SRK equation. Although there is no cubic equation of state that can be singled out as the most accurate for the prediction of all properties under all conditions, we propose a methodology for locating a two-parameter cubic equation of state associated with an optimum value of Zc. A general two-parameter cubic equation of state can be represented by the equation
P)
RT a v - b v2 + mbv + nb2
(1)
where P is the pressure, T is the absolute temperature, v is the molar volume and R is the gas constant. The coefficients m and n are the equation-of-state-dependent constants used to represent a particular two-parameter cubic equation of state (for the SRK equation, m ) 1 and n ) 0, and for the PR equation, m ) 2 and n ) -1). The location of all possible cubic equations of state on an m-n diagram has been shown by Twu et al.9 Schmidt and Wenzel10 drew a straight line starting from the SRK equation (1, 0), which represents the value of m ) 1 and n ) 0 on an m-n diagram, and passing through the points for the PR (2, -1) and Harmens11 (3, -2) equations to form a three-parameter cubic equation of state. They proposed that the relationship between m and n be m + n ) 1, which was then incorporated into eq 1 in the volume function (v2 + mbv + nb2) for a better description of liquid molar volume. Their line represents a linear relationship between m and n on the m-n plot with a negative slope of -1. Twu et al.9 found that this line did not yield the best representation of liquid densities for hydrocarbons but that it best describes the saturated liquid density for polar components when the value of m is greater than 2. After analyzing all possible equations on the m-n plot, Twu concluded that the line m - n ) 4 having a positive slope of +1 on the m-n diagram predicts liquid densities best for hydrocarbons. The m and n relationships proposed by Schmidt and Wenzel10 and Twu et al.9 both result in more accurate predictions of the liquid density than any two-parameter cubic equations of state. However, they require the liquid density data in advance to find the third parameter on their lines for each component in the system before they can start any phase-equilibrium calculations. This requirement, unfortunately, is not only inconvenient in the application of their equations, but also can create inconsistencies in the regressed Zc values from the liquid density data for the components. Furthermore, their K value predictions are no better than those of the two-parameter
cubic equations of state. As a result of these drawbacks, the use of the m-n relationship in a cubic equation of state, which results in a three-parameter cubic equation of state, never gained popularity in industrial applications. Therefore, any cubic equation of state with more than two parameters is not considered in this work. Finding an optimal value of Zc, instead of using the m-n relationship, for a two-parameter cubic equation of state is one of our aims in this paper. The word “optimal” in this work is used to refer to the best overall description of the liquid densities for hydrocarbons and polar components obtained from an equation of state. Although it is impossible to find any two-parameter cubic equation of state representing the liquid densities better than other equations for all components, there does exist an optimal point on the m-n diagram. As mentioned, the line m + n ) 1 describes the molar liquid volume best for polar components, whereas the line m - n ) 4 represents hydrocarbons best. Because the lines have different slopes and, so, are not parallel to each other, they cross at a point (2.5, -1.5) or m ) 2.5 and n ) -1.5 on the m-n diagram. We found that the point (2.5, -1.5) gives the best overall liquid density predictions, especially for heavy hydrocarbons and polar components. It is worth mentioning that Twu9 found that neither the van der Waals equation (0, 0) nor the SRK equation (1, 0) is the best for spherical molecules such as argon or methane; rather, he found that the optimal equation of state in this case is located at the point (2, -2). The optimal point (2.5, -1.5) is found between the line segment formed by the points (3, -2) and (2, -1) with equal distance to the vertexes of the triangle formed by (2, -2), (2, -1), and (3, -2). The vertexes of the triangle correspond to the spherical, PR, and Harmens equations of state, respectively. The SRK equation is not close to the optimal equation, and the vdW equation is even farther. The optimum two-parameter cubic equation of state is therefore proposed to be
P)
RT a v - b v2 + 2.5bv - 1.5b2
(2)
Equation 2 can be rewritten in another form as
P)
RT a v - b (v + 3b)(v - 0.5b)
(3)
The values of a and b at the critical temperature are found by setting the first and second derivatives of the pressure with respect to volume equal to zero at the critical point, resulting in
ac ) 0.470 507R2Tc2/Pc
(4)
bc ) 0.074 0740RTc/Pc
(5)
Zc ) 0.296 296
(6)
where the subscript c denotes the critical point. The fixed constants at the critical point for the SRK and PR equations and for our new equation of state, which is labeled TST (Twu-Sim-Tassone), are listed in Table 1 for comparison. The values of Zc from the SRK and PR equations are both larger than 0.3, whereas that
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 933 Table 1. Comparison of the Critical Constants among the SRK, PR, and TST Equations of State
Table 3. Average Absolute Deviation Percent (AAD%) for the Prediction of Vapor Pressures from the Triple Point to the Critical Point from the SRK, PR, and TST Cubic Equations of State
CEOS
a*c
b*c
ZC
SRK PR TST
0.427 480 200 0.457 235 533 0.470 507 548
0.086 640 357 0.077 796 073 0.074 074 073
0.333 333 343 0.307 401 319 0.296 296 307
Table 2. L, M, and N Parameters of the Twu r Function with the TST Cubic Equation of State component 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 n-nonadecane eicosane cyclohexane benzene acetone methanol ethanol water
Tc (K)
Pc (bar)
190.564 45.99 305.32 48.72 369.83 42.48 425.12 37.96 469.70 33.70 507.60 30.25 540.20 27.40 568.70 24.90 594.60 22.90 617.70 21.10 639.00 19.50 658.00 18.20 675.00 16.80 693.00 15.70 708.00 14.80 723.00 14.00 736.00 13.40 747.00 12.70 758.00 12.10 768.00 11.60 553.58 40.73 562.16 48.98 508.20 47.01 512.64 80.97 513.92 61.48 647.13 220.55
L
M
N
0.573 159 0.602 469 0.856 729 0.512 768 0.379 003 0.116 525 0.655 021 0.480 608 0.476 586 0.434 215 0.350 720 0.347 166 0.532 094 0.751 165 0.991 001 0.692 934 0.644 891 0.672 729 0.481 264 0.374 821 0.163 372 0.080 4681 0.794 606 0.525 320 2.741 74 0.434 175
0.982 462 0.900 708 0.937 274 0.846 110 0.817 571 0.861 045 0.829 163 0.809 353 0.796 538 0.800 647 0.791 588 0.796 276 0.784 911 0.790 331 0.837 872 0.780 273 0.787 812 0.790 676 0.781 396 0.799 312 0.818 375 0.861 052 0.925 926 0.873 161 6.356 11 0.871 528
0.600 000 0.726 375 0.617 916 1.030 32 1.373 01 3.249 60 1.121 34 1.510 03 1.590 17 1.831 41 2.161 04 2.328 98 1.816 25 1.448 54 1.225 59 1.654 16 1.856 94 1.876 55 2.454 87 3.146 76 1.948 56 3.011 48 0.936 024 1.986 50 0.098 454 0 1.661 08
from the TST equation is slightly below it and is closest to the true value. The parameter a is a function of temperature. The value of a(T) at temperatures other than the critical temperature can be calculated from
a(T) ) R(T) ac
(7)
where the alpha function, R(T), is a function only of reduced temperature, Tr ) T/Tc. We use the Twu alpha function correlation12 NM)
R(T) ) TrN(M-1) eL(1-Tr
(8)
Equation 8 has three parameters, L, M, and N. These parameters are unique to each component and are determined from the regression of pure-component vapor pressure data. The L, M, and N parameters used to obtain the correct pure-component vapor pressures for all pure components used in this paper are listed in Table 2 for use with our TST equation. The results of predicting vapor pressures and saturated liquid densities from the new cubic equation of state are shown in Tables 3 and 4, respectively. Also listed in the tables are the values predicted by the SRK and PR equations of state for comparison. The results in Table 3 show that the three cubic equations of state all predict very accurate vapor pressures from the triple point to the critical point for all pure components. This high accuracy comes from the use of a proper alpha function in these equations of state because the ability to predict vapor pressures using any cubic equation of state is controlled by the selection of an appropriate R function. It is worth pointing out that the new TST
AAD% comp
SRK
PR
TST
C1 C2 C3 NC4 NC5 NC6 NC7 NC8 NC9 NC10 NC11 NC12 NC13 NC14 NC15 NC16 NC17 NC18 NC19 NC20 DMK MeOH EtOH H2O
0.48 0.72 0.52 0.58 0.56 0.64 0.44 0.46 0.30 0.46 0.48 0.42 0.33 0.24 0.14 0.18 0.19 0.37 0.47 0.60 0.08 0.14 0.67 0.24
0.27 0.25 0.25 0.29 0.37 0.53 0.23 0.31 0.14 0.31 0.31 0.26 0.14 0.05 0.15 0.11 0.07 0.15 0.24 0.42 0.12 0.29 0.58 0.13
0.17 0.18 0.19 0.21 0.28 0.48 0.16 0.24 0.07 0.23 0.23 0.18 0.05 0.08 0.24 0.21 0.17 0.14 0.14 0.35 0.21 0.41 0.78 0.09
Table 4. Average Absolute Deviation Percent (AAD%) for the Prediction of Saturated Liquid Densities from the SRK, PR, and TST Cubic Equations of State AAD% saturated liquid density (Tr < 0.7)
saturated liquid density (to Tr ) 1.0)
comp
SRK
PR
TST
SRK
PR
TST
C1 C2 C3 NC4 NC5 NC6 NC7 NC8 NC9 NC10 NC11 NC12 NC13 NC14 NC15 NC16 NC17 NC18 NC19 NC20 DMK MEOH ETOH H2O
0.98 4.20 6.31 7.44 9.50 11.28 12.96 14.30 15.66 16.72 18.31 19.02 20.99 22.94 23.10 23.74 23.93 25.21 25.11 25.37 22.76 25.10 16.84 25.80
11.96 8.04 5.52 4.32 1.98 0.88 1.96 3.44 5.02 6.18 8.02 8.80 11.05 13.25 13.44 14.17 13.39 15.85 15.75 16.08 12.96 15.71 6.34 16.31
18.37 11.80 11.36 10.12 7.60 5.52 3.47 1.99 1.42 1.50 2.93 3.75 6.14 8.47 8.67 9.44 9.69 11.23 11.13 11.49 8.12 11.07 1.16 3.64
7.00 6.77 8.15 9.59 11.18 12.95 14.32 15.80 17.95 17.73 19.08 19.68 21.15 23.90 22.96 23.20 23.70 25.15 25.60 25.60 23.73 26.93 20.09 28.66
10.23 6.91 5.17 4.50 2.90 2.46 3.37 5.03 7.51 7.22 8.75 9.42 11.10 14.22 13.13 13.42 13.99 15.55 16.14 16.15 13.97 17.64 9.79 19.51
15.04 14.09 10.00 8.91 6.94 5.37 4.93 3.07 2.91 2.71 3.63 4.33 6.12 9.42 8.25 8.58 9.17 10.79 11.45 11.47 9.14 13.04 4.69 6.81
equation has the quality of predicting the most accurate vapor pressures (the PR prediction is the next most accurate, and the SRK prediction is the least accurate). Because the accurate prediction of vapor pressure is a prerequisite for accurate K values, from this point of view, the PR equation is better than the SRK equation, and the TST equation is better than the PR equation in phase-equilibrium calculations. As mentioned previously, the major difference among cubic equations of state is the accuracy of liquid density
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
predictions. The prediction of volumetric properties, however, is controlled by the volume function. Table 4 demonstrates that the accuracy of predicting volumetric properties differs for the different equations and also changes from one component to the next in a systemic manner. Two sets of AAD% values are given in Table 4 to show the ability of cubic equations of state to predict saturated liquid densities at both low and high temperatures. The SRK equation with (v2 + bv) better predicts the liquid density for methane, the PR equation with (v2 + 2bv - b2) is best for n-pentane to n-heptane, and the TST equation with (v2 + 2.5bv - 1.5b2) is superior for n-octane and higher carbon numbers, as well as for polar components. Consequently, these three cubic equations of state, the SRK, PR, and TST equations, cover the entire range of hydrocarbons and polar systems. Because the TST equation of state is better for the prediction of liquid density of heavy hydrocarbons and polar components, the TST equation from this point of view is more suitable than the SRK and PR equations in applications to polymer solutions when polar solvents are used.
If eq 9 is applied at zero pressure, an equation containing the liquid volume is obtained by substituting eq 12 into eq 9
[( )( )] [ ( ) (
E / AE0 A0vdw v0vdw - 1 bvdw ) ln / RT RT b v0 - 1
-
)]
/ a/vdw v/0 + w v0vdw +w a* 1 - / ln / ln / (w - u) b* v0 + u bvdw v0vdw + u
AE0 and v/0 ) v0/b are the excess Helmholtz energy and reduced liquid volume at zero pressure, respectively. As mentioned previously, the subscript vdw denotes that the properties are evaluated from the cubic equation of state using the van der Waals mixing rule for the a and b parameters. The zero-pressure volume is obtained from eq 13 by setting the pressure equal to zero and selecting the smallest root
v/0 )
{(
1 a* a* 2 -u-w - u+w2 b* b*
) [(
)
(
4 uw + Zero-Pressure CEOS/AE Mixing Rules The excess Helmholtz free energy AE can be expressed in terms of its departure function, ∆A, with respect to a van der Waals fluid mixture as
AE - AEvdw ) ∆A - ∆Avdw
(9)
The subscript vdw in AEvdw and ∆Avdw indicates that the properties are evaluated from the cubic equation of state using the van der Waals mixing rule for the a and b parameters without any binary interaction parameters
avdw ) bvdw )
∑i ∑j xixjxaiaj
[
(10)
]
1
∑i ∑j xixj 2(bi + bj)
(11)
An analytical expression for ∆A is needed for the derivation of the mixing rule. A general two-parameter cubic equation of state provides the function
Z + wb* a* ∆A 1 ) -ln(Z - b*) ln RT b* Z + ub* (w - u)
(
)
(12)
In eq 12, u and w are equation-of-state-dependent constants used to represent a general two-parameter cubic equation of state
P)
RT a v - b (v + ub)(v + wb)
(13)
For our TST equation of state, u is 3, and w is -0.5 (for the SRK equation, u ) 1 and w ) 0, and for the PR equation, u ) 2.4141 and w ) -0.4142). The parameters a* and b* in eq 12 are defined as
a* ) Pa/R2T2
(14)
b* ) Pb/RT
(15)
(16)
)] } (17)
a* b*
1/2
Equation 17 has a root as long as
a* g (2 + u + w) + 2x(u + 1)(w + 1) b*
(18)
Equations 16 and 17 represent an exact model for a CEOS/AE mixing rule. However, because the equationof-state parameter a*/b* and the zero-pressure liquid volume are interrelated by eqs 16 and 17, the exact model does not permit explicit solution of eq 16 for a*/b*, so an iterative technique is required for the solution. If eqs 16 and 17 are used to solve for a*/b*, the resulting new mixing rule will give an exact match between the excess Helmholtz free energy of the equation of state at zero pressure and that of the incorporated excess Gibbs free energy model. Nevertheless, the nonexplicit nature of the expression for the mixing rule becomes cumbersome in the evaluation of thermodynamic properties such as fugacity coefficients. This paper presents a methodology for overcoming this obstacle to obtaining an explicit expression for this new mixing rule. In this paper, the reduced liquid volume at zero pressure, v/0, is assumed to be equal to that of the van / , thus allowing for the explicit der Waals fluid, v0vdw expression for the mixing rule / v/0 ) v0vdw
(19)
Using eqs 10 and 11 for the parameters a and b in eq / can readily be calculated from that equation. 17, v0vdw Equation 16 can then be simplified to
( ) [
]
E / AE0 A0vdw bvdw a* avdw ) ln - / + C v0 RT RT b b* b vdw
with the density function Cv0 being
(
)
/ v0vdw +w 1 C v0 ) ln / (w - u) v0vdw + u
(20)
(21)
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 935 Table 5. NRTL Interaction Parameters and Results of the Prediction in Terms of Average Absolute Deviation Percentage (AAD%) for Activity Coefficients, Bubble Point Pressure, and K Values mixing rule
γ1 (%)
TST(1) TST(2)
a
γ2 (%)
P (%)
K1 (%)
K2 (%)
ethanol (1)/n-heptane (2) from 30.12 to 70.02 °C; I/2e/377, 379; I/2c/457, 458a A12 ) 521.746, A21 ) 727.003, R12 ) 0.4598 at 30.12 °C 0.08 0.04 1.28 1.43 0.20 0.08 1.40 1.47
2.42 2.44
TST(1) TST(2)
ethanol (1)/water (2) from 24.99 to 120 °C; I/1b/93, 106, 107, 108a A12 ) 13.3878, A21 ) 437.683, R12 ) 0.2945 at 24.99 °C 0.21 0.14 4.17 7.10 0.35 1.48 3.69 6.32
5.22 4.08
TST(1) TST(2)
methanol (1)/cyclohexane (2) from 25 to 55 °C; I/2a/242; I/2c/208, 209a A12 ) 644.886, A21 ) 784.966, R12 ) 0.4231 at 25 °C 0.15 0.11 1.32 1.84 0.16 0.13 1.37 1.84
2.81 2.81
TST(1) TST(2)
methanol (1)/benzene (2) from 25 to 90 °C; I/2c/188; I/2a/207, 210, 216, 217, 228a A12 ) 441.228, A21 ) 738.702, R12 ) 0.5139 at 25 °C 0.13 0.10 3.17 5.91 0.11 0.09 3.19 5.91
3.93 3.93
TST(1) TST(2)
acetone (1)/benzene (2) from 25 to 45 °C; I/3+4/194, 203, 208a A12 ) -35.4443, A21 ) 193.289, R12 ) 0.3029 at 25 °C 0.01 0.01 0.86 1.05 0.26 0.27 0.91 1.09
1.77 1.78
TST(1) TST(2)
acetone (1)/ethanol (2) from 32 to 48 °C; I/2a/323, 324, 325a A12 ) 24.3880, A21 ) 224.395, R12 ) 0.3007 at 32 °C 0.01 0.01 1.21 0.94 0.80 1.00 1.32 1.42
1.82 2.45
TST(1) TST(2)
acetone (1)/methanol (2) from 45 to 55 °C; I/2a/75, 80, 81a A12 ) 31.5237, A21 ) 180.554, R12 ) 0.3004 at 45 °C 0.00 0.00 0.48 0.35 0.62 0.47
0.84 0.76
0.92 0.93
TST(1) TST(2)
ethanol (1)/benzene (2) from 25 to 55 °C; I/2a/398, 407, 415, 417, 418, 421, 422a A12 ) 115.954, A21 ) 584.473, R12 ) 0.2904 at 25 °C 0.12 0.10 1.03 3.25 0.12 0.13 1.00 3.25
2.58 2.56
TST(1) TST(2)
methanol (1)/water (2) from 24.99 to 100 °C; I/1b/29; I/1/41, 49, 72, 73a A12 ) -23.1150, A21 ) 188.147, R12 ) 0.3022 at 24.99 °C 0.13 0.11 2.78 4.71 0.86 1.93 2.41 4.07
4.91 3.83
TST(1) TST(2)
methanol (1)/n-hexane (2) from 25 to 45 °C; I/2c/219; I/2a/252a A12 ) 823.172, A21 ) 848.519, R12 ) 0.4388 at 25 °C 0.09 0.04 1.75 1.29 0.94 0.29 2.09 1.36
1.16 1.22
Data taken from DECHEMA Chemistry Data Series by Gmehling, Onken, and Arlt; numbers correspond to volume/part/page.
Equation 20 gives the mixing rule explicitly for the equation-of-state parameter a in terms of AE0 at zero pressure as
a* ) b*
{
a/vdw b/vdw
( )]}
[
E E A0vdw bvdw 1 A0 - ln + Cv0 RT RT b
(22)
E A0vdw in eq 22 can be calculated from the following equation derived from the cubic equation of state:
E A0vdw
RT 1
)
[( [ ( ) ∑i xi ln a/vdw
(w - u) b/vdw
ln
/ v0i
-1
)( )]
/ v0vdw -1 / v0vdw + w / v0vdw +u
bi
bvdw
-
∑i xi
b/i
/ / ) v0vdw v0i
(24)
E Using eq 24 in eq 23, A0vdw can then be simplified to
a/i
calculations are always lighter than others, and eq 17 might not have real root for these lighter pure components, which occurs when the inequality in eq 18 is not satisfied. When this is the case, some sort of extrapola/ must be made. To omit the need for tion for v0i / values from extrapolation of the pure-component v0i the equation of state, we assume further that the zero/ , is pressure liquid volume of the pure components, v0i / also equal to that of the van der Waals mixture, v0vdw
ln
( )] / v0i
E A0vdw
+w
/ v0i +u
(23)
Equation 17 is used to calculate the reduced liquid / ) volume at zero pressure for both the mixture (v0vdw / and the pure components (v0i) to be used in eq 23. However, some of the pure components in the VLE
RT
)
( ) bi
∑i xi ln b
vdw
1 (w - u)
-
(
ln
)(
/ v0vdw + w a/vdw / v0vdw + u b/vdw
-
∑i xi
)
a/i b/i
(25)
Equation 22 is the mixing rule for parameter a. We can force the mixing rule for parameter b to satisfy the
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
quadratic composition dependence of the second virial coefficient. However, it has been observed that any temperature dependence whatsoever in the covolume term of van der Waals-type equations will result in anomalies in the predicted thermodynamic properties of fluids at extremely high pressures.13 Therefore, the temperature-independent linear mixing rule for the b parameter is used in this work
b)
[
1
]
∑i ∑j xixj 2(bi + bj)
(26)
Results We have chosen here the alpha function form of Twu12 for the TST cubic equation of state and the NRTL activity model for use in our mixing rules. Because AE0 in eq 22 is at zero pressure, its value is identical to the excess Gibbs free energy, GE, at zero pressure. Therefore, any activity model such as the NRTL equation can be used directly for the excess Helmholtz free energy expression AE0 in the equation. The NRTL parameters, Aij, Aji, and Rij, obtained from the lowest temperature of each binary system reported in the DECHEMA Chemistry Data Series, are used directly in the mixing rule models. These values of Aij, Aji, and Rij are then used in the mixing rules at all temperatures, where temperature T is in Kelvin. The values of the NRTL binary interaction parameters for the binary systems used in this study are given in Table 5. We considered two cases for 10 highly nonideal binary mixtures, which are traditionally described by liquid activity models and are listed in Table 5. In the first case, for VLE predictions, the assumption of eq 19 is used. The second case uses the assumptions of eqs 19 and 24. We use TST(1) to refer to our mixing rules assuming only eq 19 and TST(2) to refer to our mixing rules assuming both eqs 19 and 24. The accuracies of reproducing the activity coefficients of component i, γi, expressed in terms of average absolute deviation percentage (AAD%) from the incorporated GE model using the two different mixing rules in the TST equation, are given in Table 5. Similarly, the accuracies of the VLE predictions from the two different mixing rules, expressed in terms of AAD% in bubble-point pressure and K values of components 1 and 2, are also presented in Table 5 for the TST equation. Examining the accuracy of reproducing the activity coefficients, as given in Table 5, the TST equation of state produces accurate results for both methods. The errors in the reproduction of the activity coefficients for these systems are minimal for these two mixing rules. The ability to match the GE value derived from the equation of state with that from the incorporated GE model validates the assumption made behind our mixing rules for the liquid density. Table 5 also compares the VLE predictions for these two mixing rules. For the VLE predictions, both mixing rules give excellent agreement between the experimental data and the predictions over a wide range of temperatures and pressures using only the information in the GE model. The comparisons verify that there is little difference in the accuracy of the predictions from these two mixing rules. This indicates that the assumption of eq 24 has a negligible effect on phase-equilibrium predictions, but that it has the advantage of simplifying eq 23 to eq 25
and, more importantly, eliminating the need for ex/ values. trapolation of the pure-component v0i Conclusion An optimal two-parameter cubic equation of state has been found to allow better prediction of liquid densities for heavy hydrocarbons and polar components. The new cubic equation of state is also slightly better in predicting vapor pressures than the SRK and PR equations. Because of these features, the new equation of state is more attractive for applying to refinery systems containing pseudocomponents, strongly nonideal polar systems, and polymer solutions in phase-equilibrium calculations. We have successfully developed a CEOS/ AE mixing rule that simplifies the mixing rule and eliminates the need for extrapolation of the purecomponent liquid volumes from the equation of state. We demonstrated that our mixing rule incorporated into the TST equation of state reproduces the activity coefficients of the GE model with which it is associated and accurately predicts vapor-liquid equilibria. Nomenclature a, b ) cubic-equation-of-state parameters a*, b* ) reduced forms of parameters a and b A ) Helmholtz free energy Aij, Aji ) NRTL binary interaction parameters in Kelvin Cv0 ) zero-pressure function defined in eq 21 G ) Gibbs free energy Ki ) K value of component i defined as yi/xi L, M, N ) parameters in the Twu R function m, n ) cubic-equation-of-state constants P ) pressure R ) gas constant T ) temperature u, w ) cubic-equation-of-state-dependent constants v ) molar volume v/0 ) reduced zero-pressure liquid volume xi ) liquid mole fraction of component i yi ) vapor mole fraction of component i Z ) compressibility factor Greek Letters R ) cubic-equation-of-state alpha function Rij ) NRTL binary interaction parameters φi ) fugacity coefficient of component i in the mixture γi ) activity coefficient of component i ∆ ) departure function Subscripts 0 ) zero pressure ∞ ) infinite pressure c ) critical property i, j ) property of component i, j ij ) interaction property between components i and j vdw ) van der Waals Superscripts * ) reduced property E ) excess property
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Received for review February 19, 2001 Revised manuscript received April 6, 2001 Accepted April 30, 2001 IE0101588
