Ind. Eng. Chem. Res. 2006, 45, 1829-1839
1829
Volumetric Property Improvement for the Soave-Redlich-Kwong Equation of State Hong Lin, Yuan-Yuan Duan,* Tao Zhang, and Zhi-Min Huang Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua UniVersity, Beijing 100084, People’s Republic of China
This work presents a temperature-dependent volume translation to improve the volume properties of the SoaveRedlich-Kwong equation of state (SRK EOS) for pure fluids and mixtures. The volume translation is generalized as a function of the critical parameters and reduced temperature. The SRK EOS with the generalized volume translation can accurately represent the densities for different polar and nonpolar fluids in the saturated and supercritical region, including alkanes, olefins, alkines, cycloolefins, aromatics, alcohols, ethers, ketones, halogenated hydrocarbons, and inorganic molecules. The average relative deviation is 1.73% for pure fluids in the saturated region. The generalized volume translation SRK EOS (VTSRK EOS) has better accuracy than other methods and EOSs in the saturated and supercritical regions. Introduction The Soave-Redlich-Kwong (SRK) equation of state (EOS)1 is one of the van der Waals-type cubic EOSs that have been widely used in the chemical industry. However, the predicted constant critical compressibility factor for all fluids and the prediction error for the saturated liquid density are two inherent limitations of these EOSs. A volume translation method was proposed by Martin2 and Pe´neloux et al.3 for the cubic EOS to improve the SRK EOS for representing the critical compressibility factor and the saturated liquid density. Pe´neloux and Rauzy3 used a linear volume translation term in the SRK EOS to improve the saturated and high-pressure liquid densities for nonpolar and polar fluids. The volume translation was correlated as a function of the Rackett compressibility factor and the critical parameters. Watson et al.4 used volume translation to develop the van der Waals-711 EOS for alkanes (nC1-nC16) with the volume translation correlated with the critical parameters, the acentric factor, and the reduced temperature. Yu and Lu5 used the volume translation technique with the Peng-Robinson (PR) EOS and generalized the parameters in terms of the acentric factor. Jhaverl and Youngren6 used the volume translation concept with the PR EOS to improve volumetric predictions of reservoir fluids with the constant translation correlated as a function of molecular weight. Carrier et al.7 used group theory to calculate the volume translation parameters with the PR EOS for representing vapor pressure data in petroleum fluids. Chou and Prausnitz8 applied volume translation to the SRK EOS to represent phase equilibrium and the densities of some pure fluids in the vapor-liquid critical region. Magoulas and Tassios9 developed a translated form of the van der Waals (t-vdW) and the Peng-Robinson (t-PR) EOS for predicting vapor pressures, saturated liquid volumes, and enthalpies of vaporization for alkanes. Soave et al.10 used a simple group-contribution method to estimate the volume translation parameter in a new volume translation RK EOS. Kenney et al.11 used a hard-sphere volume translation method in the van der Waals EOS to model supercritical water oxidation. Ji and Lempe12,13 applied temperature-dependent volume translation to the SRK EOS to improve the volume prediction both inside and outside the * To whom correspondence should be addressed. Tel.: +86-106279-6318. Fax: +86-10-6277-0209. E-mail address: yyduan@ mail.tsinghua.edu.cn.
critical region for some pure polar and nonpolar fluids. The volume translation was correlated as a function of the critical parameters, the critical compressibility factor, and the reduced temperature. Tsai and Chen14 developed a volume-translated Peng-Robinson (VTPR) EOS for some polar and nonpolar fluids, but the volume translation was not generalized. The VTPR EOS satisfactorily predicted the vapor pressures and both the saturated vapor and liquid molar volumes, especially for the polar fluids (e.g., water). de Sant’Ana et al.15 compared volume translation techniques for representing the volume properties of Ungerer and Batut16 with three other volume translation methods.6,9,17 Wang and Gmehling18 improved the SRK EOS for representing the volumetric properties of petroleum fluids, including the saturated liquid density data and pressure-volume-temperature (PVT) data under supercritical conditions. The volume translation was not generalized for any of the petroleum fluids, except for the n-alkanes. Ahlers and Gmehling19 developed a constant volume translation to represent the saturated liquid densities near and far from the critical point for polar and nonpolar fluids. Yelash and Kraka20 analyzed temperature-dependent and temperature-independent volume translation methods and gave the criterion for the isotherms crossing.21 Lin and Duan22 developed a new volume translation method for the PR EOS for representing the liquid density of different polar and nonpolar fluids, including alkanes, cycloparaffins, halogenated hydrocarbons, olefins, cyclic olefins, aromatics, and inorganic molecules. All these results show that the volume translation method can be successfully applied to the cubic EOS to improve the volumetric properties. This work presents a volume translation SRK EOS (VTSRK EOS), using a generalized temperature-dependent volume translation to represent the volume properties of 198 polar and nonpolar fluids inside and outside the saturated region. Development of the Volume Translation Soave-Redlich-Kwong Equation of State (VTSRK EOS) According to Martin2 and Pe´neloux et al.,3 the VTSRK EOS can be translated by an additional term, c, using the volume translation techniques:
P)
a RT V + c - b (V + c)(V + b + c)
10.1021/ie051058v CCC: $33.50 © 2006 American Chemical Society Published on Web 01/27/2006
(1)
1830
Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006
cc ) (VSRK)c - (Vexp)c )
RTc
(31 - Z ) P c
(7)
c
where VSRK is the volume calculated by the original SRK and Vexp is the experimental value. The temperature-dependent function, f(Tr), is given by
f(Tr) ) β + (1 - β) exp(γ|1 - Tr|)
Figure 1. Variation of the volume translation parameter β in eq 8 with the critical compressiblity factor Zc for many polar and nonpolar pure fluids.
(8)
where β and γ are two parameters that can be determined by fitting experimental liquid densities. At the reduced temperature Tr ) 1.0, the f-function in eq 8 is unity and the volume translation must be equal to the critical volume translation. Therefore, the critical compressibility factor calculated by the VTSRK EOS is the experimental critical compressibility factor. The parameters in eq 8 were optimized by minimizing the objective function:
F)
∑| N i)1 N
1
(Fi)cal - (Fi)exp
|
(Fi)exp
(9)
where N is the number of experimental data points and (Fi)cal and (Fi) exp are the densities calculated from the VTSRK EOS and the experimental data, respectively. The VTSRK EOS can also be used for volumetric calculations for nonpolar and polar binary mixtures when the following mixing rules are used: 2
am )
2
∑ ∑xixjaij i)1 j)1
(10)
aij ) (1 - kij)ai1/2 aj1/2 2
Figure 2. Variation of the volume translation parameter γ in eq 8 with the critical compressibility factor Zc for many polar and nonpolar fluids.
bm )
The VTSRK EOS coefficients a and b can be calculated from the critical parameters in the same way as that for the SRK EOS:1
cm )
( )
R2Tc2 a ) 0.42748 R(T) Pc b ) 0.08664
( ) RTc Pc
(2)
(12)
2
xici ∑ i)1
(13)
where xi is the mole fraction of component i, aii and ajj are the cohesive energies, bi is the volumetric parameter of component i, and ci is the volume translation of component i, and kij is the binary interaction coefficient. Results for Pure Fluids
R(T) ) [1 + m(1 - Tr0.5)]2
(4)
m ) 0.480 + 1.574ω - 0.176ω2
(5)
where Tr ) T/Tc is the reduced temperature and ω is the acentric factor. Because the constant volume translation, c, cannot accurately represent the data over the entire saturated region,19 the predictions of the critical compressibility factor and volumetric properties by the VTSRK EOS can be improved by calculating the volume translation c from the experimental values as
where cc is the critical volume translation:
xibi ∑ i)1
(3)
where Tc and Pc are the critical temperature and pressure, respectively. The function R(T) is given by
c(T) ) ccf(Tr)
(11)
(6)
Saturated Fluids. The fit parameters β and γ for the temperature-dependent volume translation, which are listed in Tables 1-4, were optimized using the experimental saturated liquid densities for many polar and nonpolar fluids. Figures 1 and 2 show the relationship between β and γ and the critical compressibility factor. The two parameters can be generalized by
(31 - Z )] + 0.2334 1 γ ) -3.4620 exp[16.0813( - Z )] - 4.0957 3
[
β ) -3.7303 exp -60.2833
c
c
(14) (15)
The generalized VTSRK EOS is defined by eqs 1-8, eq 14, and eq 15. The saturated liquid densities calculated by the generalized VTSRK EOS are compared with the experimental data and the results are listed in Tables 1-4, which also includes the critical temperature and pressure, and the critical compress-
398 459.93 511.7 553.8 604.2 647.2
512.5 514 536.8 563 588.1 610.3 632.6 652.5 670.7 687.3 703.6 719.4
cyclopropane cyclobutane cyclopentane cyclohexane cycloheptane cyclooctane average
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol 1-decanol 1-undecanol 1-dodecanol average
8.084 6.137 5.169 4.414 3.897 3.417 3.058 2.777 2.528 2.315 2.147 1.994
5.54 4.98 4.51 4.08 3.82 3.56
4.5992 4.8718 4.2471 3.796 3.64 3.3688 3.0123 2.7358 2.4863 2.3056 2.1229 1.9657 1.8239 1.68 1.57 1.48 1.4 1.34 1.27 1.21 1.16 3.3812 3.1992
Pc (MPa)
0.2220 0.2410 0.2520 0.2580 0.2600 0.2610 0.2530 0.2540 0.2590 0.2630 0.2640 0.2390
0.2710 0.2730 0.2760 0.2730 0.2680 0.2710
0.2863 0.2793 0.2787 0.2740 0.2781 0.2695 0.2643 0.2633 0.2586 0.2550 0.2490 0.2432 0.2376 0.2320 0.2260 0.2240 0.2200 0.2190 0.2170 0.2150 0.2130 0.2700 0.2690
Zc
Based on eqs 14 and 15.
0.357-0.997 0.321-0.997 0.281-0.997 0.336-0.997 0.340-0.997 0.385-0.997 0.390-0.997 0.406-0.997 0.407-0.997 0.415-0.997 0.416-0.997 0.414-0.997
-23.8394 -19.3780 -16.9003 -13.6526 -14.4340 -12.8051 -17.2927 -17.7146 -13.5650 -14.8950 -15.1825 -19.8776
0.2295 0.2001 0.1697 0.1650 0.1570 0.1741 0.1743 0.1826 0.2082 0.2244 0.2393 0.1923 c
0.364-0.997 0.396-0.997 0.350-0.997 0.502-0.997 0.439-0.997 0.445-0.997
-11.7733 -11.5039 -11.0957 -12.9552 -14.4750 -14.6584
0.0842 0.1075 0.1090 0.1191 0.1246 0.1484
0.499-0.997 0.311-0.997 0.243-0.997 0.320-0.997 0.282-0.997 0.313-0.997 0.355-0.997 0.342-0.997 0.387-0.997 0.373-0.997 0.396-0.997 0.391-0.997 0.406-0.997 0.400-0.997 0.407-0.997 0.405-0.997 0.408-0.997 0.313-0.997 0.416-0.997 0.410-0.997 0.411-0.997 0.250-0.997 0.599-0.997
-10.3206 -12.1396 -12.1608 -12.7991 -11.0706 -13.0467 -13.7649 -14.8490 -14.4908 -16.8069 -18.3197 -20.1206 -21.4696 -21.0287 -22.5868 -25.7376 -26.2976 -27.9486 -29.5486 -30.0486 -34.7460 -13.4844 -15.4496
-0.0034 0.0742 0.1084 0.1244 0.1039 0.1479 0.1714 0.1882 0.1974 0.1982 0.2028 0.2013 0.2051 0.2211 0.2174 0.2332 0.2336 0.2214 0.2234 0.2319 0.2380 0.1214 0.1003
Tr range
γ
β
N δF ) 1/N ∑i)1 |((Fi)cal - (Fi)exp)/(Fi)exp|. b Based on optimized values.
190.56 305.33 369.83 425.13 407.82 469.65 507.43 540.26 568.83 595.65 618.45 638.76 658.2 675 693 708 723 736 747 758 768 460.43 433.78
CH4 C2H6 C3H8 nC4 iC4 nC5 nC6 nC7 nC8 nC9 nC10 nC11 nC12 nC13 nC14 nC15 nC16 nC17 nC18 nC19 nC20 isopentane neopentane average
a
Tc (K)
substance
26.56 19.06 14.95 14.30 13.27 14.93 15.09 15.40 17.12 16.91 17.56 18.02 16.93
10.54 3.52 6.13 6.43 6.86 5.56 4.78 4.83 3.74 3.83 3.25 3.75 5.27
2.57 4.35 3.73 3.11 2.88 2.33 3.16
Cycloparaffins 5.45 8.06 6.04 8.83 5.23 9.32 4.56 9.82 4.16 9.72 3.29 10.66 4.79 9.40 Alcohols 17.32 9.33 4.82 3.61 2.66 4.20 4.00 4.23 6.41 6.08 6.76 7.31 6.39
3.68 2.44 2.19 2.28 2.76 2.24 2.29 2.21 2.29 2.20 2.25 2.36 2.42 2.78 2.48 3.00 2.91 4.27 2.83 6.41 8.81 2.46 3.41 3.09
4.64 6.42 7.87 9.36 8.20 11.15 13.28 14.18 15.81 16.12 17.37 18.22 19.44 20.81 21.36 22.64 22.87 25.04 24.61 27.80 30.48 9.38 9.31 16.36
Alkanes 9.20 6.65 5.29 4.40 5.21 3.22 2.38 2.91 4.85 5.05 6.45 7.35 8.74 10.35 11.23 12.43 12.76 14.90 14.85 18.33 21.35 4.33 5.36 8.59
PT
SRK
PR
2.92 2.14 2.11 2.13 2.08 3.39 2.74 2.84 4.86 6.22 7.31 4.96 3.64
2.33 1.21 1.95 0.91 2.26 1.63 1.72
1.39 1.20 1.83 1.68 1.88 1.97 2.11 2.86 2.53 2.97 3.17 3.91 4.52 4.63 3.15 5.24 5.37 7.18 4.54 7.22 8.04 1.89 4.00 3.62
refs 12 and 13
δF (%)a
3.37 3.33 2.87 1.73 1.60 1.16 2.46 1.85 2.45 3.72 4.84 3.97 2.78
2.20 1.99 1.91 0.48 2.20 0.59 1.56
0.56 0.45 1.24 0.52 1.54 0.60 0.65 0.83 0.90 0.76 1.25 2.31 2.61 1.90 1.58 2.85 2.59 1.94 1.80 1.52 2.26 1.13 3.32 1.53
ref 22
4.18 3.29 1.83 0.80 0.70 0.45 0.72 0.63 0.36 0.40 0.44 1.06 1.24
0.57 1.33 1.08 0.31 0.51 0.34 0.69
0.25 0.30 0.49 0.36 0.44 0.45 0.45 0.55 0.55 0.63 0.79 0.96 1.13 2.14 1.43 2.73 2.54 2.20 2.22 2.28 2.70 0.58 0.66 1.17
VTSRKb
4.47 4.20 3.67 2.24 2.32 1.19 2.67 2.00 1.94 3.06 4.15 3.63 2.96
2.34 2.54 1.71 0.68 2.59 0.40 1.71
0.56 0.59 0.91 0.38 1.02 0.57 0.61 0.78 0.75 0.64 1.09 1.91 2.19 2.32 1.73 3.19 3.11 2.95 2.45 2.55 2.91 1.88 3.34 1.67
VTSRKc
29 29 29 29 29 29 29 29 29 29 29 29
29 29 29 29 29 29
23 24 25 26 27 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 28 28
critical data
reference
29 29 29 29 29 29 29 29 29 29 29 29
29 29 29 29 29 29
23 24 25 26 27 28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 28 28
FL
Table 1. Critical Data of Pure Fluids and Comparison of Saturated Liquid Densities Calculated by the PR, SRK, PT, and VTSRK EOSs, Ji and Lempe,12,13 and Lin and Duan:22 Alkanes, Cycloparaffins, and Alcohols
Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006 1831
507 560.4 598 632
308.3 402.4 440 481.2 516.2 547 574 598.05 619.85
cyclopentene cyclohexene cycloheptene cyclooctene average
acetylene methylacetylene ethylacetylene 1-pentyne 1-hexyne 1-heptyne 1-octyne 1-nonyne 1-decyne average
6.138 5.63 4.6 4.17 3.62 3.21 2.88 2.61 2.37
4.8 4.35 4.01 3.68
5.041 5.47 4.6 4.5 4.3299 4.02 4.21 4 4.1 3.56 3.69 3.66 3.447 3.53 3.42 3.21 2.92 2.68 2.33 2.22 2.03 1.93 1.77 1.66 1.57 1.48 1.41 1.34 1.28 1.22
Pc (MPa)
0.2680 0.2760 0.2620 0.2890 0.2720 0.2730 0.2670 0.2610 0.2540
0.2790 0.2720 0.2710 0.2710
0.2810 0.2710 0.2810 0.2670 0.2700 0.2780 0.2720 0.2750 0.2740 0.2750 0.2730 0.2720 0.2600 0.2860 0.2560 0.2720 0.2670 0.2660 0.2480 0.2530 0.2460 0.2410 0.2380 0.2360 0.2500 0.2490 0.2490 0.2480 0.2480 0.2470
Zc
0.1354 0.1499 0.1232 0.1029 0.1145 0.1416 0.1543 0.1618 0.1723
0.1063 0.1322 0.0970 0.1311
0.0756 0.1377 0.0881 0.1594 0.1379 0.1128 0.1414 0.1267 0.1294 0.1304 0.1439 0.1303 0.1833 0.0324 0.1875 0.1578 0.1559 0.1620 0.2071 0.1925 0.2022 0.2030 0.2113 0.2160 0.2119 0.2293 0.2201 0.2203 0.2244 0.2268
β
N δF ) 1/N ∑i)1 |((Fi)cal - (Fi)exp)/(Fi)exp|. b Based on optimized values.
282.34 393.15 364.9 444 425.374 419.5 435.5 417.9 428.6 464.8 475 474.2 465 452.7 470 504 537.3 567 594 617 638 658 675 692 708 722 736 748 760 771
ethene allene propylene 1,2-butadiene 1,3-butadiene 1-butene cis-2-butene isobutene trans-2-butene 1-pentene cis-2-pentene trans-2-pentene 2-methyl-1-butene 3-methyl-1-butene 2-methyl-2-butene 1-hexene 1-heptene 1-octene 1-nonene 1-decene 1-undecene 1-dodecene 1-tridecene 1-tetradecene 1-pentadecene 1-hexadecene 1-heptadecene 1-octadecene 1-nonadecene 1-eicosene average
a
Tc (K)
substance
c
7.71 9.80 7.31 9.97 8.70 11.95 10.78 10.17 5.38 8.79 10.09 11.40 12.57 14.30 10.60
Cycloolefins 0.282-0.997 5.42 0.312-0.997 4.00 0.371-0.997 5.92 0.347-0.997 3.96 4.83 Alkines 3.68 3.48 3.90 7.89 4.67 3.62 2.59 2.14 2.88 3.87
-12.5435 -13.4314 -15.0460 -13.9966
-13.2781 -11.7121 -15.8522 -11.1582 -13.8691 -15.1417 -16.2876 -15.1749 -16.4950
Based on eqs 14 and 15.
0.610-0.997 0.435-0.997 0.345-0.997 0.357-0.997 0.283-0.997 0.360-0.997 0.347-0.997 0.381-0.997 0.378-0.997
6.53 10.41 7.50 12.04 10.62 9.01 10.68 9.60 9.83 10.25 9.66 9.67 14.64 8.18 16.34 10.88 11.57 12.32 18.30 15.99 17.83 16.98 18.80 19.69 19.98 20.78 20.73 20.95 21.28 21.68 14.09
0.386-0.997 0.385-0.997 0.255-0.997 0.321-0.997 0.412-0.997 0.222-0.997 0.319-0.997 0.330-0.997 0.404-0.997 0.243-0.997 0.267-0.997 0.291-0.997 0.301-0.997 0.243-0.997 0.309-0.997 0.274-0.997 0.296-0.997 0.310-0.997 0.332-0.997 0.441-0.997 0.359-0.997 0.369-0.997 0.378-0.997 0.383-0.997 0.387-0.997 0.392-0.997 0.393-0.997 0.394-0.997 0.397-0.997 0.398-0.997
SRK
-11.8641 -13.0206 -12.4405 -13.6728 -13.3309 -12.4076 -12.1242 -11.8602 -12.7091 -12.9987 -14.4425 -13.3892 -14.2205 -11.2529 -15.5792 -14.3179 -14.7496 -14.1770 -16.2368 -17.2663 -18.8271 -20.7071 -21.8993 -21.1792 -19.7593 -17.5332 -17.5332 -17.7510 -17.7510 -17.9724
PR Olefins 6.63 3.74 5.53 2.67 3.67 4.31 3.60 4.29 4.19 3.45 3.71 4.05 3.71 4.95 6.06 2.94 2.66 2.43 7.80 5.06 7.06 6.03 8.14 9.08 9.48 10.37 10.36 10.60 10.89 11.38 5.96
Tr range
γ
4.10 2.34 2.44 6.32 5.05 4.86 4.66 4.70 4.18 4.29
2.74 2.19 4.18 3.05 3.04
2.54 3.42 2.04 2.44 2.58 1.98 2.50 2.40 2.57 1.98 2.20 2.54 5.77 2.64 6.02 2.07 2.74 3.32 4.75 2.45 2.24 2.89 2.66 2.64 2.70 2.84 2.94 3.32 3.55 3.90 2.95
PT
1.61 2.82 3.40 4.37 1.69 2.49 2.56 2.99 3.52 2.83
1.78 1.87 3.44 1.87 2.24
0.83 1.38 2.19 2.13 1.41 2.99 1.92 1.91 1.36 3.28 3.13 1.85 2.15 4.62 2.08 3.03 2.45 2.50 2.67 2.47 3.53 4.84 5.00 5.07 5.12 5.59 5.72 5.83 6.18 6.44 3.32
refs 12 and 13
δF (%)a
1.03 2.74 4.10 4.76 0.90 0.94 1.06 2.19 3.20 2.32
1.16 0.44 3.65 0.64 1.47
0.50 0.44 1.67 0.56 0.54 2.08 1.11 1.31 0.80 1.98 1.22 0.47 1.00 4.42 1.93 1.18 0.57 0.61 1.43 1.22 1.67 4.54 3.23 2.86 2.11 2.63 2.66 2.56 2.92 3.06 1.78
ref 22
0.47 0.21 0.63 4.12 0.55 0.44 0.58 1.13 1.13 1.03
0.46 0.44 1.19 0.38 0.62
0.23 0.41 1.08 0.45 0.37 1.09 0.35 0.31 0.28 1.15 0.85 0.52 0.87 4.12 2.17 0.56 0.61 0.60 1.18 0.69 1.00 2.43 1.65 1.82 2.50 1.57 2.23 2.26 2.23 2.29 1.26
VTSRKb
0.71 2.09 4.25 5.42 1.79 0.80 1.36 2.44 3.21 2.45
0.84 0.57 3.98 0.92 1.58
0.27 0.63 1.53 0.55 0.73 1.63 0.65 0.74 0.32 1.48 1.08 0.74 1.26 4.73 2.42 0.94 0.94 0.74 1.60 1.20 1.61 4.35 2.90 2.61 2.44 2.95 2.97 2.91 3.20 3.35 1.78
VTSRKc
29 29 29 29 29 29 29 29 29
29 29 29 29
29 28 29 28 28 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29
critical data
reference
29 29 29 29 29 29 29 29 29
29 29 29 29
29 28 29 28 28 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29
FL
Table 2. Critical Data of Pure Fluids and Comparison of Saturated Liquid Densities Calculated by the PR, SRK, PT, and VTSRK EOSs, Ji and Lempe,12,13 and Lin and Duan:22 Olefins, Cycloolefins, and Alkines
1832 Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006
508.2 535.5 560.95 561.08 587.61 582.82 602 606.6 611.4 627.7 623.8 632.7 652.5 648.1 640 643.7
acetone methyl ethyl ketone 3-pentanone 2-pentanone 2-hexanone 3-hexanone 4-heptanone 3-heptanone 2-heptanone 3-octanone 4-octanone 2-octanone 2-nonanone 3-nonanone 5-nonanone 4-nonanone average
4.701 4.15 3.74 3.694 3.287 3.32 2.92 2.92 2.94 2.704 2.704 2.64 2.41 2.453 2.32 2.453
4.895 4.108 4 3.732 3.541 3.511 3.609 3.209 3.2 4.051 3.05 2.95 2.97 2.604 1.5789
Pc (MPa)
0.2330 0.2490 0.2690 0.2380 0.2540 0.2590 0.2530 0.2510 0.2510 0.2570 0.2590 0.2490 0.2490 0.2550 0.2440 0.2570
0.2680 0.2640 0.2610 0.2640 0.2590 0.2590 0.2630 0.2650 0.2650 0.2690 0.2700 0.2650 0.2660 0.2530 0.2450
Zc
0.2401 0.2334 0.2459 0.1843 0.2084 0.2075 0.2107 0.2174 0.2084 0.2162 0.2112 0.2174 0.2259 0.2252 0.2374 0.2122
0.1471 0.1723 0.1847 0.1806 0.1908 0.1954 0.1841 0.1881 0.1953 0.2080 0.1119 0.2088 0.2010 0.2015 0.2205
β
N δF ) 1/N ∑i)1 |((Fi)cal - (Fi)exp)/(Fi)exp|. b Based on optimized values.
562.05 591.75 648 630.3 617 616.2 617.15 631 638.35 748.35 650 664.54 660 679.9 774.26
benzene toluene phenylethene o-xylene m-xylene p-xylene ethylbenzene cumene n-propylbenzene naphthalene isobutylbenzene sec-butylbenzene tert-butylbenzene n-pentylbenzene dodecylbenzene average
a
Tc (K)
substance
c
23.59 19.56 16.63 17.51 17.08 16.39 17.04 18.53 17.24 17.62 16.30 18.12 19.08 17.98 21.57 16.07 18.14
0.360-0.997 0.357-0.997 0.426-0.997 0.358-0.997 0.378-0.997 0.381-0.997 0.409-0.997 0.394-0.997 0.397-0.997 0.416-0.997 0.406-0.997 0.408-0.997 0.415-0.997 0.417-0.997 0.425-0.997 0.393-0.997
Ketones 13.52 8.72 5.60 6.65 6.22 5.42 6.08 7.87 6.23 6.86 5.19 7.29 8.45 7.17 11.27 4.86 7.34
SRK 11.52 13.17 14.40 14.01 15.35 15.73 13.92 13.67 13.96 14.81 8.23 15.28 13.98 16.77 19.50 14.29
PR
Aromatics 0.505-0.997 3.21 0.309-0.997 2.41 0.381-0.997 3.25 0.401-0.997 2.83 0.373-0.997 4.35 0.472-0.997 4.59 0.297-0.997 2.85 0.288-0.997 2.51 0.279-0.997 2.49 0.452-0.997 3.58 0.348-0.997 5.00 0.304-0.997 4.31 0.333-0.997 2.37 0.299-0.997 5.94 0.360-0.997 8.98 3.91
Tr range
Based on eqs 14 and 15.
21.4762 16.5621 12.8426 19.0898 15.6121 14.3654 17.5923 15.0967 18.2503 13.9762 16.0049 19.0169 17.6182 16.3040 18.6583 17.9110
-13.6637 -13.6546 -14.9290 -13.3077 -13.9645 -15.0519 -14.2191 -14.7486 -15.2848 -12.4389 -15.8830 -12.9698 -15.3453 -16.2137 -19.3140
γ
13.62 8.60 4.54 5.67 4.11 3.29 3.08 4.96 3.14 2.92 2.32 3.14 3.08 2.73 5.62 2.66 4.59
3.06 3.41 5.37 2.62 3.68 4.11 2.93 2.32 2.26 3.71 6.56 5.14 3.97 2.46 4.05 3.71
PT
3.22 3.45 6.98 5.71 2.77 3.18 2.96 2.71 3.19 3.72 3.16 3.42 3.27 3.43 3.19 4.11 3.65
1.17 2.32 1.94 2.15 2.24 1.60 2.76 3.35 3.95 4.58 3.27 4.60 3.79 3.20 4.65 3.04
refs 12 and 13
δF (%)a
2.03 1.44 5.97 4.36 0.83 1.62 0.69 1.19 0.89 2.29 1.34 0.80 0.98 1.81 2.32 1.40 1.87
0.60 0.70 0.75 1.08 0.87 0.90 0.74 1.16 1.15 3.99 3.16 3.01 1.50 0.88 1.13 1.44
ref 22
2.53 0.79 0.25 1.31 0.63 0.45 0.68 0.70 0.74 0.46 0.49 0.78 0.72 0.55 1.41 0.92 0.84
0.48 0.54 0.70 0.39 0.52 0.75 0.65 0.64 0.69 0.25 1.30 0.55 0.58 0.93 1.04 0.67
VTSRKb
2.65 2.14 5.41 3.78 0.78 1.52 0.64 1.36 0.69 2.14 1.37 0.73 1.38 1.89 2.66 1.38 1.91
0.73 0.70 0.77 0.77 0.82 0.94 0.75 1.06 1.30 3.42 3.56 2.69 1.88 1.04 0.96 1.43
VTSRKc
29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29
29 29 28 29 29 29 29 29 29 28 29 29 29 29 28
critical data
reference
29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29
29 29 28 29 29 29 29 29 29 28 29 29 29 29 28
FL
Table 3. Critical Data of Pure Fluids and Comparison of Saturated Liquid Densities Calculated by the PR, SRK, PT, and VTSRK EOSs, Ji and Lempe,12,13 and Lin and Duan:22 Aromatics and Ketones
Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006 1833
Tc (K)
416.25 510 351.26 369.3 385.12 471.15 301.88 340.08 227.5 451.6 299.3 556.35 536.4 345.86 602 624 645 665 345.86 386.41 650.15 523.4 545 292.8 353.15 551 487.8 487.5 418.85 353.1 293.02 456.9 461.1 395.43 339.4 391.74 374.25 477.3 410.3 571 345.05 572 593 503.15 489
substance
methyl chloride dichloromethane difluoromethane chlorodifluoromethane dichlorodifluoromethane trichlorofluoromethane chlorotrifluoromethane bromotrifluoromethane carbon tetrafluoride dichlorofluoromethane trifluoromethane carbon tetrachloride chloroform 1,1,2-trifluoroethane 1,1,2-trichloroethane 1,1,1,2-tetrachloroethane 1,1,2,2-tetrachloroethane pentachloroethane 1,1,1-trifluoroethane 1,1-difluoroethane 1,2-dibromoethane 1,1-dichloroethane 1,1,1-trichloroethane hexafluoroethane chloropentafluoroethane 1,1,2,2-tetrachlorodifluoroethane 1,2-dibromotetrafluoroethane 1,1,2-trichlorotrifluoroethane 1,2-dichlorotetrafluoroethane chloropentafluoroethane hexafluoroethane 2,2-dichloro-1,1,1-trifluoroethane 1,2-dichloro-1,1,2-trifluoroethane 2-chloro-1,1,1,2-tetrafluoroethane pentafluoroethane 1,1,2,2-tetrafluoroethane 1,1,1,2-tetrafluoroethane 1,1-dichloro-1-fluoroethane 1-chloro-1,1-difluoroethane trichloroethylene octafluoropropane 1,2-dichloropropane 1-iodopropane 1-chloropropane 2-chloropropane
6.6793 6.0795 5.782 4.99 4.136127 4.407627 3.87927 4.01727 3.7527 5.18 4.83227 4.5596 5.4716 3.7581 4.48 4.02 4.09 3.68 3.761 4.5168 5.4769 5.061 4.2962 2.979 3.1573 3.34 3.393 3.392227 3.25727 3.12 3.0537 3.674 3.741 3.6243 3.6173 4.64 4.059227 4.25 4.041 4.91 2.6801 4.24 5.03 4.5799 4.54
Pc (MPa) 0.2680 0.2650 0.2406 0.2729 0.2750 0.2790 0.2774 0.2769 0.2771 0.2709 0.2582 0.2720 0.2930 0.2418 0.2520 0.2520 0.2480 0.2460 0.2533 0.2523 0.2650 0.2791 0.2660 0.2740 0.2785 0.2640 0.2711 0.2751 0.2748 0.2678 0.2770 0.2682 0.2713 0.2653 0.2678 0.2697 0.2583 0.2709 0.2669 0.2650 0.2790 0.2590 0.2960 0.2700 0.2760
Zc 0.1813 0.2132 0.2624 0.1466 0.0996 0.1078 0.0907 0.0800 0.0497 0.1369 0.2042 0.1293 -0.0545 0.2003 0.2174 0.1942 0.2213 0.2164 0.2153 0.2255 0.1709 0.1190 0.1382 0.1197 0.0750 0.1793 0.1198 0.1281 0.1085 0.0978 0.0825 0.1520 0.1247 0.1388 0.1469 0.1307 0.1843 0.1227 0.1558 0.1419 0.0643 0.1985 -0.0001 0.1070 0.0982
β Tr range PR
Halogenated Hydrocarbons -12.5903 0.428-0.997 2.32 -12.6815 0.353-0.997 4.42 -17.4009 0.398-0.997 13.86 -12.8958 0.325-0.997 3.46 -13.5758 0.312-0.997 5.57 -12.3592 0.350-0.997 5.34 -12.9111 0.315-0.997 6.00 -12.9109 0.323-0.997 6.45 -14.0774 0.404-0.997 7.90 -12.5862 0.314-0.997 3.74 -12.9972 0.401-0.997 5.65 -12.5758 0.458-0.997 4.08 -11.1581 0.401-0.997 5.86 -18.2057 0.477-0.997 8.64 -17.0418 0.399-0.997 6.44 -15.4922 0.329-0.997 5.26 -15.3651 0.364-0.997 8.25 -15.7134 0.373-0.997 8.18 -13.9395 0.477-0.997 7.13 -13.6653 0.404-0.997 8.26 -13.2433 0.438-0.997 2.42 -13.3942 0.340-0.997 5.01 -14.5782 0.450-0.997 3.46 -15.2763 0.598-0.997 4.78 -12.2070 0.496-0.997 6.54 -13.4781 0.508-0.997 2.64 -13.5036 0.342-0.997 4.42 -12.5201 0.492-0.997 4.26 -12.7971 0.435-0.997 5.05 -15.3110 0.496-0.997 5.37 -14.9829 0.607-0.997 6.83 -13.6543 0.368-0.997 2.97 -14.3378 0.455-0.997 4.31 -14.8469 0.252-0.997 3.33 -14.3178 0.516-0.997 3.11 -14.1925 0.477-0.997 3.91 -14.6054 0.460-0.997 3.81 -13.0734 0.360-0.997 4.31 -13.3534 0.353-0.997 2.82 -14.3701 0.333-0.997 3.27 -14.6114 0.377-0.997 7.39 -13.9907 0.306-0.997 4.59 -12.1812 0.295-0.997 10.82 -14.4468 0.308-0.997 5.01 -13.0785 0.327-0.997 5.61
γ 13.17 15.47 23.92 10.61 7.73 8.17 7.36 6.94 5.41 10.57 16.50 10.29 7.10 19.35 17.33 16.12 19.00 18.80 17.95 18.92 13.20 8.36 10.89 9.36 7.63 14.18 9.45 9.99 8.80 8.77 7.05 11.57 9.72 10.60 11.49 10.30 15.04 9.68 11.96 11.08 5.77 15.44 2.22 8.42 7.81
SRK 6.74 7.93 14.75 2.17 2.62 2.76 2.93 3.25 4.73 2.46 6.84 2.65 3.35 10.41 8.02 7.31 9.79 9.97 8.60 9.19 4.88 3.26 2.39 3.59 5.23 3.23 3.11 3.31 3.99 4.26 5.24 2.30 4.70 2.52 3.26 3.74 3.22 2.55 2.34 2.47 7.88 6.29 8.96 3.11 3.55
PT 1.78 3.91 3.18 2.29 1.65 1.66 1.42 1.59 2.53 1.86 2.16 0.99 4.57 3.17 2.87 2.68 2.65 2.65 1.54 2.44 1.39 2.29 1.99 1.44 0.56 1.29 1.54 1.23 0.94 3.52 2.71 1.85 1.77 2.23 1.11 1.43 1.61 1.60 1.85 2.16 1.42 2.62 0.93 2.09 1.46
refs 12 and 13
δF (%)a
1.86 3.14 5.50 1.27 0.87 1.29 0.57 1.00 2.56 0.61 1.62 0.53 4.18 2.86 0.67 1.61 1.11 1.39 1.33 2.17 0.70 1.50 1.72 1.08 0.58 0.89 0.83 1.22 0.46 3.68 2.05 0.54 1.24 1.82 0.60 0.99 0.96 0.79 0.62 1.69 1.44 0.91 2.46 2.21 0.60
ref 22 0.42 0.37 2.14 0.36 0.42 0.29 0.36 0.4 0.34 0.4 0.61 0.32 5.59 2.36 0.81 0.98 0.97 1.2 0.77 0.75 0.49 0.37 0.54 0.38 0.21 0.43 0.39 0.24 0.28 0.55 0.78 0.37 0.42 0.73 0.46 0.29 0.64 0.32 0.39 0.53 0.37 0.72 1.31 0.56 0.35
VTSRKb 1.66 3.10 5.29 0.82 1.37 0.84 0.75 1.31 3.18 0.58 1.71 0.59 6.51 2.67 0.98 1.53 1.33 1.51 1.90 2.57 0.74 1.24 1.93 1.47 0.89 0.77 1.43 0.48 0.72 4.01 3.02 0.52 1.45 2.57 0.94 1.42 0.93 1.12 0.66 1.94 1.91 1.05 4.13 2.88 0.86
VTSRKc
28 28 30 27 32 32 32 33 34 35 36 28 28 28 28 28 28 28 27 27 28 28 28 28 28 28 28 32 32 27 27 37 38 39 40 41 42 41 43 28 28 28 28 28 28
critical data
reference
Table 4. Critical Data of Pure Fluids and Comparison of Saturated Liquid Densities Calculated by the PR, SRK, PT, and VTSRK EOSs, Ji and Lempe,12,13 and Lin and Duan:22 Halogenated Hydrocarbons and Other Compounds
28 28 30 31 31 31 31 31 31 31 31 28 28 28 28 28 28 28 31 31 28 28 28 28 28 28 28 31 31 31 27 31 31 31 31 31 31 27 31 28 28 28 28 28 28
FL
1834 Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006
a
δF ) 1/N
N ∑i)1 |((Fi)cal
- (Fi)exp)/(Fi)exp|. Based on optimized values.
c
-9.8553 -10.9052 -10.3212 -9.9603 -10.5265 -11.1636 -8.4652 -10.5501 -11.2371 -13.8286 -11.2852 -12.4286 -13.9951 -14.3909 -21.4983 -14.3962 -13.7300 -10.6416 -10.2589 -11.1551
SRK
4.75 4.02 5.14 6.42 3.33 3.98 8.53 3.97 9.41 12.73 7.09 10.61 12.17 10.99 28.30 22.74 14.62 6.77 4.85 7.50 13.39
9.40
12.23
11.08 12.27 10.77 9.38 11.97 23.18 14.75 10.17 7.58 9.03 22.74 16.94 16.84 18.67 16.75 13.35 12.49 13.22
5.40
Based on eqs 14 and 15.
-0.1632 -0.0424 -0.0043 0.0265 -0.0061 -0.0125 -0.5232 0.0061 0.1221 0.1799 0.0774 0.1231 0.1435 0.1763 0.2663 0.2612 0.1726 0.0499 -0.0046 0.0588
PR 3.02 2.69 3.53 4.88 2.65 13.02 3.71 3.69 6.29 5.29 12.50 6.07 5.85 8.01 5.60 2.31 2.76 2.56
Other Compounds 0.586-0.997 14.47 0.577-0.997 10.81 0.568-0.997 9.28 0.566-0.997 8.04 0.362-0.997 9.43 0.523-0.997 9.92 0.512-0.997 20.24 0.396-0.997 9.02 0.420-0.997 4.51 0.466-0.997 2.15 0.511-0.997 6.62 0.598-0.997 4.45 0.723-0.997 3.70 0.378-0.997 2.64 0.434-0.997 19.06 0.496-0.997 12.74 0.499-0.997 3.26 0.518-0.997 7.36 0.531-0.997 9.25 0.600-0.997 7.28
Tr range 0.399-0.997 0.492-0.997 0.469-0.997 0.582-0.997 0.528-0.997 0.374-0.997 0.289-0.997 0.279-0.997 0.493-0.997 0.618-0.997 0.309-0.997 0.476-0.997 0.360-0.997 0.407-0.997 0.542-0.997 0.340-0.997 0.420-0.997 0.364-0.997
γ -15.9387 -14.2599 -14.0877 -14.0465 -15.1585 -20.9864 -15.5339 -13.5480 -12.6268 -14.4305 -15.5556 -17.8757 -18.2408 -17.5605 -15.1486 -13.8813 -13.6368 -12.7266
6.19 b
0.3000 0.2910 0.2880 0.2860 0.2880 0.2890 0.3050 0.2870 0.2760 0.2690 0.2840 0.2740 0.2740 0.2780 0.2294 0.2545 0.2490 0.2830 0.2870 0.2850
β 0.1445 0.1539 0.1384 0.1152 0.1548 0.2350 0.1861 0.1356 0.0830 0.1099 0.2576 0.2050 0.1996 0.2108 0.1998 0.1779 0.1547 0.1680
overall average
2.653 4.898 5.50195 5.84037 5.043 3.4 1.313 5.1724 7.71 7.8841 8.96291 7.245 7.383 21.7 22.064 11.333 8.31 8.552 8.31 5.6742
Zc 0.2657 0.2657 0.2700 0.2758 0.2666 0.2320 0.2570 0.2700 0.2780 0.2775 0.2440 0.2480 0.2470 0.2440 0.2550 0.2650 0.2630 0.2650
8.56
44.4 150.86 209.35 289.74 154.58 126.2 33.19 144.12 417.15 430.75 373.53 309.57 304.21 730.15 647.1 405.4 324.65 363.15 423.85 455
2.926 3.64 3.925 3.502 3.2 3.61 3.82 3.9 3.9 2.7775 3.19 4.07 4.07 3.72 3.2728 4.5191 4.5505 4.5191
Pc (MPa)
average
Ne Ar Ke Xe O2 N2 H2 F2 Cl2 O2S H2S N2O CO2 H2O2 H2O NH3 HCl HBr HI COCl2
average
Tc (K)
375.95 427.2 447.57 412.44 398.07 641 537 520.6 507 388.38 663 694 695 725 516.73 721.15 560.09 632.35
substance
1,1,1,2,3,3,3-heptafluoropropane 1,1,1,3,3-pentafluoropropane 1,1,2,2,3-pentafluoropropane 1,1,1,2,3,3-hexafluoropropane 1,1,1,3,3,3-hexafluoropropane 1,4-dichlorobutane 1-chlorobutane 2-chlorobutane 2-chloroisobutane octafluorocyclobutane 1,5-dichloropentane 1,3-dichlorobenzene 1,4-dichlorobenzene 1,2,4-trichlorobenzene perfluorobenzene iodobenzene fluorobenzene chlorobenzene
Table 4. (Continued)
PT
4.32
5.43
4.76 3.64 3.83 4.71 3.37 3.67 3.56 5.52 3.22 3.13 4.12 3.50 3.39 17.98 14.04 8.74 3.63 3.86 4.48
5.19
3.82 3.79 4.52 6.44 4.18 12.71 4.71 3.02 3.88 6.16 10.76 8.85 8.21 6.52 3.03 3.92 3.05 3.71
2.78
2.28
2.52 1.64 1.00 0.39 1.06 1.28 7.67 0.74 0.90 1.94 0.87 0.67 0.55 4.84 3.78 6.92 6.57 0.28 1.00 1.04
2.06
2.11 1.25 1.26 0.60 1.46 3.61 2.74 1.92 0.79 0.52 4.83 3.60 3.96 3.13 1.35 2.42 1.59 1.94
refs 12 and 13
δF (%)a
1.79
2.71
5.97 1.23 0.47 0.78 0.57 0.87 16.27 0.49 1.38 1.70 1.60 0.50 0.97 3.72 6.98 3.97 4.16 0.67 0.50 1.48
1.44
1.62 0.72 0.56 0.81 0.94 1.69 1.11 0.56 0.57 1.03 3.87 1.68 2.22 1.55 0.88 0.73 1.38 0.86
ref 22
0.91
0.84
1.00 0.20 0.23 0.24 0.20 0.20 1.36 0.19 0.29 0.34 0.19 0.35 0.38 0.45 5.30 3.24 1.91 0.24 0.22 0.26
0.70
0.50 0.41 0.32 0.26 0.31 2.20 0.85 0.58 0.26 0.27 1.22 1.09 1.00 1.15 0.67 0.43 0.54 0.36
VTSRKb
1.81
1.86
2.04 0.48 0.80 1.30 0.50 0.41 2.04 0.48 0.98 1.26 1.46 0.43 0.57 3.19 7.98 7.99 2.92 0.66 0.54 1.23
1.64
2.09 0.93 0.91 0.79 1.16 2.19 1.25 1.08 0.85 0.83 4.14 1.26 1.64 1.44 0.88 0.58 1.34 0.81
VTSRKc
29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28
31 27 27 27 27 28 28 28 28 27 28 28 28 28 28 28 28 28
critical data
reference FL
29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28
31 27 27 27 27 28 28 28 28 27 28 28 28 28 28 28 28 28
Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006 1835
1836
Ind. Eng. Chem. Res., Vol. 45, No. 5, 2006
Table 5. Comparison of Densities for Pure Fluids under Supercritical Conditions δF (%) PR
1.01 1.05 1.50 3.00
9.79 9.19 5.09 2.33 6.60
1.01 1.05 1.50 3.00
10.82 10.42 7.00 3.57 7.95
N2 (data from ref 47) 2.31 3.94 2.70 2.32 2.65 2.65 1.76 2.27 1.79 0.68 0.87 0.67 1.77 2.43 1.95
25.78 14.04 1.22 0.38 10.36
3.30 3.23 2.06 0.70 2.32
2.46 2.46 1.79 0.66 1.84
1.01 1.05 1.50 3.00
3.62 3.35 1.02 0.24 2.06
CO2 (data from ref 48) 7.38 2.72 3.29 7.31 2.69 3.66 5.59 1.79 4.59 2.17 0.61 1.99 5.61 1.95 3.38
24.81 9.32 2.27 0.82 9.31
3.14 3.25 3.37 1.19 2.74
3.17 3.29 3.42 1.21 2.77
average
average
SRK
PT
ref 22
ref 17
Argon (data from ref 46) 2.20 2.75 2.34 21.55 2.22 2.67 2.32 11.64 1.84 1.29 1.77 1.03 1.08 0.41 1.07 0.17 1.84 1.78 1.88 8.60
1,1,1,2-Tetrafluoroethane (data from ref 49) 1.01 2.51 10.51 2.58 95.18 44.49 1.05 2.61 10.27 2.73 28.68 14.62 2.56 10.39 2.67 61.93 29.56 1.01 1.05
average
this work
Tr
average
average
refs 12 and 13
1,1,1-Trifluoroethane (data from ref 50) 4.12 13.18 5.25 33.83 48.28 4.18 12.85 5.20 18.37 13.75 4.15 13.02 5.23 26.10 31.02
2.55 2.50 1.59 0.84 1.87
3.72 3.71 3.72 3.55 3.57 3.56
ibility factor. The critical parameters of the heavy compounds (beyond 10 carbon atoms) in Tables 1-4 are probably obtained by empirical extrapolation rather than by true measurements, because these compounds undergo rapid thermal degradation at their critical point. Many volume translation methods have been proposed to improve the volume calculations of the SRK EOS, but some are only applicable to certain materials and have not been generalized. Therefore, the generalized VTSRK EOS presented here is only compared with the temperature-dependent translation method given by the VTPR,22 original PR,44 PT,45 and SRK1 EOSs and by Ji and Lempe’s method12,13 in Tables 1-4. The average relative deviation of the experimental saturated liquid data from the calculated values of the present generalized VTSRK EOS is 1.81%, which is better than the original PR44 (6.19%), PT45 (4.32%) and SRK1 (13.39%) EOSs, and Ji and Lempe’s method12,13 (2.78%). The results of the generalized VTSRK EOS are as good as the VTPR EOS22 (1.79%) and show distinct improvements for the original SRK1 EOS for alkanes, olefins, alkines, cycloolefins, alcohols, aromatics, ketones, halogenated hydrocarbons, and inorganic molecules. Supercritical Fluids. For supercritical fluids, the volume translation parameter c is temperature independent and can be represented by eq 8 at Tr ) 1.5. Therefore, the f-function for the volume translation is given as
f(Tr) ) β + (1 - β) exp(0.5γ)
(16)
The volumetric properties of the supercritical fluids can then be calculated with the VTSRK EOSs, using eqs 1-7 and eqs 14-16. The calculated densities are compared with the standard EOSs45-49 and the average relative deviations are listed in Table 5. The results show that the proposed VTSRK EOS more accurately represents the supercritical density, compared to the original PR and SRK EOS. The VTSRK EOS has almost the same accuracy as the results of Wang and Gmehling18 but with a wider application range. Ji and Lempe’s method12,13 and the VTPR22 EOS can predict liquid densities well for nonpolar fluids
Figure 3. Relationship between the reduced temperature (Tr) and the compressiblity factor (Zc), and the temperature interval of (b - c) < 0.
in the supercritical region but are not proper for the polar fluids. The temperature-dependent volume translation parameter c is not used for the cubic EOS to predict the liquid densities in the supercritical region. If the constant c is used for the VTPR22 EOS as eq 16, good results would be obtained. The pressure range for each reduced temperature Tr is up to 60 MPa in Table 5. Therefore, the f-function for the volume translation proposed here for the entire temperature region is given as
f(Tr) )
{
β + (1 - β) exp[γ(1 - Tr)] (for Tr e 1) (for Tr > 1) β + (1 - β) exp(0.5γ)
(17)
where the β and γ can be calculated by eqs 14 and 15. Results for Binary Mixtures Introduction of the volume translation parameter c did not change the accuracy of the vapor-liquid equilibrium (VLE) calculations, because c was canceled when calculating the fugacity. The generalized VTSRK EOS also has better precision in calculating the liquid densities for various types of binary mixtures. Table 6 lists the average relative deviations of the experimental data from the generalized VTSRK, VTPR, PR, PT, and SRK EOSs and Ji and Lempe’s method12,13 for liquid densities of mixtures at atmospheric pressure. Some substances listed in Table 6 were not used to generalize eqs 14 and 15, including 2-propanol, methyl methacrylate, and diethyl ether. The binary interaction coefficients kij for these mixtures are equal to zero. The results in Table 6 show that the average relative deviation of the present VTSRK EOS is 0.
Many temperature-dependent translations have been checked by the Yelash and Kraska criterion,20 and the results show that all of the temperature-dependent translations can lead to the crossed isotherms. However, the crossed isotherm is only observed at very high pressure in the liquid phase and the pressure is far away from the saturated pressure. For argon, the isotherm line Tr ) 0.6 crossed the isotherm lines Tr ) 0.7, 0.8, and 0.9 at pressures of 360, 150, and 55 MPa while the saturated pressure was 0.136 MPa. For CH4, the isotherm line Tr ) 0.55 crossed the isotherm lines Tr ) 0.65, 0.75, and 0.85 at pressures of 720, 310, and 110 MPa while the saturated pressure is 0.053 MPa. For 1,1,1,2-tetrafluoroethane, the isotherm line Tr ) 0.55 crossed the isotherm lines Tr ) 0.65, 0.75, and 0.85 at pressures of 2020, 648, and 165 MPa and the saturated pressure is 0.0095
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MPa. When the temperature is very close to the critical temperature, the isotherm crossing can be avoided at p epc. The temperature-dependent volume translation correction is only an empirical approach, but it is still useful in the improvement of the constant compressibility factor and liquid volumetric properties prediction for the original EOS. Conclusions A new temperature-dependent volume translation was developed to improve the volume prediction properties of the Soave-Redlich-Kwong (SRK) equation of state (EOS) for pure fluids and mixtures. The volume translation was generalized as a function of the critical parameters of the pure fluids and reduced temperature. The generalized volume translation SRK EOS (VTSRK EOS) accurately represents the critical compressibility factor and the liquid densities of pure fluids and mixtures. Symbols a ) cohesive energy parameter in the SRK equation of state (Pa m6/mol2) b ) volumetric parameter in the SRK equation of state (m3/ mol) c ) temperature-dependent volume translation parameter in the SRK equation of state EOS ) equation of state f ) temperature-dependent function for the volume correction F ) objective function k ) binary interaction coefficients N ) number of data points P ) pressure (Pa) R ) molar gas constant; R ) 8.314471 J mol-1 K-1 T ) temperature (K) V ) molar volume (m3/mol) VTSRK ) volume translation Soave-Redlich-Kwong equation of state Z ) compressibility factor Greek Letters R ) temperature-dependent function for the PR EOS β, γ ) fitted parameters in the temperature-dependent volume correction function δ ) average relative deviation (%) F ) density Subscripts c ) critical point exp ) experimental value i ) component i j ) component j m ) mixture r ) reduced Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 50225622) and the Foundation for the Author of National Excellent Doctoral Dissertation of the People’s Republic of China (No. 200336). Literature Cited (1) Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197-1203. (2) Martin, J. J. Equations of State-Applied Thermodynamics Symposium. Ind. Eng. Chem. 1967, 59 (12), 34-52.
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ReceiVed for reView September 20, 2005 ReVised manuscript receiVed December 4, 2005 Accepted January 4, 2006 IE051058V
