Gaussian-like Volume Shifts for the Peng−Robinson Equation of State

Ind. Eng. Chem. Res. 1998, 37, 1663-1672

1663

Gaussian-like Volume Shifts for the Peng-Robinson Equation of State Wayne D. Monnery,*,† William Y. Svrcek,† and Marco A. Satyro‡ The University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, and SEA++ Inc., Calgary, Alberta, Canada T3G 2J1

The original form of the Peng-Robinson (PR) cubic equation of state does not predict liquid densities that are accurate enough to be used for process calculations. Liquid densities predicted by the PR equation have been improved by incorporating a Gaussian-like volume translation which is a strong function of temperature near the critical point but becomes a constant value away from the critical point, without any discontinuities in the critical region. This simple method provides liquid densities for 283 nonpolar, polar, and hydrogen-bonding fluids with an overall average absolute deviation of 0.70%, which compares favorably with existing volume translation and dedicated liquid density calculation methods. Introduction The use of the Peng-Robinson (PR) equation of state to provide thermodynamic properties for process calculations is well-known, and with proper care, the equation can be used to model phase behavior and thermophysical properties of fluid mixtures encountered in natural gas, refinery, petrochemical, and chemical applications. As originally proposed, like other cubic equations of state, predicted liquid-phase densities from the PR equation are not accurate enough for process design purposes. As shown by Peneloux et al. (1982), the overall average deviation in liquid densities for 233 compounds was about 5.2%. Although liquid density models, such as the modified Rackett equation (Spencer and Danner, 1972) or the COSTALD model (Hankinson and Thomson, 1979), can be used to predict liquid densities within about 1%, they are typically only applicable in the subcritical range. As a result, there is a discontinuity where the density calculations switch to the equation of state, i.e., in moving from the liquid to the dense phase. However, liquid densities calculated by a cubic equation of state were improved significantly by using a constant-volume translation, calculated at a fixed temperature. Peneloux et al. (1982) obtained values of the volume translation from experimental liquid density data for 233 compounds and generalized these values using the modified Rackett compressibility, ZRA. Overall, the average deviation in calculated liquid densities was reduced to about 1.7%, without affecting the vaporliquid equilibrium. Although this procedure works well at temperatures away from the critical point, deviations increase substantially beyond a reduced temperature of about 0.70.8 (Peneloux et al., 1982). As shown and discussed, the volume shift increases strongly with temperature * Corresponding author. Phone: 403-220-5751. Fax: 403282-3945. E-mail: [email protected]. † The Univeristy of Calgary. ‡ SEA++ Inc.

when close to the critical point (Soave, 1984; Watson et al., 1986; Mathias et al., 1989) As such, Watson et al. (1986) correlated the volume shift with an exponential function. Unfortunately, this function is only valid up to the critical point, thus causing a discontinuity as well as some unrealistic PVT behavior, as shown by Hnedkovsky and Cibulka (1990). Indeed, depending on how the volume shift is incorporated in the equation of state, the temperature dependency of the parameters may result in physically unrealistic negative predictions for heat capacities (Trebble and Bishnoi, 1986). Mathias et al. (1989) also proposed a method to increase the volume shift near the critical point, based on adding a term which is a function of the bulk modulus (inverse isothermal compressibility), which must be calculated. It should be noted that, in order for the method to predict the critical shift, it requires the critical volume as a parameter. As is shown, this method gives remarkably good results for the entire temperature range from the triple point up to the critical point for low molecular weight hydrocarbons, nitrogen, and water and is applicable to both the vapor and liquid phases. Unfortunately, Mathias et al. (1989) did not provide parameter values for a large number of components so that the method could be widely used, although a welldefined calculation procedure was suggested. The purpose of this work was to improve liquid densities predicted by Peng-Robinson for a wide variety of compounds with a simple volume translation, without discontinuities in the critical region. Like Mathias et al. (1989), this is an extension of the concept of Peneloux et al. (1982). However, it should be noted that it was not deemed imperative to accurately predict the critical volume itself. Based on this premise, a Gaussian-like function is used to model the temperature dependency of the volume shift, which provides a small relatively constant correction for temperatures away from the critical point but provides an enhanced correction which is continuous in the critical region. An evaluated database with 283 components is used in this study to determine the three parameters in the Gaussian-like volume translation function. A compari-

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1664 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Table 1. Database Components, Parameters, and Results component air argon boron trifluoride diborane bromochlorodifluoromethane dichlorodifluoromethane phosgene trichlorofluoromethane carbon tetrachloride carbon tetrafluoride chlorodifluoromethane dichlorofluoromethane chloroform trifluoromethane hydrogen cyanide dichloromethane difluoromethane methyl chloride methyl iodide formamide nitromethane methane methanol methyl mercaptan carbon monoxide carbonyl sulfide carbon dioxide carbon disulfide 1,2-dibromotetrafluoroethane chlorotrifluoroethylene chloropentafluoroethane 1,2-dichlorotetrafluoroethane 1,1,2-trichlorotrifluoroethane trichloroacetyl chloride tetrafluoroethylene acetylene 1,1-dichloroethylene 1,1,1,2-tetrafluoroethane 1-chloro-1,1-difluoroethane 1,1,1-trichloroethane 1,1,2-trichloroethane 1,1,1-trifluoroethane acetonitrile ethylene 1,2-dibromoethane 1,1-dichloroethane 1,2-dichloroethane 1,1-difluoroethane ethylene oxide acetic acid methyl formate ethyl chloride N-methylformamide ethane dimethyl ether ethanol dimethyl sulfoxide dimethyl sulfide ethyl mercaptan dimethylamine ethylamine hexafluoroacetone octafluoropropane methylacetylene propionitrile cyclopropane propylene 1,2-dichloropropane acetone 1,2-propylene oxide 1,3-propylene oxide ethyl formate methyl acetate propionic acid n-propyl chloride propane

c1 -0.31468 -0.30917 -0.28934 -0.38171 -0.39146 -0.39456 -0.39525 -0.41437 -0.40894 -0.36911 -0.31564 -0.37464 -0.33422 -0.31399 -0.48608 -0.33660 -0.36444 -0.26514 -0.40040 -0.52683 -0.45226 -0.32869 -0.39584 -0.33165 -0.32395 -0.28297 -0.25793 -0.40256 -0.19851 -0.36920 -0.42946 -0.43231 -0.44607 -0.49818 -0.28249 -0.31050 -0.41349 -0.33329 -0.36523 -0.40730 -0.42972 -0.39943 -0.54879 -0.36896 -0.34405 -0.40544 -0.34835 -0.38398 -0.34473 -0.49287 -0.36138 -0.20364 -0.55936 -0.35613 -0.35366 -0.39445 -0.51656 -0.34615 -0.27520 -0.33143 -0.27473 -0.40069 -0.46662 -0.32209 -0.56079 -0.35900 -0.38673 -0.40412 -0.45381 -0.43402 -0.40195 -0.38454 -0.38374 -0.52190 -0.19639 -0.38523

c2 1.24923 1.27282 1.37346 1.02554 1.00437 0.99544 0.99094 0.94640 0.96283 1.06352 1.25803 1.05488 1.17267 1.27489 0.86954 1.18917 1.11037 1.50208 0.97302 0.80323 0.90969 1.19628 1.02553 1.19250 1.21351 1.39365 1.53608 0.96950 1.86591 1.06649 0.91033 0.90647 0.87898 0.77737 1.39833 1.27753 0.94474 1.19936 1.08004 0.96795 0.93754 1.00747 0.77307 1.06440 1.15588 0.96824 1.13804 1.04927 1.16469 0.83843 1.11041 1.95019 0.75544 1.10410 1.11610 1.02040 0.80043 1.14633 1.43478 1.19843 1.44868 0.98607 0.83313 1.22881 0.75199 1.09867 1.01378 0.99208 0.89761 0.93446 1.00541 1.04295 1.04546 0.79116 1.98309 1.02043

c3 0.72576 0.72864 0.67219 0.72556 0.70896 0.71523 0.72287 0.71780 0.70114 0.72579 0.68699 0.70620 0.68487 0.67437 0.69541 0.67894 0.69103 0.66571 0.74689 0.67794 0.70057 0.71815 0.68135 0.70268 0.72746 0.63707 0.68977 0.73577 0.14877 0.74118 0.72059 0.71061 0.70900 0.75844 0.65132 0.70795 0.74269 0.67238 0.73157 0.70386 0.69246 0.65604 0.68313 0.71626 0.66219 0.71804 0.70563 0.68491 0.66240 0.68438 0.66992 0.55696 0.68259 0.71535 0.71177 0.66970 0.68342 0.67111 0.62164 0.66464 0.68216 0.70628 0.72918 0.69447 0.68368 0.70525 0.72756 0.67246 0.68484 0.67941 0.67797 0.67087 0.68068 0.68682 0.39049 0.71497

m 0.34566 0.37838 1.05616 0.55619 0.65641 0.63672 0.68255 0.65994 0.66010 0.65427 0.69531 0.66503 0.72625 0.77137 1.15906 0.69160 0.80888 0.60893 0.81216 0.96139 1.01008 0.40753 1.23300 0.68692 0.43979 0.52502 0.68386 0.53119 0.72422 0.90032 0.74354 0.74165 0.71678 0.85276 0.68955 0.66453 1.00327 0.84958 0.74657 0.70362 0.72628 0.74829 0.94175 0.50510 0.63404 0.68248 0.87108 0.81582 0.64259 1.17682 0.72830 0.63654 0.94793 0.52868 0.68176 1.26260 0.64570 0.68161 0.63442 0.80936 0.74357 0.86541 0.83917 0.69613 0.90063 0.55927 0.57112 0.73109 0.82202 0.80669 0.66046 0.76845 0.87740 1.16088 0.70863 0.59907

n 10-1

1.122 × -2.473 × 10-2 -3.216 × 10-1 7.169 × 10-2 -1.849 × 10-2 3.621 × 10-2 -5.477 × 10-2 -1.727 × 10-2 1.484 × 10-2 -7.138 × 10-2 2.557 × 10-2 8.900 × 10-2 -1.339 × 10-1 -5.717 × 10-2 -1.162 -1.055 × 10-1 -1.892 × 10-1 -2.545 × 10-2 -8.193 × 10-1 3.331 × 10-2 -7.504 × 10-1 -6.214 × 10-2 -4.053 × 10-1 -4.031 × 10-1 7.284 × 10-2 -9.517 × 10-3 9.977 × 10-2 5.022 × 10-2 1.164 × 10-1 -7.910 × 10-1 1.740 × 10-2 3.537 × 10-2 1.730 × 10-1 -3.546 × 10-1 1.217 × 10-1 -4.118 × 10-2 -1.275 -3.652 × 10-3 -1.665 × 10-1 -2.757 × 10-2 1.808 × 10-1 1.404 × 10-2 -4.789 × 10-1 1.577 × 10-2 2.678 × 10-1 2.110 × 10-1 -4.442 × 10-1 -3.771 × 10-1 1.406 × 10-1 -8.058 × 10-1 1.318 × 10-1 1.184 × 10-1 1.031 × 10-1 -1.024 × 10-2 -4.421 × 10-2 3.519 × 10-2 8.050 × 10-1 -1.032 × 10-1 1.150 × 10-1 6.220 × 10-2 2.854 × 10-1 2.311 × 10-1 4.084 × 10-2 -2.301 × 10-3 -3.266 × 10-1 4.576 × 10-2 8.602 × 10-2 1.123 × 10-1 -1.514 × 10-2 -2.187 × 10-1 7.776 × 10-2 1.325 × 10-1 -1.061 × 10-1 8.839 × 10-2 1.862 × 10-2 2.812 × 10-2

FDEVa

VPDEVb

1.23 1.34 0.38 1.01 0.58 0.71 0.97 0.73 0.49 1.08 0.34 0.57 0.43 0.41 2.76 0.35 1.05 0.33 1.46 2.73 1.51 1.07 1.43 0.49 1.24 0.36 0.31 1.38 1.01 0.60 0.80 0.63 0.62 1.47 0.47 0.39 1.12 0.33 0.47 0.46 0.57 0.74 2.60 1.89 0.33 0.70 0.36 0.72 0.64 1.84 0.55 0.95 2.24 0.80 0.56 0.84 1.44 0.34 0.50 0.36 0.36 0.51 1.08 0.40 2.06 0.62 0.95 0.39 1.10 1.01 0.80 0.45 0.46 1.61 0.83 0.70

0.40 0.24 2.03 1.23 0.24 0.23 0.20 0.28 0.43 0.30 0.26 0.34 0.40 0.26 1.09 0.81 0.71 0.24 1.71 0.26 1.46 0.19 0.21 1.49 0.21 0.20 0.15 0.41 0.38 1.36 0.61 0.61 0.36 1.23 0.24 0.77 3.23 0.24 0.37 0.41 0.42 0.08 0.06 0.18 0.31 0.19 0.91 0.21 0.17 1.06 0.19 0.11 0.45 0.25 0.74 0.78 0.84 0.54 0.09 1.40 0.14 0.69 0.26 0.37 0.36 0.28 0.24 0.06 0.05 0.58 0.39 0.15 0.96 0.90 0.31 0.31

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1665 Table 1 (Continued) component isopropyl alcohol 1-propanol isopropylamine n-propylamine trimethylamine octafluorocyclobutane furan thiophene pyrrole 1,3-butadiene vinyl acetate cis-2-butene trans-2-butene cyclobutane isobutene 1-butanal 1,2-epoxybutane methyl ethyl ketone tetrahydrofuran n-butyric acid 1,4-dioxane ethyl acetate isobutyric acid methyl propionate n-propyl formate tetrahydrothiophene n-butyl chloride isobutyl chloride pyrrolidine n-butane isobutane 1-butanol 2-butanol 2-methyl-1-propanol 2-methyl-2-propanol diethyl sulfide tert-butylamine tetramethylsilane pyridine glutaronitrile cyclopentene cyclopentane 2-methyl-1-butene 2-methyl-2-butene 3-methyl-1-butene 1-pentene cis-2-pentene trans-2-pentene methyl isopropyl ketone 2-pentanone 3-pentanone n-butyl formate ethyl propionate isopropyl acetate methyl n-butyrate n-pentanoic acid n-propyl acetate 1-chloropentane isopentane neopentane n-pentane ethyl propyl ether 2-methyl-2-butanol 3-methyl-1-butanol 1-pentanol 2-pentanol ethylene glycol monopropyl ether n-pentylamine hexafluorobenzene monochlorobenzene fluorobenzene iodobenzene nitrobenzene benzene phenol aniline

c1 -0.40451 -0.36636 -0.38245 -0.33099 -0.38005 -0.45173 -0.34269 -0.38474 -0.35401 -0.35575 -0.45840 -0.38637 -0.39314 -0.36041 -0.41680 -0.42529 -0.42433 -0.44430 -0.30804 -0.52002 -0.31805 -0.42248 -0.56807 -0.41628 -0.41999 -0.43093 -0.39825 -0.37852 -0.31511 -0.40679 -0.41744 -0.28118 -0.39866 -0.32698 -0.20839 -0.31021 -0.43518 -0.44631 -0.34667 -0.72961 -0.41452 -0.41409 -0.41740 -0.44215 -0.43557 -0.39194 -0.31384 -0.19594 -0.20829 -0.45529 -0.43335 -0.42679 -0.45590 -0.45250 -0.45121 -0.54311 -0.46322 -0.45136 -0.25141 -0.44936 -0.40451 -0.45933 -0.41129 -0.29030 -0.20430 -0.33132 -0.46215 -0.37571 -0.44208 -0.35631 -0.36985 -0.35481 -0.46657 -0.38199 -0.48189 -0.31729

c2 0.99220 1.09159 1.03331 1.20236 1.03832 0.86494 1.15628 1.02549 1.13009 1.11384 0.88311 1.02072 1.00185 1.08386 0.94547 0.94659 0.95005 0.91087 1.27873 0.78755 1.23558 0.95137 0.73115 0.96486 0.95644 0.93145 1.00545 1.04947 1.23018 0.96697 0.94052 1.41890 1.00448 1.22027 1.91428 1.26412 0.90257 0.87973 1.14376 0.60430 0.94574 0.95128 0.96113 0.91016 0.89891 1.00907 1.25671 1.92800 1.91341 0.88508 0.92856 0.93942 0.88357 0.88871 0.89162 0.75379 0.87064 0.89094 1.53607 0.87006 0.97611 0.84966 0.96484 1.34752 1.94199 1.20255 0.87194 1.05614 0.90939 1.11604 1.07269 1.12135 0.86807 1.03469 0.80356 1.25534

c3 0.66201 0.65199 0.69284 0.64844 0.70363 0.72262 0.68410 0.70315 0.65182 0.68567 0.67802 0.70059 0.71098 0.62571 0.71499 0.67603 0.67570 0.69661 0.62008 0.68922 0.62730 0.68306 0.68847 0.68006 0.67384 0.62939 0.67263 0.65231 0.61480 0.70928 0.70522 0.63378 0.67482 0.61625 0.68539 0.60986 0.69180 0.70325 0.70388 0.70900 0.71959 0.68052 0.65821 0.66107 0.71872 0.69671 0.67942 0.23041 0.72630 0.68559 0.69154 0.67585 0.68754 0.67580 0.67610 0.69103 0.68622 0.68844 0.47417 0.71787 0.69291 0.73759 0.66015 0.51931 0.53888 0.62539 0.69882 0.68640 0.68535 0.66570 0.67742 0.66885 0.72793 0.69457 0.74093 0.66491

m 1.20084 1.13021 0.71657 0.66106 0.70584 0.86151 0.62317 0.65112 0.75670 0.65454 0.84832 0.67407 0.72303 0.65757 0.66690 0.79476 0.68807 0.84283 0.71853 1.36215 0.75433 0.90511 1.26386 0.87753 0.79733 0.63436 0.77163 0.73284 0.71537 0.67281 0.63997 1.03652 0.99236 1.00458 1.01699 0.84821 0.70276 0.64746 0.70108 1.40791 0.68595 0.68866 0.65997 0.80478 0.74538 0.70781 0.73449 0.73829 1.08710 0.84906 0.83284 0.91776 0.93383 0.82351 0.92074 1.33512 0.92604 0.95631 0.71126 0.66175 0.74697 0.99926 0.71617 0.98869 1.10751 0.89286 1.30435 0.97439 0.94800 0.73722 0.72985 0.71530 1.27360 0.67783 1.11122 0.89999

n 10-1

5.713 × 6.666 × 10-1 3.786 × 10-1 7.117 × 10-1 -1.422 × 10-1 1.668 × 10-1 2.992 × 10-1 7.407 × 10-2 3.109 × 10-1 1.450 × 10-2 2.077 × 10-1 2.587 × 10-2 -1.528 × 10-1 -5.446 × 10-2 -1.637 × 10-2 -6.750 × 10-2 2.014 × 10-1 5.504 × 10-3 -4.398 × 10-2 -2.285 × 10-1 1.616 × 10-1 1.548 × 10-2 -1.186 × 10-1 2.334 × 10-2 1.748 × 10-1 2.104 × 10-1 2.921 × 10-2 -3.601 × 10-4 3.041 × 10-1 -3.032 × 10-3 3.070 × 10-2 1.007 1.122 1.169 1.288 -2.379 × 10-1 4.186 × 10-1 3.381 × 10-1 1.537 × 10-1 -1.130 -1.011 × 10-1 -1.133 × 10-1 3.082 × 10-1 -1.516 × 10-1 -1.787 × 10-1 4.499 × 10-2 1.465 × 10-2 1.286 × 10-2 -1.151 1.324 × 10-1 2.236 × 10-1 7.921 × 10-2 6.140 × 10-2 4.612 × 10-1 1.669 × 10-2 6.223 × 10-2 6.612 × 10-2 -5.524 × 10-1 -1.770 × 10-3 2.875 × 10-2 -1.054 × 10-2 -6.850 × 10-1 2.007 1.267 6.233 × 10-1 1.616 -1.244 -8.006 × 10-2 -1.456 × 10-2 4.099 × 10-2 6.025 × 10-2 1.332 × 10-1 -1.442 5.920 × 10-2 -5.326 × 10-1 1.207 × 10-1

FDEVa

VPDEVb

0.59 0.44 0.43 0.31 0.44 0.74 0.35 0.48 0.38 0.35 0.81 0.49 0.56 1.57 0.77 0.61 0.64 0.77 0.92 1.28 0.99 0.51 1.66 0.48 0.49 0.61 0.36 0.34 1.02 0.59 0.64 1.26 0.40 0.70 1.25 0.92 0.59 0.49 0.33 2.78 0.75 0.51 0.43 0.61 0.71 0.38 0.38 0.97 0.70 0.59 0.50 0.40 0.55 0.50 0.53 1.18 0.60 0.47 0.83 0.73 0.38 0.83 0.37 1.05 1.31 0.83 0.58 0.35 0.48 0.37 0.33 0.39 0.77 0.39 1.56 0.57

1.28 0.56 0.17 0.69 0.44 0.21 0.08 0.42 0.02 0.22 0.23 0.20 0.42 0.05 0.34 0.59 0.31 0.14 0.76 0.81 0.22 0.83 1.77 0.87 0.38 0.30 0.30 0.51 0.10 0.32 0.36 0.73 0.45 0.07 0.38 1.09 0.59 0.21 0.19 0.55 0.75 0.85 0.40 0.54 0.59 0.17 0.37 0.25 2.59 0.03 0.43 0.17 0.48 0.74 0.85 0.51 0.47 1.40 0.29 0.24 0.32 1.71 1.60 0.64 0.39 1.26 5.29 0.53 0.55 0.49 0.46 0.15 3.34 0.45 2.56 0.29

1666 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Table 1 (Continued) component 4-methylpyridine diethyl oxalate cyclohexane 2,3-dimethyl-1-butene 2-ethyl-1-butene 1-hexene trans-2-hexene methylcyclopentane 2-methyl-1-pentene 2-methyl-2-pentene 4-methyl-1-pentene cyclohexanol methyl isobutyl ketone n-butyl acetate tert-butyl acetate ethyl n-butyrate isobutyl acetate n-propyl propionate cyclohexylamine 2,2-dimethylbutane 2,3-dimethylbutane n-hexane 3-methylpentane diisopropyl ether di-n-propyl ether 1-hexanol benzonitrile benzyl chloride m-cresol o-cresol o-toluidine p-toluidine cycloheptane 1,1-dimethylcyclopentane cis-1,2-dimethylcyclopentane trans-1,2-dimethylcyclopenta cis-1,3-dimethylcyclopentane trans-1,3-dimethylcyclopenta ethylcyclopentane 1-heptene cis-2-heptene cis-3-heptene methylcyclohexane ethyl isovalerate n-heptanoic acid isopentyl acetate 2,2-dimethylpentane 2,3-dimethylpentane 2,4-dimethylpentane 3,3-dimethylpentane 3-ethylpentane n-heptane 2-methylhexane 3-methylhexane 2,2,3-trimethylbutane ethylbenzene o-xylene p-xylene phenetole 1,1-dimethylcyclohexane cis-1,2-dimethylcyclohexane trans-1,2-dimethylcyclohexan cis-1,3-dimethylcyclohexane trans-1,3-dimethylcyclohexan cis-1,4-dimethylcyclohexane trans-1,4-dimethylcyclohexan ethylcyclohexane 2-ethyl-1-hexene 1-octene n-propylcyclopentane n-octanoic acid 2,2-dimethylhexane 2,3-dimethylhexane 2,4-dimethylhexane 3,3-dimethylhexane 3,4-dimethylhexane

c1 -0.25991 -0.39520 -0.35456 -0.34236 -0.44204 -0.19775 -0.38111 -0.42646 -0.37396 -0.43301 -0.30915 -0.41420 -0.41651 -0.40958 -0.40278 -0.50727 -0.45461 -0.43642 -0.48493 -0.46469 -0.25390 -0.20004 -0.43847 -0.47233 -0.45282 -0.41881 -0.51303 -0.48959 -0.43131 -0.47320 -0.56629 -0.50085 -0.45638 -0.24990 -0.40131 -0.25067 -0.50831 -0.49701 -0.43274 -0.42543 -0.46928 -0.46268 -0.44229 -0.45148 -0.48074 -0.42632 -0.45162 -0.42324 -0.45034 -0.48436 -0.43200 -0.40257 -0.40568 -0.42303 -0.43104 -0.38513 -0.35367 -0.43990 -0.47208 -0.47267 -0.52476 -0.48058 -0.41650 -0.48793 -0.48709 -0.40225 -0.47671 -0.57403 -0.49988 -0.39925 -0.52293 -0.44322 -0.43995 -0.39607 -0.39521 -0.43994

c2 1.53476 1.00103 1.10403 1.15635 0.90856 2.00726 1.04817 0.92213 1.06791 0.92623 1.25778 0.96618 0.96122 0.97604 0.97410 0.79908 0.88254 0.91817 0.79863 0.84203 1.51478 1.97240 0.89743 0.82826 0.86740 0.94657 0.79574 0.82687 0.92933 0.82431 0.73183 0.81231 0.85848 1.47112 0.98182 1.47681 0.75868 0.77864 0.91177 0.94061 0.85807 0.86870 0.88834 0.86963 0.83547 0.93782 0.87208 0.93170 0.87652 0.80538 0.91334 0.99236 0.97818 0.93847 0.90625 1.03733 1.12842 0.91208 0.85313 0.85096 0.77455 0.83750 0.95978 0.82673 0.82787 0.99298 0.84306 0.66174 0.80526 0.99448 0.77111 0.89216 0.90027 0.99425 1.00174 0.89906

c3 0.71308 0.78344 0.65418 0.66449 0.65838 0.58170 0.65806 0.70290 0.65131 0.66178 0.58408 0.65678 0.70212 0.71045 0.67153 0.68998 0.67285 0.69298 0.74258 0.70897 0.43755 0.49069 0.70485 0.71293 0.70896 0.67072 0.69887 0.68167 0.66164 0.72160 0.68998 0.69658 0.71143 0.33432 0.70938 0.35812 0.73847 0.73184 0.69450 0.66786 0.68912 0.67621 0.70754 0.73651 0.73054 0.72883 0.69653 0.66563 0.68283 0.72533 0.70356 0.67445 0.67686 0.65527 0.68314 0.67490 0.66310 0.67705 0.70019 0.66954 0.67053 0.66338 0.65916 0.67272 0.67120 0.64915 0.65950 0.75271 0.67053 0.67752 0.70494 0.69109 0.68008 0.68367 0.62310 0.68380

m 0.84698 1.89129 0.67107 0.66302 0.59877 0.78212 0.71190 0.70788 0.67621 0.65189 0.70424 0.74608 1.06123 0.95154 0.78452 0.97260 0.94744 1.00925 0.98411 0.73405 0.72921 0.83705 0.78719 0.81170 0.95626 1.05703 1.01480 0.77737 0.98474 1.01323 1.06871 1.10252 0.73795 0.81266 0.77736 0.79342 0.76027 0.78951 0.77714 0.85643 0.78460 0.77407 0.72467 0.82206 1.44932 1.27751 0.81932 0.79794 0.80487 0.76051 0.82096 0.87071 0.84540 0.83466 0.74262 0.81372 0.83057 0.85032 1.01553 0.67243 0.64509 0.67714 0.66232 0.64236 0.63970 0.67692 0.66862 1.03867 0.90876 0.85316 1.38436 0.85710 0.87653 0.86670 0.84111 0.86313

n 10-1

-1.649 × -2.339 8.505 × 10-2 2.789 × 10-1 6.538 × 10-1 2.228 × 10-2 2.775 × 10-1 4.496 × 10-2 3.187 × 10-1 4.888 × 10-1 1.316 × 10-1 1.089 -7.001 × 10-1 8.810 × 10-2 5.123 × 10-1 -1.234 × 10-1 2.607 × 10-1 2.036 × 10-2 -5.210 × 10-1 -7.052 × 10-2 4.785 × 10-2 -1.215 × 10-1 -5.834 × 10-2 2.869 × 10-1 -2.649 × 10-1 8.237 × 10-1 -6.271 × 10-1 3.179 × 10-1 2.255 × 10-1 -8.785 × 10-2 -3.396 × 10-1 -2.969 × 10-1 -2.335 × 10-2 -2.261 × 10-1 -6.426 × 10-2 -1.292 × 10-1 8.076 × 10-2 -1.099 × 10-1 -1.891 × 10-2 -1.417 × 10-4 1.269 × 10-1 1.855 × 10-1 -8.223 × 10-3 7.230 × 10-1 -8.756 × 10-2 -1.138 -1.150 × 10-1 5.226 × 10-2 6.079 × 10-2 3.682 × 10-2 3.885 × 10-2 6.279 × 10-2 4.360 × 10-2 5.005 × 10-2 7.325 × 10-3 1.771 × 10-2 -1.940 × 10-2 -4.695 × 10-2 -2.115 × 10-1 2.739 × 10-1 4.324 × 10-1 2.876 × 10-1 3.653 × 10-1 4.576 × 10-1 4.531 × 10-1 2.830 × 10-1 4.019 × 10-1 -6.525 × 10-1 5.537 × 10-2 -1.753 × 10-2 3.520 × 10-1 5.153 × 10-2 9.759 × 10-3 3.942 × 10-2 1.507 × 10-3 1.686 × 10-2

FDEVa

VPDEVb

0.67 0.47 0.44 0.55 0.44 0.91 0.33 0.48 0.34 0.39 0.69 0.39 0.34 0.30 0.32 0.72 0.42 0.36 1.22 0.63 0.87 0.94 0.46 0.59 0.48 0.39 1.03 0.69 0.44 0.88 1.34 0.88 0.59 2.14 0.74 2.06 1.23 1.04 0.41 0.34 0.47 0.42 0.47 0.42 0.45 0.33 0.41 0.42 0.38 0.68 0.37 0.33 0.38 0.36 0.36 0.32 0.42 0.39 0.50 0.44 0.72 0.48 0.30 0.52 0.52 0.34 0.44 1.84 0.49 0.38 0.52 0.36 0.36 0.33 0.59 0.37

0.93 4.45 0.15 0.06 0.49 0.31 0.10 0.31 0.11 0.16 0.06 1.97 1.80 0.61 0.42 0.45 0.68 0.26 0.64 0.35 0.24 0.59 0.52 0.76 1.04 0.84 1.34 0.30 1.29 0.56 0.97 0.53 0.44 0.44 0.43 0.49 0.19 0.37 0.40 0.37 0.44 0.09 0.47 1.12 2.73 3.51 0.96 0.29 0.30 0.26 0.51 0.29 0.51 0.33 0.35 0.32 0.27 0.30 1.18 0.14 0.05 0.08 0.05 0.06 0.06 0.10 0.05 1.60 0.20 0.34 1.39 0.34 0.28 0.30 0.33 0.31

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1667 Table 1 (Continued) component 3-ethylhexane 2-methyl-3-ethylpentane 2-methylheptane 3-methylheptane 4-methylheptane 2,2,3,3-tetramethylbutane 2,2,3-trimethylpentane 2,2,4-trimethylpentane 2,3,3-trimethylpentane 2,3,4-trimethylpentane di-n-butyl ether octamethylcyclotetrasiloxane cumene mesitylene n-propylbenzene 1,2,4-trimethylbenzene n-propylcyclohexane 3,3-diethylpentane 2-methyloctane 3-methyloctane 4-methyloctane n-nonane 2,2,3,3-tetramethylpentane 2,2,3,4-tetramethylpentane 2,2,4,4-tetramethylpentane 2,3,3,4-tetramethylpentane 2,2,5-trimethylhexane naphthalene 1,2,3,4-tetrahydronaphthalen n-butylbenzene p-cymene 1,2,4,5-tetramethylbenzene cis-decahydronaphthalene 1-decene n-decane 1-decanol 2-methylnaphthalene n-undecane biphenyl diphenyl ether diphenylmethane dibutyl sebacate perchloryl fluoride deuterium oxide hydrogen iodide hydrogen (para) water hydrogen selenide krypton nitrogen nitrous oxide neon oxygen ozone sulfur trioxide a

c1 -0.38526 -0.44800 -0.45701 -0.43314 -0.41716 -0.56498 -0.47635 -0.43481 -0.48669 -0.47085 -0.39052 -0.46980 -0.20695 -0.48001 -0.40556 -0.47114 -0.48643 -0.28370 -0.49662 -0.47089 -0.46578 -0.52864 -0.50339 -0.48755 -0.27356 -0.50845 -0.45701 -0.41976 -0.43032 -0.45685 -0.50722 -0.49875 -0.45032 -0.52086 -0.56476 -0.47288 -0.49230 -0.60968 -0.54147 -0.50835 -0.52785 -0.64443 -0.37461 -0.17075 -0.35500 -0.31930 -0.19963 -0.28529 -0.31703 -0.32392 -0.29679 -0.25498 -0.18062 -0.32532 -0.30341

c2

c3

1.03456 0.88371 0.87641 0.92285 0.95762 0.69054 0.82622 0.90104 0.80518 0.83590 1.01615 0.84313 1.92774 0.83837 0.98528 0.85336 0.82541 1.36336 0.80883 0.85072 0.85958 0.76321 0.77678 0.80666 1.45851 0.76698 0.86482 0.95355 0.92994 0.87714 0.79540 0.80821 0.87629 0.77291 0.71765 0.84559 0.81889 0.66926 0.70948 0.79259 0.76500 0.61663 1.04672 2.38685 1.10543 1.22708 2.03032 1.37118 1.24107 1.21383 1.33449 1.54659 2.14958 1.23409 1.31618

0.65398 0.66675 0.67124 0.65574 0.65990 0.74290 0.68168 0.67672 0.70111 0.69260 0.70294 0.70792 0.69807 0.68491 0.69830 0.68422 0.66479 0.50554 0.69043 0.69429 0.68983 0.68872 0.70325 0.69537 0.65451 0.71501 0.69390 0.67721 0.66564 0.70215 0.69311 0.70646 0.70116 0.70025 0.69966 0.70896 0.69619 0.70190 0.74304 0.71694 0.70606 0.82461 0.72902 0.48845 0.71611 0.70628 0.38268 0.65517 0.71923 0.72859 0.69432 0.71570 0.42673 0.65984 0.83737

m 0.90162 0.84901 0.90562 0.91316 0.90570 0.71598 0.80408 0.81336 0.79940 0.83154 1.01886 1.13994 0.84445 0.93640 0.86927 0.93635 0.65176 0.90999 0.99047 0.99025 0.99204 0.99798 0.75463 0.81049 0.84897 0.85079 0.89025 0.84087 0.83964 0.95907 0.90432 1.02564 0.83484 1.09200 1.07514 1.25527 0.99924 1.12289 0.92014 1.08983 1.06656 1.93169 0.63130 0.95908 0.40051 0.08567 0.91132 0.46303 0.37973 0.43692 0.50850 0.33545 0.41166 0.71531 0.65805

n 10-3

-7.121 × 2.935 × 10-2 7.829 × 10-2 -3.958 × 10-3 3.069 × 10-2 1.138 × 10-1 3.037 × 10-2 1.730 × 10-2 -4.934 × 10-4 2.090 × 10-2 1.362 × 10-1 3.994 × 10-1 3.124 × 10-2 5.179 × 10-2 2.704 × 10-2 -9.691 × 10-2 6.176 × 10-1 -2.420 × 10-1 -4.298 × 10-2 -1.082 × 10-1 -1.172 × 10-1 6.501 × 10-2 1.806 × 10-1 9.726 × 10-2 -6.140 × 10-2 -1.020 × 10-1 2.840 × 10-2 -1.238 × 10-1 6.910 × 10-2 -9.504 × 10-2 1.259 × 10-2 -1.643 × 10-1 -1.643 × 10-1 -1.752 × 10-1 1.397 × 10-3 3.186 × 10-1 -4.726 × 10-1 2.149 × 10-2 -6.956 × 10-2 -5.097 × 10-1 -1.884 × 10-1 -4.085 × 10-1 3.589 × 10-3 -3.027 × 10-1 2.124 × 10-1 -2.657 × 10-1 -2.285 × 10-1 1.678 × 10-2 -8.122 × 10-3 -4.905 × 10-3 4.945 × 10-1 -9.257 × 10-2 5.645 × 10-3 -1.559 × 10-1 2.068

FDEVa

VPDEVb

0.66 0.37 0.35 0.36 0.36 0.65 0.43 0.56 0.51 0.43 0.43 0.42 1.13 0.46 0.29 0.42 0.40 0.83 0.39 0.33 0.32 0.50 0.51 0.42 1.33 0.58 0.36 0.32 0.34 0.33 0.50 0.48 0.37 0.42 0.60 0.34 0.50 0.75 1.25 0.46 0.51 0.74 0.99 0.04 1.11 1.62 0.04 1.45 1.03 1.27 0.51 1.26 1.84 0.81 0.68

0.44 0.31 0.17 0.39 0.36 0.44 0.33 0.33 0.30 0.28 0.54 0.74 0.44 0.28 0.12 0.43 0.20 1.01 0.95 1.10 1.03 0.15 0.18 0.11 0.89 0.85 0.20 1.09 0.10 0.48 0.47 0.65 0.77 1.09 0.40 1.95 1.80 0.41 1.12 0.77 0.50 0.33 0.13 0.87 0.11 0.09 0.18 0.82 0.23 0.19 0.49 0.09 0.13 0.44 2.35

FDEV ) 100|FCALC - FDATABASE|/FDATABASE. b VPDEV ) 100|VPCALC - VPDATABASE|/VPDATABASE.

son between the proposed procedure and the use of a temperature-independent Peneloux volume shift or liquid density model is presented, and the effect of using a temperature-dependent volume translation on the behavior of pressure-volume isotherms is investigated. Volume Translation Model The Peng-Robinson cubic equation of state is given by

P)

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

(1)

where a and b are component-dependent parameters;

b is independent of temperature and a is comprised of the product of a temperature-independent term and a temperature-dependent term. The temperature-independent parameters are determined by applying the Gibbs critical point criteria. These parameters are determined as follows:

b ) 0.07780RTc/Pc

(2)

a ) acβ(T)

(3)

ac ) 0.45724R2Tc2/Pc

(4)

The temperature function β(T) is determined by

1668 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998

Figure 1. Nitrogen saturation envelope of the Gaussian volume translation. Temperatures are from 63.1 K to Tc. The average error in molar volume for the vapor and liquid phases is equal to 2.71%.

matching vapor pressures predicted by the equation of state with experimental values and has been correlated with several functional forms. In this work, the equation proposed by Mathias (1983) is used to calculate β(T):

β(T) ) [1 + m(1 - xTr) - n(1 - xTr)2]2

(5)

The predicted liquid volume was then corrected using the volume translation concept of Peneloux et al. (1982) combined with a temperature-dependent Gaussian-like enhancement, which is only significant in the critical region:

vCORR ) vPREOS + c c ) c1 +

c4

x2πc2

[ (

exp -0.5

(6)

)]

T r - c3 c2

2

(7)

As is shown, the volume translation requires three parameters, which have been regressed for 283 compounds. The dimensional constant c4 is set to 1 m3/kmol in this work. Database and Results The database consists of 283 pure components, with critical properties, acentric factor, liquid density, and vapor pressure data taken from the DIPPR database (Daubert and Danner, 1994). The database was originally compiled by Satyro (1997), with the components chosen according to the following criteria: 1. A maximum amount of experimental data.

2. Critical temperature and pressure data accurate to within 5%. 3. Liquid density data accurate to within 2%. 4. Vapor pressure data accurate to within 5%. The accuracy values stated above are based on the quality indicators given in the database. However, comparisons by Monnery (1995) and Satyro (1997) support the stated accuracy values and consistency of the data. Furthermore, a qualitative criteria was that the database represents as wide a variety of nonpolar, polar, and hydrogen-bonding compounds as possible within the previous constraints. The resulting database is shown in Table 1. Overall, the temperature range of the data was from a reduced temperature of 0.60 to 0.95 K. Since the volume translation at lower temperatures is approximately constant, it was not deemed necessary to use temperatures any lower than Tr ) 0.6. Using eqs 1-7, molar volumes were calculated and the parameters c1, c2, and c3 were regressed to minimize the error between calculated and database values of liquid density. Although not the primary purpose of this work, by also utilizing fugacity relationships, parameters m and n were simultaneously regressed to minimize the difference between calculated and database values of vapor pressure. Note that the method also calculates saturated vapor and dense phase volumes. The resulting parameter values of c1, c2, c3, m, and n for each compound are given in Table 1. The average absolute deviations between predicted and database values of liquid density and vapor pressure for each compound are also given in Table 1. As is shown, the average absolute deviations for liquid densities are excellent, with an overall average for all 283 compounds of 0.70%, which compares favorably with the constant-

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1669

Figure 2. Propane saturation envelope of the Gaussian volume translation. Temperatures are from 235.47 K to Tc. The average error in molar volume for the vapor and liquid phases is equal to 2.46%.

Figure 3. Water saturation envelope of the Gaussian volume translation. Temperatures are from 283.2 K to Tc. The average error in molar volume for the vapor and liquid phases is equal to 3.23%.

volume translation of Peneloux et al. (1982). For methane, ethane, propane, isobutane, n-butane, nitrogen,

water, and carbon dioxide, the results in this work are comparable to those of Mathias et al. (1989). The

1670 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998

Figure 4. Nitrogen PVT map predicted using the Gaussian volume translation.

Figure 5. Propane PVT map predicted using the Gaussian volume translation.

saturated liquid and vapor densities for nitrogen, propane, and water which compared well to those generated by the NIST Standard Reference Database 12 (1992) are shown in Figures 1-3, and the method can also be used for the saturated vapor phase.

In addition, according to DIPPR (1982) for 255 compounds, the overall average absolute deviations for the modified Rackett equation (Spencer and Danner, 1972) and the COSTALD method (Hankinson and Thomson, 1979) are 0.7% and 0.8%, respectively, for

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1671

Figure 6. Water PVT map predicted using the Gaussian volume translation.

hydrocarbons and 1.2% and 2.1%, respectively, for nonhydrocarbons. Although the database from the DIPPR (1982) study and the one used in this study are not identical, they are similar and overall the average absolute deviations in this work compare favorably with those stated for the modified Rackett equation and the COSTALD method. A concern with temperature-dependent volume translation methods is the possibility of unrealistic PVT behavior and thermodynamic properties as discussed by Trebble and Bishnoi (1986) and Hnedkovsky and Cibulka (1990). Isotherms for nitrogen, propane, and water are given in Figures 4-6. As is shown, the subcooled and saturated liquid volumes are properly predicted using the temperature-dependent volume translation. Nevertheless, for high pressures and temperatures, the magnitude of the volume translation correction is too large. For nitrogen, propane, and water, reasonable dense gas volumes are predicted only up to the vicinity of the critical pressure. As such, the method needs to be improved and then applied to mixtures.

mended for the dense state at pressures above the critical pressure. Nomenclature a ) Peng-Robinson parameter, J2/mol2‚K/kPa ac ) Peng-Robinson parameter, J2/mol2‚K/kPa b ) Peng-Robinson parameter, m3/kmol c ) volume translation parameter, m3/kmol c1, c4 ) Gaussian parameters, m3/kmol c2, c3 ) Gaussian parameters m ) Mathias-Copeman parameter n ) Mathias-Copeman parameter NC ) number of components in the mixture P ) pressure, kPa R ) universal gas constant, J/mol‚K T ) temperature, K v ) molar volume, m3/kmol VP ) vapor pressure, kPa β ) Mathias-Copeman parameter Subscripts

Conclusions A temperature-dependent volume translation with a Gaussian-like critical enhancement, which has no discontinuities in the critical region, has been developed for the Peng-Robinson equation of state. The predicted liquid densities for 283 compounds have an overall average absolute deviation of 0.70%, which compares favorably with temperature-independent volume translations and dedicated liquid density models. Isotherms show that good results can be obtained for subcooled and saturated conditions. The use of the proposed volume translation expression is not recom-

c ) critical DEV ) deviation r ) reduced

Literature Cited Daubert, T. E.; Danner, R. P. DIPPR Data Compilation of Pure Compound Properties, Version 9, NIST/SRD Database 11, 1994. DIPPR Documentation Report No. 802-4-82; Design Institute for Physical Properties, American Institute of Chemical Engineers: New York, 1982. Hankinson, R. W.; Thomson, G. H. A New Correlation for Saturated Liquid Densities of Liquids and Their Mixtures. AIChE J. 1979, 25, 653.

1672 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Hnedkovsky, L.; Cibulka, I. On a Temperature Dependence of the van der Waals Volume Parameter in Cubic Equations of State. Fluid Phase Equilib. 1990, 60, 327. Mathias, P. M. A Versatile Phase Equilibrium Equation of State. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 385. Mathias, P. M.; Naheiri, T.; Oh, E. M. A Density Correction for the Peng-Robinson Equation of State. Fluid Phase Equilib. 1989, 47, 77. Monnery, W. D. Viscosity Prediction from a Modified Square Well Intermolecular Potential Model. Ph.D. Dissertation, The University of Calgary, Calgary, Aberta, Canada, 1995. NIST Standard Reference Database 12; Fluid Mixtures Data Center, Thermophysics Division, National Institute of Standards and Technology: Boulder, CO, 1992. Peneloux, A.; Rauzy, E.; Freze, R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7. Satyro, M. A. An Equation of State Based On Significant Structure Theory. Ph.D. Dissertation, The University of Calgary, Calgary, Aberta, Canada, 1997.

Soave, G. Improvement of the van der Waals Equation of State. Chem. Eng. Sci. 1984, 39, 357. Spencer, C. F.; Danner, R. P. Improved Equation for Prediction of Saturated Liquid Density. J. Chem. Eng. Data 1972, 17, 236. Trebble, M. A.; Bishnoi, P. R. Accuracy and Consistency Comparisons of 10 Cubic Equations of State for Polar and Nonpolar Compounds. Fluid Phase Equilib. 1986, 29, 465. Watson, P.; Cascella, M.; May, D.; Salerno, S.; Tassios, D. Prediction of Vapor Pressures and Saturated Molar Volumes with a Simple Cubic Equation of State: Part II: The van der Waals 711 EOS. Fluid Phase Equilib. 1986, 27, 35.

Received for review September 10, 1997 Revised manuscript received January 13, 1998 Accepted February 6, 1998 IE970640J