726
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
A c k n o w l e d g- m e n t
van = specific viscosity, (v - vo)/vo ~. .
The authors are grateful to Occidental Research Corporation for permission to publish this work. Thanks are extended to Richard Price and Stephen Webster for their technical assistance.
7.1
Nomenclature
L = length of spindle Leff = effective length of spindle M = torque R = gas constant Ri = spindle radius T = absolute temperature x , = volume fraction of solid x , , ~= volume fraction for which viscosity tends to infinity B , E , K , n = constants Greek L e t t e r s
i. = shear rate 7 = apparent viscosity of t o= viscosity of medium
shear stress
no= angular velocity Literature Cited Back, A. L., Rubber Age, 639 (Jan 1959). Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena", Chapter 3, Wiley, New York, N.Y., 1960. Brookfield Engineering Laboratories, Inc., Stoughton, Mass., Brookfield Synchro-Lectric Viscometer Instruction Manual. Eubank, P. T., Fort, B. F., ISA Trans., 6(14), 296 (1976). Jeffrey, D. J., Acrivos, A., AIChE J., 22(3), 417 (1976). Jinescu, V. V.. Int. Chem. f n g . , 14, 397 (1974). Krieger, I.M., Dougherty, T. J., Trans. SOC. Rheol., 3, 137 (1959). Krieger, I. M., Maron, S.H.,J. Appl. Phys., 23(1), 147 (1952). Middleman, S., "The Flow of High Polymers, Contlnuum and Molecular Rheology", Interscience, New York, N.Y., 1968. Rosen, M. R., J . Colloid Interface Sci., 36(3), 350 (1972). Rosen, M. R.,J . Colloid Interface Sci., 39(2), 413 (1972). Sikdar, S.K., Or& F., AIChE J., 23(3), 380 (1977). Thomas, D. G., J. ColloidSci., 20, 267 (1965). Ting, A. P., Luebbers, R. H., AIChE J., 3(1), 111 (1957). Wood, J. H., Catacalos, G., Lieberman, S.V., J. pharm. Sci., 52(3), 296 (1963).
slurry
Received for review December 18, 1978 Accepted June 27, 1979
A Comparison of Eight Equations of State to Predict Gas-Phase Density and Fugacity Ramanathan R. Tarakad, Calvin F. Spencer,* and Stanley B. Adler Pullman Kellogg, A Division of Pullman Inc., Houston, Texas 77046
In recent years a number of new equations of state, many of them modifications of the original Redlich-Kwong equation, have been proposed. Eight different equations, some of them generalized and others more specific, are evaluated here for their ability to predict gas-phase density and fugacity. Both pure components and mixtures are considered. Polar systems, including aqueous-gas and sour-gas mixtures, are emphasized. Included in the study are the original Redlich-Kwong, the Redlich-Kwong-Chueh, the Redlich-Kwong-Soave, the Barner-Adler-Joffe, the virial, the Nakamura-Breedveld-Prausnitz, the Redlich-Kwong-deSantis, and the Redlich-Kwong-Guerreri equations of state. In terms of its predictive ability for gas-phase density, the original Redlich-Kwong equation is found to be as reliable as some of its modifications. At low and moderate pressures polar systems are adequately represented by the virial equation; however, at high pressures, none of the equations is in general reliable.
Introduction
Estimates of the density and fugacity are required in numerous design calculations. Density is directly required for sizing process equipment like tanks, pipes, columns, and pumps. It is also used in calculations for almost all separation operations. Further, since density is a measure of molecular interactions which govern the variation of thermophysical properties, it is often indirectly required as an input parameter for correlations to predict other physical properties. Fugacity is important in representing vapor and liquid-phase nonidealities in vapor-liquid equilibrium calculations. Specifically, the fugacity coefficient, di, is directly encountered while calculating the vapor-liquid equilibrium constant Ki through the single equation of state approach
or the two (multi) equation of state approach
For the most part, gas-phase density and fugacity are obtained from equations of state, many of which have evolved from the familiar two-parameter Redlich-Kwong (1949) equation. Typical examples include the ChuehPrausnitz (1967) modification and the Soave (1972) modification of the original formulation of Redlich and Kwong. Most modifications have emphasized improvement in predicting Ki by improving the prediction for liquid-phase nonideality. Their superiority over the Redlich-Kwong equation for vapor-liquid equilibrium calculations has been repeatedly demonstrated. There are, however, instances where a two-equation approach is more viable, and the main function of the equation of state is to account for vapor-phase nonideality. In such calculations the liquid-phase nonideality is represented through an activity coefficient model like the Margules, Wilson, NRTL, or UNIFAC equations, and the vapor phase through an equation of state. The ability of the equation of state to predict vapor fugacities then takes on special significance. Moreover, if the system contains polar components, the two-equation of state approach is almost always used. For such systems, the vapor phase has to be represented, quite often, by an equation of state developed
0019-7882/79/1118-0726$01.00/00 1979 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
727
Table I. Equations Evaluated in This Study no.
eauation of state
1.
the original two-parameter Redlich-Kwong (1949) equation the Chueh modification of the Redlich-Kwong equation the Joffe equation of state
2.
3.
abbreviation
RKC
See Chueh and Prausnitz (1967)
JBA
In the present study this always refers to the Barner-Adler (1970) modification of the original Joffe (1947) equation. Many versions of this equation are now in use. In this paper we have followed the development of Graboski and Daubert (1978). In this study second virial coefficients were predicted either by the method of Tarakad and Danner (1977) or by the method of Tsonopoulos (1974, 1975). For the systems studied here, the two methods give essentially equivalent results.
4.
the three-parameter Soave (197 2) modification of the Redlich-Kwong equation
RKS
5.
the virial equation
virial
6.
the Nakamura-BreedveldPrausnitz (1976) extended hard-sphere equationa the deSantis, Breedveld, Prausnitz ( 1 9 7 4 ) modification of the Redlich-Kwong equation for aqueous-gas mixtures the Guerreri-Prausnitz ( 1 9 7 3 ) modification of the RedlichKwong equation for systems containing ammonia and water
NBP
7.
8.
comments
RK
RKD
RKG
We thank Professor J. M. Prausnitz for providing us with a copy of the computer program for the Nakamura-BreedveldPrausnitz equation.
for nonpolar mixtures. Information about the relative merits of equations of state under such conditions is important, and this subject does not appear to have been investigated in detail. Therefore, the overall objectives in undertaking this study are as follows: (a) to evaluate the merits of equations of state that have been commonly used to calculate density and fugacity in the gas phase; (b) to compare these equations with others that have appeared more recently in the literature; (c) to get an idea of the magnitude of errors to be expected when some of the common equations of state are used for systems containing polar species; (d) to provide some insight as to what equation is most appropriate in a given design situation. The following discussion is by no means an exhaustive study of the broad field of equations of state as applied to gases. Rather, the main advantages and drawbacks of some of the equations presently available have been studied in line with the above objectives. Equations Studied The eight equations evaluated in this study are listed in Table I. The first five are generalized. The latter three, though more specific, are applicable to many important systems and conditions that are not expected to be within the scope of the more general equations. These specialized equations, therefore, are useful in filling some of the major gaps in the generalized equation of state framework. Several other useful equations were eliminated from consideration in order to keep the study within reasonable bounds. Undoubtedly the final choice has been influenced by factors like the authors’ familiarity with certain equations and applicability to systems that are of particular interest to Kellogg. It is also appropriate to emphasize at this stage that we have not included the BWR equation in this evaluation. The reliability and accuracy of the BWR equation have been demonstrated through
numerous studies (e.g., Starling (1973), Lin and Hopke (1974), Adler et al. (1977)). For systems where the necessary constants and interaction coefficients are available, the BWR equation is usually reliable. Unfortunately, these constants are available only for certain hydrocarbons and some of the common nonhydrocarbons, and the BWR equation would not be applicable to many of the systems that were evaluated in this study. BWR constants for some polar systems have been successfully generated through the popular multiproperty regression method. However, such an approach is, in general, quite time consuming and expensive, and does not always guarantee success. Data Base Though there are deficiencies for certain classes of systems, the amount of gas-phase volumetric data available in the literature is large, and the retrieval and processing of all these data would, by itself, constitute a major project. The data set used for the present study is only a representative one. The systems included in this data set, along with the temperature and pressure ranges, and literature source(s) of each are given in Table 11. It should be emphasized that though we have chosen data that are generally considered reliable, we have not made a detailed study evaluating their accuracy. The majority of cases encountered in practical design calculations involve mixtures rather than pure components. However, some pure component data are also included. An equation of state that is unable to adequately represent the PVT properties of a component in the pure state, cannot, in general, be expected to handle mixtures containing that component. Thus, the PVT representation of pure compounds gives an immediate feel for the capability of an equation to handle various types of systems. Also, data on fugacities of components in mixtures are extremely scarce. Conclusions regarding the ability of an equation to predict gas-phase fugacity have to be based,
728
Ind. Eng. Chern. Process Des. Dev., Vol. 18, No. 4, 1979
Table 11. Temperature and Pressure Rangesa and Literature Sources for Systems Considered in This Study system
temp range, "F press. range, psia
methane propane ethylene acetylene benzene carbon dioxide hydrogen sulfide sulfur dioxide ammonia water nitrous oxide methanol ethanol 1-propanol 2-propanol hydrochloric acid methane-ethane methane-n-butane methane-n-decane methane-hydrogen
-200-2000 -60-800 -1 60-1000 -1 13-1 1 5 100-800 -80-1500 - 120-6 00 0-480 -40-600 32-1600 -22-302 70-463 32-400 77-540 26-584 131-158 100 220-340 160-460 -210-90
1.0-15000 1.0-4250 1.0-7 500 1.0-13 00 1.0-3000 1.0-20000 1.0-4000 1.0-4500 1.0-16000 1.0-5500 192-4630 0.7-117 5 0.1-200 0.4-738 0.1-690 1257-2712 200-3000 300-10000 1000-5180 59-7327
methane-nitrogen methane-hydrogen sulfide ethane-propane ethane-hydrogen
-260-200 40-220 10-400 70-300
10-1500 200-10000 100-10000 43-3790
ethane-methanol propane- benzene nitrogen-n-butane carbon dioxide-propane carbon dioxide-n-butane hydrogen sulfide-n pentane ammonia-isooc tane ethylene-carbon dioxide ethylene-chloroform acetylene-ammonia hydrogen sulfide-nitrogen ethyl ether-1-butanol methanol- 1-butanol 2-butanol-1-butanol nitrogen-ammonia nitrogen-hydrogen-ammonia hydrogen-ammonia-nitrogen-argon-methane mixtures of hydrogen-ammonia-propane argon-water methane-water n -butane-water carbon dioxide-water hydrogen sulfide-water ammonia-water
167-212 400-460 310 100-160 100-400 40-340 294-506 104-257 77-21 2 122-302 68-212 338-533 392-538 428-540 208-248 302-752 32-167 129-251 752 100-752 100-305 752-11 1 2 85-340 -6 0-4 00
294-882 1000-1 0000 406-10000 188-950 200-10000 20-10000 100-1600 99-7450 294-1483 147-1323 250-3500 210-642 312-1133 244-627 21750-36260 974-8700 985-5805 162-1567 5800-43500 200-14500 52-638 5800-20300 325-5000 0.3-520
1. N,-C,-C0,-C,-C,
Natural Gas Systems 100-200 233-1614
2. N,-C,-C0,-C, -H,S-C,
100
194-1640
100 3. N,-C,-C0,-C,-H,S-C,-i-C,-C4-C,-C6-C,A
400-1000
4. C0,-H,S-N,-C,-C,-C,-i-C,-C, 40-175 40-175 5. C0,-H,S-N,-C, -Cz-C3-X4-C4 6. C0,-C,-C,-C,-i-C,-C,-i-C, -C,-C,, 88-200 91-150 7. C0,-C,-C,-C,-i-C,-C, 143-230 8. C0,-H,S-N,-C,-C,-C,-i-C,-C,-i-C,-C,-C,, 9. H,S-C,-C, 102-160 10. C0,-H,S-C,-C, 100-160 100-160 11. c0,-c,-c, 12. C0,-H,S-N,-C, -C, 100-176 50-175 13. C0,-H,S-N,-C,-C,
600-5000 56 5-5021 500-3500 500-1000 3000-5015 1026-7026 1026-7026 1026-7026 200-4000 664-1594
lit. source Canjar and Manning (1967)b Canjar and Manning (1967)b Canjar and Manning (1967)b Canjar and Manning (1967)b Canjar and Manning (1967)b Canjar and Manning ( 1967)b Starling (1973)b Canjar and Manning (1967)b Canjar and Manning (1967)b Canjar and Manning ( 1967)b Couch and Kobe (1961) Kudchadker (1968); Smith (1948) Reid and Smith (1951) Cosner et al. (1961) Eng. Sei. Data ( 1976)b Thomas (1962) Sage and Lacey (1950) Sage and Lacey (1950) Sage and Lacey (1950) Solbrig and Ellington (1963); Chuang et al. (1976) Bloomer e t al. (1955) Sage and Lacey (1955) Sage and Lacey (1955) Solbrig and Ellington (1963); Mihara e t al. (1977) Ma and Kohn (1964) Sage and Lacey (1955) Evans and Watson (1956) Sage and Lacey (1955) Sage and Lacey (1955) Sage and Lacey (1955) Kay and Warzel(1958) Sass et al. (1967) Shim and Kohn (1964) Khodeeva (1964) Robinson e t al. (1975) Kay and Donham (1955) Kay and Donham (1955) Kay and Donham (1955) Maslennikova et al. (1976) Kuznetsava e t al. (1973) Linshits et al. (1970) Antezana and Cheh (1975, 1976) Lentz and Franck (1969) Welsch (1973); Sage and Lacey (1955)' Sage and Lacey (1955)' Franck and Todeheide (1959) Sage and Lacey (1955)' Macriss et al. ( 1964)' Robinson and Jacoby (1965) (mix tures 7,8,9,11,12)d Robinson and Jacoby (1965) (mixtures 13,14,15,18,19)d Robinson and Jacoby (1965) (mixture 26)d McLeod (1968) (mixture l)d McLeod (1968) (mixture 3)d McLeod (1968) (mixture 22)d McLeod ( 1 9 6 8 ) (mixture 25)d Simon and Briggs (1964) Satter and Campbell (1963) Buxton and Campbell (1967) Buxton and Campbell (1967) Wichert (1970) Wichert (1970)
a The temperature and pressure ranges indicate the range of the data used in this study. The data in the literature source These are critically evaluated tabulations of pure component data. For the purpose of this often cover a wider range. study, they have been regarded as experimental data. No compressibility factors o r fugacity coefficients are reported for these systems. The literature sources were used only t o obtain the equilibrium composition of the saturated vapor mixture. These refer t o the mixture identification numbers in the literature source.
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
often, on its ability to reproduce PVT data for mixtures.
Evaluation Against Experimental Data The eight equations of state were evaluated against the experimental data described above. Not all eight equations were tested against all the data. Some of the equations are limited as to the type of compounds they can handle, or to the pressure range for which they are applicable. The NBP equation, for example, was developed to include common polar compounds a t high pressures. However, it is now applicable only to the components for which equation of state parameters and interaction constants are available. The virial equation, on the other hand, though applicable to a wide variety of compounds, is limited to low and moderate pressures. Nevertheless, from a practical standpoint, there are numerous instances where such equations could be useful and perhaps be indispensable. The RKD equation is primarily meant for waterhydrocarbon systems. Similarly, the RKG equation is designed for systems containing ammonia and water. These equations were studied because questions pertaining to hydrocarbon-water and ammonia-water systems arise repeatedly in many process-design calculations. The equation of state parameters were either taken from the literature or were estimated via well-accepted equations. It is often possible to adjust the parameters so as to best fit the data at hand. Such a procedure, though commonly used in fitting data, would defeat the purpose of a study where we are interested in the predictive capability of an equation. For each experimental point tested, the gas-phase compressibility factor and the fugacity coefficient of each component in the system were predicted. Experimental fugacity coefficients for mixtures are scarce. However, by comparing the different fugacity predictions, it is possible to get an idea of the variation to be expected when using different equations of state. Presentation of the Results Graphical presentation of the results has been emphasized. The advantages of this approach are easy to see. In studies comparing correlations against experimental data the common practice is to tabulate percent average, percent bias, and percent root-mean-square deviations. In a study like the present one, such tabulations may not always be meaningful. At low and moderate pressures, the compressibility factor usually varies between 0.9 and 1.0. At the critical point it is usually about 0.27. A 20% error in predicting the critical compressibility factor through an equation of state is often considered acceptable. On the other hand, even a 2% error at low pressures is generally considered poor. Thus, an average percent deviation has little significance unless one knows the exact distribution of the experimental gas-density data over the P-T space. Most of these drawbacks can be overcome when the results are shown graphically. Tabulated results are given wherever they are appropriate. All comparisons are based on the compressibility factor rather than the density. The compressibility factor varies over a smaller range (usually 0.25 to 1.5), whereas the density varies over several orders of magnitude. More important, however, the deviation of the compressibility factor from unity gives an immediate idea about the extent of nonideality in the gas phase. For the same reasons, fugacity coefficients rather than fugacities are compared. Discussion of Results-Pure Compounds Nonpolar and Mildly Polar Compounds. The hydrocarbons studied in this category are methane, propane,
729
ethylene, acetylene, and benzene. Most equations of state are expected to do well for hydrocarbons since polarity effects are either absent or are minimal. The RK, RKS, and the JBA equations all predict compressibility factors over most of the PVT space with less than 2 % deviation. In the critical region 10 to 20% errors are to be expected, while a t the critical point itself, the error could be as high as 50%. In the immediate vicinity of the critical point, the Redlich-Kwong equation appears to be somewhat more reliable than RKS or JBA. For the saturated vapor, as may be expected, the JBA equation is slightly superior. The virial equation gives reliable results within its range of applicability. Along the saturation line the deviation in the compressibility factor seldom exceeds 2% up to a fifth of the critical density. Even at densities that are one-third to one-half the critical, the compressibility factor is generally reliable to within 5%. A t higher densities the deviations become increasingly large, and the virial equation truncated after the second or third virial coefficient, fails. At high temperatures in the supercritical region, the applicability of the virial equation extends to considerably higher pressures. Thus, for methane at 2000 O F and 4000 psia the compressibility factor predicted by the virial equation deviates from the experimental value by less than 0.5%. The other compounds included in this group of nonpolar and mildly polar substances are C02, HzS,SOz,and N20. (The term mildly polar is used here rather loosely, and refers to compounds where the effects of polarity may be assumed to be minimal.) These compounds, despite their dipole and quadrupole moments, exhibit gas-phase behavior that is quite similar to pure hydrocarbons. The characteristics described earlier for pure hydrocarbons are found to be generally true for C02, H2S,SO2, and N20. Figure 1 compares experimental and predicted compressibility factors of SO2. These are typical of the results obtained both for these mildly polar inorganics and for the hydrocarbons described in this section. A study was also made of the Redlich-Kwong-Chueh (RKC) equation for the systems in this group. For pure compounds, the difference between the RKC and the RK equations is that the parameters Qa and Qb are allowed to vary from compound to compound, rather than being fixed. For the compounds studied here, the RKC equation gives a slight improvement over RK. The improvement is somewhat more pronounced for the saturated vapor than for the superheated vapor. This is expected because the RKC Qa and Qb are adjusted based on the vapor pressure curve. However, in the critical region no improvement is noticed, although the critical compressibility factor itself is predicted better. The pure component fugacity coefficients, 4, were also evaluated. For a pure component the fugacity coefficient is defined by
where the symbols have their usual meaning. The fugacity coefficient involves an integration from zero pressure to the system pressure. Consequently, errors in 2 at low pressures cumulate when integrated over the pressure range 0 to P. An accurate prediction of the compressibility factor at a given point does not necessarily imply an accurate fugacity coefficient-the fugacity coefficient could still be significantly in error if the equation of state predicts erroneous compressibility factors a t low pressures. Conversely, the calculated fugacity coefficient at pressure P could be quite reliable even if the error in Z is large,
730
Ind. Eng. Chem.
Process Des. Dev., Vol. 18, No. 4,
1979
PRESSURE. p i a
00
I W
1
2m
I
I
1
1
1
I
I
400
603
803
lax,
2oto
40cO
6033
I 8030 K 03
Figure 1. Experimental and predicted compressibility factors of sulfur dioxide.
12
1.0
0.8
EXPERIMENTAL 06
- I - RK - - VIRIAL - a - JBA 0
0.4
-0-
RKS
0.2 100
a,
Figure 2. Experimental and predicted compressibility factors of ammonia.
provided 2's at lower pressures have been consistently well predicted. For hydrocarbons and mildly polar materials at low and moderate pressures, the equations of state evaluated here seldom deviate by more than 170from the experimental compressibility factor. Resultingly, despite the large errors in 2 around the critical, in most instances the fugacity coefficient is estimated quite accurately right up to the critical point. This is illustrated in Table I11 where estimated values of 2 and $ for carbon dioxide at the critical point are compared with the experimental values. Despite the large variations in the predicted Z:s, the predicted 4:s are all quite close to the experimental value. Even the virial equation, with a 113% deviation in Z, predicts $, to within 1070,mainly because of its excellent accuracy a t low and moderate pressures. Highly Polar and Associating-Type Compounds. The compounds evaluated in this category are ammonia, water, some low molecular weight alcohols, and hydrochloric acid. These compounds are characterized by their high dipole moments which result in intermolecular attractive forces other than of the dispersive type. The effect of these forces, though small at reduced temperatures exceeding unity, is quite significant at low temperatures. All alcohols, besides being highly polar, tend to associate in the vapor phase, even at low pressures. Conventional equations of state like the Redlich-Kwong equation have
Table 111. Predicted Values of Z, and @ c for Carbon Dioxidea % dev
- eq
RK RKC JBA RKS virial
zccalcd
0.3333 0.2730 0.1823 0.3333 0.5851
inZc 21.6 -0.4 -33.5 21.6 113.5
% dev @CCd&
0.666 0.647 0.643 0.666 0.628
in@c -3.9 -6.6 -7.2 -3.9 -9.4
no provision to account for polarity and association. Consequently, at system conditions where such forces have a large effect on the thermodynamic behavior of a compound, these equations can give erroneous answers. In the saturation region the RK, RKS, and JBA equations give larger deviations for ammonia and water than for nonpolar compounds. The virial equation, on the other hand, gives accurate predictions up to fairly high saturation pressures-up to about one-fifth of the critical density. Figure 2 shows the compressibility factors of saturated ammonia as predicted by the various equations. The experimental and predicted fugacity coefficients for saturated ammonia are compared in Figure 3. Figure 2 also shows compressibility factors for superheated ammonia at 400 and 600 O F , as calculated from the
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979 731
SATURATED VAPOR
I-
4u
-EXPERIMENTAL
LL
0.6-W
- I-
8 ED 9 ?i
RK
-0-VIRIAL
-6- JEA -0-
RKS
CRITICAL W I N 1
PRESSURE, p i a
I 20
O
I
I
I
40
60
1
80 100
I
I
200
400
I
I
I
I
600 Eo0 1000
4600
2ooo
Figure 3. Experimental and predicted fugacity coefficients of ammonia.
SATURATED VAPOR
1.0
a
an-;
e t_
i
i 0.6-2
-EXPERIMENTAL -0-
VlRlAL
-4-
JBA
-0-
RKS
Y CRITICAL POINT
I
I
I
20
40
60
PRESSURE, p i a I I
83100
I
I
i
200
403
603
I
800K
Figure 4. Experimental and predicted compressibility factors of 1-propanol.
different equations. In the highly superheated region, all the equations tested give surprisingly good results. Deviations in the compressibility factor seldom exceed 1% except in the critical region and at very high pressures (8000 psia and above). The alcohols included in the present study are methanol, ethanol, 1-propanol, and 2-propanol. The RK and RKS equations often deviate from experimental values of the saturation compressibility factor by 3-470 even at pressures of 25 psia or less. The JBA equation is somewhat better, except for methanol where the deviations are large. The virial equation is clearly superior to the other equations. Even at high pressures, the virial equation generally gives better predictions than the RK, RKS, and JBA equations. Figures 4 and 5 show compressibility factors and fugacity coefficients for saturated 1-propanol. It should be especially noted that these figures extend to pressures of 10 psia or less. Unlike Figures 1 through 3, where the different equations of state give essentially correct compressibility factors at pressures less than 100 psia, in Figure 4 significant discrepancies can be seen at considerably lower pressures. Figure 5 also illustrates how errors in predicting low pressure compressibility factors can affect fugacity predictions at higher pressures. The virial and
JBA equations which predict better low-pressure 2 s (see Figure 4) also predict better 4’s for almost the whole range. Unfortunately, all the hydrochloric acid data are in the high-pressure, critical region. None of the equations is expected to do well in this region, and a comparison is both unfair and meaningless. Nevertheless, it should be noted that even at low pressures hydrochloric acid associates to a great extent-much more so than the alcohols. Probably, the best way to handle such systems would be to use the virial equations with virial coefficients predicted from the correlation of Hayden and O’Connell (1975). The same procedure should be used for organic acids like acetic acid. Finally, although we have not evaluated the NBP equation for pure compounds, the authors (Nakamura et al. 1976) have shown that the equation gives consistently good results for pure components to which the equation of state constants are available. Likewise, the RKD equation was not evaluated for water and the RKG equation was not included in the comparison for ammonia. Discussion of Results-Mixtures Mixtures Containing Nonpolar or Mildly Polar Compounds Only. In terms of number of systems, this is the biggest category in this study. The systems included are methane-hydrogen, methane-nitrogen, methane-
732
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
SATURATED VAPOR
1.0c
09-
EY
;
8
-
EXPERIMENTAL
a 0.8- s Y
0.7
-
0.6*
I
-a-
RK
-o -
VlRlAL
-A-
JBA
-0-
RKS
I
I
t
PRESSURE, p i a I I I
I
1
4 6 8 0 20 40 6omm Figure 5. Experimental and predicted fugacity coefficients of 1-propanol.
I 200
CRITICAL POINT I I I
400
600800
Table IV. Comparison o f the Redlich-Kwong, Redlich-Kwong-Chueh, Joffe-Barner-Adler, and Redlich-Kwong-Soave Equations for Compressibility Factors of Nonpolar and Mildly Polar Gas Mixtures % dev in compressibility factor"
no. of system points methane-hy drogen 90 methane-nitrogen 55 methane-ethane 46 methane-n -butane 55 methane-n decane 22 methane-hydrogen sulfide 46 ethane-hydrogen 87 ethane-propene 43 propane-benzene 28 nitrogen-n -butane 37 ethylene-carbondioxide 46 carbondioxide-propane 47 carbon dioxide-n-butane 46 hydrogen sulfide-nitrogen 30 hydrogensulfide-n-pentane 35 a
Note: av =
Z
av
RK bias
rnax
av
RKC bias max
av
1.3 0.7 2.8 3.0 2.8 2.7 0.8 3.0 4.0 3.4 3.1 15.2 5.0 3.1 3.4
-0.3 0.1 2.7 2.7 2.7 -0.4 -0.2 2.5 3.5 -1.2 -0.2 -14.0 -4.0 -3.1 2.5
-6.3 9.7 15.1 10.7 12.7 9.6 -1.6 8.9 9.7 9.3 -11.5 -62.5 -24.0 -7.1 11.7
1.4 0.7 3.0 4.7 3.9 2.7 0.4 3.5 7.1 3.4 4.2 2.9 5.5 1.7 7.5
-0.2 0.4 2.8 4.6 3.9 2.0 -0.2 2.0 7.1 3.4 3.4 -0.4 5.1 -1.7 7.4
1.4 0.4 2.3 6.9 2.7 3.6 1.7 5.8 9.5 3.0 3.2 9.8 3.5 3.0 4.4
ID l/N; bias = ZDIN where D
= (Zakd
-
-6.2 11.4 15.7 12.1 14.7 13.3 -1.4 -22.3 14.4 13.6 18.5 11.9 29.9 -4.5 26.9
JBA bias max
RKS bias max 17.2 0.7 10.1 4.0 3.9 12.2 -0.2 2.9 1.1 1.1 18.4 4.2 1.3 12.0 4.2 10.8 4.6 -0.3 -47.3 4.6 15.7 4.4 2.5 1 3 . 0 4.4 18.7 1.6 14.0 3.9 3.8 2.8 10.9 1.4 10.5 2.8 11.4 4.3 -1.7 -47.8 4.3 11.8 -3.6 -40.4 5.9 5.9 12.6 13.1 2.9 2.7 -0.6 20.1 6.1 3.2 11.5 6.1 -17.9 -6.5 61.9 3.4 -1.6 5.6 29.0 5.6 -0.8 -40.1 2.0 -3.0 -6.2 0.6 0.6 19.7 1.7 -32.0 5.6 5.3 av
Zexptl)/Zexptlx 100; N = no. of points.
ethane, methane-n-butane, methane-n-decane, methane-hydrogen sulfide, ethane-hydrogen, ethane-propene, propane-benzene, nitrogen-n-butane, ethylene-carbon dioxide, carbon dioxide-propane, carbon dioxiden-butane, hydrogen sulfide-nitrogen, and hydrogen sulfide-n-penh e . Several natural gas mixtures that were evaluated are treated separately. The RK, RKC, RKS, JBA, virial, and NBP equations were evaluated. For each system, several compositions were studied, when such data were available. The relative merits of these equations vary from system to system, and often, for the same system, from composition to composition. The following discussion is based on general observations for the 15 systems listed above, but many exceptions can be found. In general, the deviations in predicting compressibility factors of mixtures are about two times the deviations for the corresponding pure components at the same conditions. This is especially true a t moderate and high pressures. A t low pressures the interactions between unlike molecules are less significant, and mixture properties can be predicted to about the same reliability as pure compounds. By and large, the deviations increase when the mixture contains molecules of widely different sizes. Thus, most equations give better results for the CH4-H,, CH,-Nz, CH4-HzS, and CH4&H6 mixtures than for the CH4-C4Hlo
or CH,-C,,,Hzz mixtures. Properties of mixtures containing same molecular types (e.g., paraffin-paraffin) are more correctly predicted than properties of mixtures with different molecular types (e.g., paraffin-aromatic). Table IV gives an overall comparison of the RK, RKC, RKS, and JBA equations for d 15 systems evaluated. The RK, RKC, and RKS equations give more or less equivalent overall results. In fact, in most instances the original Redlich-Kwong (RK) gives better predictions than the modified versions of Chueh or Soave (RKC and RKS). The JBA equation tends to give poor results at very high pressures. At other conditions it is superior or equivalent to the other equations. The virial equation and the NBP equation were also evaluated against an appropriate portion of the data set. As indicated earlier in this paper, both equations are limited in scope; the virial equation is valid only at low and moderate pressures, while the NBP equation can be applied only to mixtures containing compounds for which the necessary parameters are available. Within its region of applicability, the predictions from the virial equation are consistently reliable. This is true regardless of whether the vapor is in the saturation state or the superheated state. The NBP equation generally gives accurate predictions at high and very high pressures. However, since the
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
733
SATURATED VAPOR
1 - lWF
0.8
NOTE: JBA C O I N C I D E S WITH V I R I A L .
-
-
Y
SATURATED VAPOR T 100' F
U
=
0.6-f 0
m
r
-" Y
-
-
-.-
EXYERIHENTAL
0'4-i
RK
-I-
RKC
C
-
--o-vylr,i~t
4
I 0.0
m
JBh
-0-
0.2 ?
- - RKI 0
I
I
100
203
I J
PRESSURE, p r i a
I 403
I I I 6038301m
PRESSURE.
PI.
ZOXI
Figure 6. Experimental and predicted fugacity coefficients of nbutane in carbon dioxide. 19 '.
SATURATED VAPOR
04
0.8
T- 1WF
9 0.7 '4
\
a
-0
I.
-1-
RK
-m-
RUC
-0-
VIRIAL
-0-
JBA
0.4
q 1a L
a31I
C R I T I C A L POINT
I
JBA
-0-
RKS
PRESURE. p r a
R K S O V E R L A P S RKC NBP F A I L E D
203
VIRIAL
Figure 9. Predicted fugacity coefficients of n-decane in methane.
RKS
-0-
I
-
-0-
PRESSURE. p i a I I I
I ,I
I
W n
m400m6038001doo
Figure 7. Experimental and predicted compressibility factors of the carbon dioxide-propane system.
correlation is based on a perturbed hard-sphere model, it is not surprising that it performs better at higher pressures. Such high pressures are generally encountered only in the supercritical region. A t low temperatures and low to moderate pressures, especially in the typical vapor-liquid equilibrium region, the NBP equation does not seem to offer a significant advantage over the Redlich-Kwong-type equation. Figure 6 compares experimental and predicted data for the fugacity coefficients of n-butane in carbon dioxide. This is one of the few systems for which fugacity coefficients of individual components in a mixture have been derived from extensive PVT data. Figure 7 compares experimental and predicted compressibility factors for the carbon dioxide-propane system-one of the feu! systems in this study where the RKC, RKS, and JBA equations are superior to the RK equation. The predicted fugacity coefficients of propane in carbon dioxide for the RKC and RK equations are shown in Figure 8. Again note that the differences in the fugacity coefficient 4 are much smaller than the corresponding differences in the compressibility factor, 2. Finally, Figure 9 compares the predicted fugacity coefficients of n-decane in methane, for the saturated vapor. Because the vapor phase is almost all methane, the
predicted compressibility factors are all quite accurate. (See Table IV). However, as shown in Figure 9, there is considerable discrepancy between the various predicted fugacity coefficients of n-decane. Mixtures Containing One Highly Polar Component. The following systems were used to evaluate compressibility factor predictions: ammonia-isooctane, acetylene-ammonia, ethane-methanol, ethylene-chloroform, nitrogen-ammonia, nitrogen-hydrogen-ammonia, and hydrogen-ammonia-nitrogen-argon-methane. Aqueous-gas mixtures, although they fall into this general category, are discussed separately later. In addition, the fugacity coefficient data of Antezana and Cheh (1975, 1976) for the hydrogen-ammonia-propane system were also evaluated. The RKC equation is not discussed separately for this category because the modified Q2,'sand Q { s and the interaction constants, k i i s , are not available for the polar components. For the ammonia-isooctane, ethane-methanol, ethylene-chloroform, and the hydrogen-ammonia-nitrogenargon-methane systems, the data are in the saturation region. For these systems except the last one, the RK, RKS, JBA, and virial equations all give poor predictions. Many of the experimental points are in the critical region. Even so, the predictions in general are considerably poorer than for nonpolar systems. Typical results are plotted in Figure 10 for the ammonia-isooctane system. Deviations in the compressibility factor generally range from 10 to 30% or higher. Compressibility factors tend to be underpredicted, although for the ethylenechloroform system the trend is erratic. At pressures of 250 psia or less, the predictions are generally of acceptable accuracy. Unfortunately, not many data exist at lower pressures to make
734
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
SATURPITED VAPOR
\
04-
-0-
VIRIAL
-0-
JEC
b
“;j0 ‘L
\ \
\ o x\ \
-0-RKS
L O
P R E S S U R E . psia‘x
I
I
I
I
I
an exact quantitative evaluation. However, it can be safely assumed that the virial equation will be the more accurate method at low pressures. The RK, JBA, and NBP equations all gave excellent results for the saturated hydrogen-ammonia-nitrogenargon-methane system. Average deviations are about 1.5%. The RKS equation gave an average deviation of 5.6%. For the acetylene-ammonia, nitrogen-ammonia, and the nitrogen-hydrogen-ammonia systems all the compressibility data are for the superheated vapor. For acetylene-ammonia, deviations range from less than 2% at 147 psia to about 14% at 1030 psia, by the RK, RKS, JBA, and virial equations. The nitrogen-hydrogen-ammonia data are all in the high-pressure region-1000 to 9000 psia-at highly su-
percritical temperatures. The RK equation gives the best predictions for this system over almost the entire range, with very few deviations exceeding 2%. The virial equation, truncated after the second virial coefficient, gives accurate predictions up to about 5000 psia, but fails at higher pressures, especially at low temperatures. Despite the fact that the NBP equation was developed for such systems at high pressures, it is not superior to the RK equation in predicting compressibility factors for this system. Figure 11 is a typical comparison of experimental and predicted compressibilities. The data for the nitrogen-ammonia system are at extremely high pressures-about 20 000 psia. The NBP equation was the best with an average error of 1.5%. The RK and RKS equations averaged 3.5% deviation. Again the JBA equation did poorly for this high-pressure system, with an average deviation of 13.6%. Experimental and predicted fugacity coefficients of hydrogen and ammonia in systems containing hydrogen, ammonia, and propane are compared in Table V. The results in this table are typical of those obtained for this system. The RK and virial equations give consistently better predictions. In regions of high nonideality in the vapor phase (see for example, fugacity of ammonia in hydrogen in Table V), the virial equation is clearly superior to the others. Binary Mixtures Where Both Components Are Highly Polar. Three systems were evaluated in this category-the ethyl ether-1-butanol, methanol-1-butanol, and 2-butanol-1-butanol systems. All data are in the form of compressibility factors in the saturation region. Figure 12 compares experimental and predicted compressibility factors for the ethyl ether-1-butanol system. Though it predicts compressibility for the ethyl ether-1-butanol system fairly accurately, the JBA equation fails for the alcohol-alcohol systems. Overall, the Soave equation gives the best predictions for the alcohol-alcohol systems. Most deviations are less than 5%, except in the critical region. The RK equation gives deviations of 10 to 15% except in the immediate vicinity of the critical point. For the alcohol-alcohol systems, the virial equation gives deviations ranging from 0 to 6% in the noncritical region. The data for the polar-polar binaries studied here
1.25.
1-15.
1.10.
.05
Figure 11. Experimental and predicted compressibility factors of the nitrogen-hydrogen-ammonia mixture.
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979 735
E e 71
M
A
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
736
SATURATED VAPOR MOLE FRACTION ETHYL ETHER
-
0.744
o w w w
-EXPERIMENTAL ---Flu
- - VlRlAL 0
t
-0-
JBA
-0-
RKS
0.3
PRESSURE. m a
0.21
I
I
I
m
250
‘FOINT CRITICAL
I
c
5x)
450
350
3
Figure 12. Experimental and predicted compressibility factors of the ethyl ether-1-butanol mixture.
“7 T p
--
75PF
5800 p i a
1.0C
E
0.8-Y
i ii
t
-EXPERIMENTAL -EXPERIMENTAL
0.6-E
8
-I -
RK R KD
I
_ _ -NBP MOLE FRACTION ARGON
0.2
0
-
1
I
I
I
0.2
0.4
0.6
0.8
I
I
Figure 13. Experimental and predicted compressibility factors of argon-water mixtures.
are all in the 200 to 1100 psia range-a region where the densities are somewhat high for the virial equation to be reliable. Nevertheless, except for the ethyl ether-1-butanol system, where deviations are in the 3 to 25% range, the predictions from the virial equation are quite reliable. Aqueous-Gas Mixtures. Most of the systems under this category can be classified in the general category of polar-nonpolar mixtures. However, because of their special importance, aqueous-gas systems have received more attention than other polar-nonpolar systems. In fact, equations have been proposed to specifically handle mixtures containing water. It is, therefore, appropriate to discuss aqueous-gas systems separately. The following systems were studied: hydrogen sulfide-water, carbon dioxide-water, argon-water, methane-water, n-butane-water, and ammonia-water. The RK, RKD, NBP, and the virial equations were evaluated. In addition, the RKG equation was also studied for the ammonia-water system. Unlike in the earlier categories, actual experimental data are extremely scarce except in the very high pressure (5000 psia and above) range. Thus, the study is more a comparison among equations of state, rather than a comparison against experimental data. The primary objective is to get an idea of the variation to be expected when employing different equations for this important but poorly understood class of systems.
joooooooooooo
*
Ind. Eng. Chern. Process Des. Dev., Vol. 18, No. 4, 1979 ,0.1
0.4
-0
02
ammonia solutions at high pressure, the fugacity coefficients of water in the equilibrium vapor can be considerably different. Because the RKG equation is specifically meant for the ammonia-water system, it is probably a more accurate representation of reality than the RK or RKD equations. The NBP equation often gives completely erroneous predictions, especially at low pressures and low temperatures, and these results are therefore not included in Figure 16. Natural Gas Mixtures. Fourteen natural gas mixtures of different composition were evaluated. Because of their importance in natural gas processing, sour natural gases of varying hydrogen sulfide and carbon dioxide content were emphasized. The percent deviations in the compressibility factor prediction for the various equations are listed in Table VI. Also listed are the maximum hydrogen sulfide and carbon dioxide content of each mixture. Overall, the JBA equation appears to give the best results. Among the Redlich-Kwong type equations, the RKC gives the best predictions, though the original RK is better or equivalent in many cases. The RKS predictions are again found to be consistently poorer. The NBP equation, evaluated for four mixtures, gives fairly reliable predictions. However, it offers little advantage over the JBA or RKC equations. A number of methods specifically designed to correlate compressibility factors of sour and sweet natural gases are available in the literature. Examples include the Robinson-Jacoby (1965) modification of the RK equation, the Buxton-Campbell (1967) method using the Curl-Pitzer charts with modified mixing rules, and the McLeodCampbell (1969) method based on the refractive index. A detailed study of these and other methods has been done by Wichert (1970). Comparison of the results given by Wichert with those in Table VI indicates that some of the specialized methods mentioned above would give better results in many instances. However, from an overall standpoint, especially when high-pressure (say 2000 to 7000 psia) data are considered, the JBA or RKC equations give predictions that are equivalent or better.
(
'\I ''
RK
-I-
\
- VlRlAL
____
I
'
'
\
'\
RKD NBP
I
I
I
400
6M
I
I
800 1003
40m
xxx)
Figure 14. Predicted fugacity coefficients of water in saturated hydrogen sulfide-water mixtures.
Figure 13 comparing experimental and predicted compressibility factors for the argon-water system is typical of a number of systems evaluated in the superheated vapor state. In all these cases the NBP equation gave the most accurate results. The RKD equation is superior to the RK equation only in the regions of high water concentration. (In a separate study, we have found that the RKD equation generally gives poor results around the critical temperature of water. Thus, at temperatures between 700 and 800 O F , for pressures exceeding 700 psia, the RK equation is preferred over the RKD equation.) Figures 14 and 15 compare predicted fugacity coefficients of water in the hydrogen sulfide-water and methanewater systems. Again, unfortunately, no comparison against "experimental" data is possible. Figure 16 compares predicted fugacity coefficients of water in the saturated ammonia-water mixture in the vapor phase. For the vapor in equilibrium with dilute ammonia solutions (10% by weight in the figure) the fugacities predicted by the RK, RKD, and RKG equations are all quite close to each other. However, for concentrated I.o
-
=e>--
.._ -.. *I
0.9
"\
o-o--
o+=o>o=,
x\ X,
0.8
I\
-
- ... 0 .
'
\
X 'X\
0.7
\
O\)\
\
\
'x
I\
>
\o
\
\
I,$
c-" 2
\
RK
-0-
VIRIAL
---
'\
\
\
\
\
\
\
\
$1 -A-
'\
\o
\
i
/
RKD
---- NBP 0.4
'O>\ \
-
-
\
x\Y
IL
0.5
\
\
x\
Y "
Y
"9
\
-g -
%\\, \
c z
0.6
\o\ \\
X\
IOO'F
";a,
\ \
w = 3 z
-
T
o >.,'
s ''
-
\'
\
\
\
\
\
\
\
\
\ \
'\
\
f
\
f
\
\
'. '.
'\
\
\ \
\\ PRESSURE. p s i a
0.3 200
I 400
I 600
I 800
737
\
$\
\
X,
I
I
1000
2000
Figure 15. Predicted fugacity coefficients of water in saturated methane-water mixtures.
\
\ \
4000
I
I
6000
BOO0
I'
738
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
Table VII. Recommended Hierarchy of Equations for Estimating Gas-Phase Density and Fugacity for Different Types of Systems in Different Regions of the PVT Spacea type of system pure compounds nonpolar, mildly polar highly polar mixtures only nonpolar and mildly polar components are present one highly polar component present more than one polar component present aqueous-gas mixtures water-hydrocarbon and water-carbon dioxide systems containing ammonia and water natural-gas mixtures
subcritical temp away from critical region
critical region
supercritical temp away from critical region
RK, RKC, JBA, virial virial
RK, RKC, JBA RK
RK, RKC, virial virial, RK
RK, RKC, JBA, virial
RK, RKC
RK, RKC, virial, NBP
RK, virial
RK
RK, virial, NBP
virial
RK
RK, virial, NBP
RK, virial
RKD, RK
NBP, RKD, virial, RK
RKG
RKG
RKG, RK
JBA, RKC, RK
JBA, RKC, RK
(1)In the critical region, errors in density are generally large, and errors in fugacity coefficients could be significant. ( 2 ) Whenever the virial equation is used, it should be remembered that there is an upper bound on the pressure, above which the two-term virial equation is either poor or completely fails. For most systems, t h e limit may be taken as P, < ( T , - 0.65) for T , < 1;P, < (1.05T, - 0.35) for T , > 1, where P, = P/zyiPCi;T,= T/zyiTcT These limits were established by plotting a large number of predicted compressibility factor deviations, and then isolating that region where the deviation was less than 2% for pure components, 3% for nonpolar o r mildly polar mixtures, and 5% for highly polar mixtures. a
1
D SATURATED VAPOR
0 SUPERHEATED VAPOR-
1.00-
3 0-
0,95-
A
1
0
e
PURE COMPOUNDS
SATURATED VAPOR SUPERHEATED VAPORJ M’XTURES
.
5 0.90-;
e. e
I 0
i c
I;
Y
s t ;
0.85-
0.60-
3 0
e
15-f
l
a
v
I
RKO
-a-RKG
MOLE FRACTION AMMONIA IN VAPOR
o,700
I
I
I
I
0.2
0.4
0.6
0.8
I .o Figure 16. Fugacity coefficients of water in ammonia-water mixtures at saturation, as predicted by three different modifications of the Redlich-Kwong equation.
Conclusions and Recommendations As shown throughout the paper, seldom does a given equation of state perform equally well in the subcritical, critical, and supercritical regions. Even among the eight equations tested here, the preferred equation varies, depending on what region of the PVT space is considered. Further, the type of compound (nonpolar, mildly polar, or highly polar) and the type of mixture (nonpolarnonpolar, polar-nonpolar, or polar-polar) can also influence what is the most appropriate equation of state. Table VI1 gives recommended equations for estimating gas-phase compressibility factors and fugacity coefficients in different regions of the P-T space for difference classes of systems. These recommendations are based on the
25
Figure 17. Solid line representing the upper pressure limit where deviations in compressibilityfactor as predicted by the two-term virial equation exceed specified levels (2% for pure compounds; 3% for nonpolar and polar-nonpolar mixtures; 5% for polar-polar mixtures).
general conclusions drawn from this study. In actual design calculations, especially when polar components are present, verification against experimental data when such data exist is always advised. Table VI1 is meant to serve as a broad guideline when there is no recourse to experimental information. Given as a footnote in Table VI1 is the suggested upper pressure limit for the two-term virial equation. The upper limit of pressure at which the virial equation begins to fail depends both on the number of terms employed and on the temperature level. The limits for the two-term virial equation were established by identifying for each system at a given reduced temperature the reduced pressure at
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
which the compressibility deviates by more than an arbitrary amount. This arbitrary deviation was set as 2% for pure compounds, 3% for nonpolar and polar-nonpolar mixtures, and 5% for polar-polar mixtures. Figure 17 shows the plot obtained by this procedure. Each point indicates a system of unique composition such that, for the temperature corresponding to that point, the deviations in the compressibility factor are within the above limits at reduced pressures less than or equal to that shown. The solid line was drawn so that nearly all the points lie above it. For convenience, the solid line can be represented by PI = (TI- 0.65) (for T I C 1) P, = (1.O5Tr- 0.35) (for TII1) These limits are merely given as a useful guide. As can be seen from Figure 17, the two-term virial equation is often valid at considerably higher pressures; in a few instances it may fail at pressures lower than those recommended. Also, the range of applicability can be considerably extended if one is willing to accept larger deviations. Literature Cited Adler, S.B., Spencer, C. F., Ozkardesh, H., Kuo, C. M., Am. Chem. SOC.Symp. Ser., No. 60, 151 (1977). Antezana, F. J., Cheh, H. Y., Ind. Eng. Chem. Fundam.. 14, 224 (1975). Antezana, F. J., Cheh, H. Y., Ind. Eng. Chem. Fundam., 15, 95 (1976). Barner, H. E., Adler, S. B., Ind. Eng. Chem. Fundam., 9, 521 (1970). Bloomer, 0.T., Eakin, B. E.,Ellington, R. T., Gami, D. C., Inst. Gas Techno/. Chicago, Res. Bull., No. 21 (1955). Buxton, T. S.,Campbell, J. M., SOC.Pet. Eng. J . , 80 (March 1967). Canjar, L. N., Manning, F. S., "Thermodynamic Properties and Reduced Correlations for Gases", Gulf Publishing Go., Houston, Texas, 1967. Chuang, S.,Chappelear, P. S.,Kobayashi, R., J . Chem. Eng. Data, 21, 403 (1976). Chueh, P. L., Prausnitz, J. P., AIChE J . , 13, 896 (1967). Cosner, J. L.. Gagiiardo, J. E., Storvick, T. S.,J . Chem. Eng. Data, 6, 360 (1961). Couch, E. J., Kobe, A., Hirth, L. J., J . Chem. Eng. Data, 6, 229 (1961). deSantis, R., Breedveld, G. J. F., Prausnitz, J. M., Ind. Eng. Chem. Process Des. Dev., 13. 374 (1974). Eng. Sci. Data Unit, Chem. Eng. Ser., 4, No. 76012 (1976). Evans, R. B., Watson, G. M., Chem. Eng. Data Ser., 1(1), 67 (1956). Franck, E. V., Todeheide, K.. Z . Phys. Chem. (Frankfurt am Main), 22, 232 (1959). Graboski, M. S., Daubert, T. E., Ind. Eng. Chem. Process Des. Dev., 17, 443 (1978). Guerreri, G., Prausnitz, J. M.. Process Techno/. Int., 18(4). 209 (1973).
739
Hayden, J. G., O'Connell, J. P., Ind. Eng. Chem. Process D e s . D e v . , 14, 209 (1975). Joffe, J., J . Am. Chem. SOC.,69, 540 (1947). Kay, W. B., Donham, W. E., Chem. Eng. Sci., 4, 1 (1955). Kay, W. B., Warzel, M. F., AIChE J., 4, 296 (1958). Khodeeva, S. M., Russ. J . Phys. Chem., 36, 693 (1964). Kudchadker, A. P., W.D. Thesis, Texas A & M University, College Station, Texas, 1968. Kuznetsava, T. S.,Kazarnovskaya, D. B., Kazarnovskii, Ya. S., Tr. Gas. Nauchno-lssled, Proektn. Inst. Azotn. Prom-sti. Prod. Org. Sint., No. 19, 5 (1973). Lentz, v. H., Franck, E. U., Ber. Bunsenges, Phys. Chem., 73, 28 (1969). Lin, C-J., Hopke, S. W., AIChE Symp. Ser., 70(140), 37 (1974). Llnshits, L. R., Rodkina, I.B., Tsiklis, D. S.,Russ. J . Phys. Chem., 44, 456 (1970). Ma, Y. H., Kohn, J. P., J . Chem. Eng. Data, 9, 3 (1964). Macriss, R. A., et al., Inst. Gas Techno/. Chicago, Res. Bull., No. 34 (1964). Maslennikova, V. Ya.. Subbotina, L. A., Tsiklis, D. S.,Russ. J . Phys. Chem., 50, 874 (1976). McLeod, W. R., Ph.D. Thesis, University of Oklahoma, Norman, Okla., 1968. McLeod, W. R., Campbell, J. M., Oil Gas J . , 67, 115 (1969). Mihara, S.,Sagara, H., Arai, Y., Saito, S.,J . Chem. Eng. Jpn., 10, 395 (1977). Nakamura, R., Breedveld, G. J. F., Prausnitz, J. M., Ind. Eng. Chem. Process Des. Dev., 15, 557 (1976). Redlich, O., Kwong, J. N. S.,Chem. Rev., 44, 233 (1949). Reid, R. C., Smith, J. M., Chem. Eng. Prog., 47, 415 (1951). Robinson, D. B., Hamaiiuk, G. P., Krishnan, T. R., Bishnoi, P. R., J . Chem. Eng. Data, 20, 153 (1975). Robinson, R. L., Jacoby, R. H., Hydrocarbon Process., 44(4), 41 (1965). Sage, B. H.,Lacey, W. N., "Monograph on API Research Project 37Thermodynamic Roperties of the Lighter Paratfin Hydrocarbons and Nitrogen", American Petroleum Institute, New York, N.Y., 1950. Sage, B. H., Lacey, W. N., "Monograph on API Research Project 37-Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide and Carbon Dbxide", American Petroleum Institute, New York, N.Y., 1955. Sass, A., Dodge, B. F.. Bretton, R. H., J . Chem. Eng. Data, 12, 168 (1967). SaFer, A., Campbell, J. M., Pet. Trans., A I M , 333 (Dec 1963). Shim, J., Kohn, J. P., J . Chem. Eng. Data, 9, 1 (1964). Simon, R., Briggs, J. E., AIChE J., 10, 548 (1964). Smith, J. M., Chem. Eng. Prog., 44, 521 (1948). Soave, G., Chem. Eng. Sci., 27, 1197 (1972). Solbrig, C. W., Ellington, R. T., Chem. Eng. Prog. Symp. Ser., 59(44), 127 (1963). Starling, K. E., "Fluid Thermodynamic Properties for Light Petroleum Systems", Gulf Publishing Co., Houston, Texas, 1973. Tarakad, R. R., Danner, R. P., AIChE J., 23, 685 (1977). llwmas, W., "Progess in International Research on Thermodynamic and Transport Properties", Second Symposium on Thermophysical Properties, p 166, The American Society of Mechanical Engineers, New York, N.Y., 1982. Tsonopoulos, C., AIChE J . , 20, 263 (1974); 21, 827 (1975). Welsch, H., Doctor of Natural Science Dissertation, University of Karlsruhe, Germany, 1973. Wichert, E., M. Eng. Thesis, University of Calgary, Alberta, 1970.
Received f o r review March 8, 1979 Accepted June 13, 1979
