I n d . Eng. Chem. Res. 1990,29, 659-666 Tanabe Seiyaku Co. L-Phenylalanine, Jpn. Kokai Pat. 59/74,994, 1984; Chem. Abstr. 1984, 101, 108947g. Ton, J. S.; Vineyard, B. D. Novel Synthesis of L-Phenylalanine. J . Org. Chem. 1984,49, 1135. Toyo Soda. Addition Compounds of Halobenzenes and Olefins and Jpn. Kokai Hydrolysis of Alpha-halo-beta-phenylpropionitriles. Pats. 561145,225 and 561158,732, 1981; Chem. Abstr. 1982, 96, 122,385g. Uzuki, T. Selective Deacylation of N-Acyl-D,L-Amino Acids. U S . Patent 3,907,638, 1975. Vineyard, B. D.; Knowles, W. S.; Saback, M.; Bachman, G. L.; Weinkauff, D. J. Asymmetric Hydrogenation. J . Am. Chem. SOC. 1977, 99, 5946. Wakamatsu, H.; Uda, J.; Yamakami, N. Synthesis of N-Acyl Amino Acids by a Carbonylation Reaction. J . Chem. SOC.,Chem. Commun. 1971, 1540. Wake, S.; Shimizu, T.; Beppu, M. Puriciation of D,L-Phenylalanine. Jpn. Kokai Pat. 621142,143, 1987; Chem. Abstr. 1987, 107, 237,29211.
659
Ware, E. The Chemistry of the Hydantoins. Chem. Reu. 1950,46, 403. Yamada, S.; Hongo, C.; Yoshioka, R.; Chibata, I. Method for the Racemization of Optically Active Amino Acids. J . Org. Chem. 1983, 48, 843. Yamagishi, T.; Yatagai, M.; Hatakeyama, H.; Hida, M. Asymmetric Hydrogenation of Dehydroamino Acids. Bull. Chem. SOC. Jpn. 1984, 57, 1897. Yamskov, I. A.; Tikhonov, V. E.; Davankov, V. A. Polymeric Catalysts for Racemization of Optically Active Amino Acids. Polym. Sci. USSR 1979, 21, 1838; Chem. Abstr. 1979, 91, 193,857~. Yamskov, I. A.; Tikhonov, V. E.; Davankov, V. A.; Ryzhov, M. G.; Vauchskii, Yu. P.; Vel'ts, A. A. Racemic Amino Acid Esters. USSR Pat. 929,629, 1982; Chem. Abstr. 1982, 97, 145284h. Zen, S.Synthetic Studies of the Derivat,ives of Nitroacetic Acid. Bull. Chem. SOC.Jpn. 1963, 36, 1143; 1973, 46, 337. Received f o r review May 22, 1989 Accepted December 12, 1989
Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 2. Representation of the Vapor Pressures and of the Molar Volumes Marek Rogalski, B r u n o C a r r i e r , R o l a n d Solimando, and A n d r e P i k e l o u x * Laboratoire de Chimie-Physique, Facults des Sciences de Luminy, 13228 Marseille Cedex 9, France
A modified version of the volume-corrected Peng-Robinson equation of state, yielding accurate hydrocarbon vapor pressures from the triple point up t o the critical point, is proposed. By use of this equation with a suitable volume correction, it is possible to correctly calculate the PVT properties of subcritical and supercritical fluids. Due to these features, it is a particularly useful tool for phase equilibrium calculations such as those performed in petroleum engineering design. This paper is a continuation of the study on the correlation and prediction of thermodynamic properties of hydrocarbons using a cubic equation of state. Recently a modified version of the Peng-Robinson equation was reported, which is suitable for calculating hydrocarbon vapor pressures from the triple point up to a pressure of 2 bar (Carrier et al., 1988). To make this method useful for chemical engineering design calculations, it was necessary to extend it to include the high-pressure range. Petroleum process design involves calculations of phase equilibria at temperatures ranging from 273 to 600 K. At any temperature in this range, the components of the petroleum fractions will be in different reduced states, depending on their chemical nature. This is illustrated in Table I, where four groups of compounds are defined in terms of their volatility. The components of the first two groups are near-critical or supercritical at any of the temperatures considered, and those of the last two groups are always subcritical and often well below their normal boiling temperature. The amount and type of experimental data available differ from one group to another. The first two groups contain compounds with known vapor pressures, critical parameters, and volumetric properties. The same applies to many (but not all) of the compounds in the third group. With the fourth group, there is a great shortage of experimental data. In the case of the Cll-C16 fraction, the critical parameters are often missing, and the existing data are rather inaccurate, although the vapor pressures below Tb have often been fairly accurately determined. The situation is even worse with the C16+ fraction. Little if anything is known about the physical properties. The only means of determining the critical parameters is to use available prediction methods.
Any equation of state proposed for dealing with petroleum mixtures should make the best use of the experimental information available, which implies that it is necessary to use different approaches for light and to heavy components. In this paper, we propose an extension of the method described by Carrier et al. (1988). Here we deal first with the vapor pressures of the hydrocarbons of group I11 from the triple point up to the critical point. The problems associated with calculating the liquid volumes of the hydrocarbons of groups I11 and IV are further discussed. Lastly, the possibility of applying the method to components of groups I and I1 is examined. Representation of Vapor P r e s s u r e The present method is based on the following equation of state: P = RT(D - 6) - a/ii(ii + y6), with y = 4.82843 (1) Parameter a( T ) was determined depending on the pressxe range studied and on the value of the pseudocovolume ( b ) used. Low Vapor Pressure: Method A l . The use of the function a( 2') proposed by Carrier-Rogalski-PBneloux (1988) (2) UCRP = aTb(l + mI(1 - (T/Tb)'") - m2(1 - T/Tb)) with
ml = --1.98255 + 7.15530m, m2 = m1/2 - m
(3) makes it possible to correctly represent vapor pressures lower than 2 bar. uTbcorresponds to the value of uCRP at the normal boiling temperature. In this method, each component is characterized by three parameters. The first
0888-5885/90/2629-0659$02.50/0 0 1990 American Chemical Society
660 Ind. Eng. Chem. Res.. Vol. 29, No. 4, 1990 Table I. Classification of Petroleum Fraction Components on the Basis of Their Volatility” group
compounds
conditions
I I1
CH4, Nz C02, C2Hs, C3Hs, i-C4Hl0,C4HIO
111 IV
c5-C10
supercritical near-critical subcritical subcritical (often subatmospheric)
C11+
available info about critical parameters vapor pressure 1A 2A 1A 2A 1B 2B 1C or 1D 2C or 2D
Critical parameters (Tc,Pc): lA, known with good accuracy; lB, in most cases known with acceptable accuracy; some possible errors in the critical pressure; l C , experimental values are inaccurate if available; l D , unknown. Vapor pressure: 2A, known up to the critical point; 2B, usually well-known up to the normal boiling point; with many compounds experimental data up to the critical point have been reported; 2C, in many cases known up to the normal boiling point; 2D, low-pressure data on a few compounds.
is the normal boiling temperature, Tb. The second, the pseudocovolume ( b ) ,is calculated with a group contribution method (Carrier et al. (19881, eq 10, Table 11) and is denoted as b,. The third parameter, m (see eq 3), can be determined with experimental subatmospheric vapor pressure data (corresponding values of m0 for numerous hydrocarbons are listed in the paper by Carrier et al. (1988), Table I) or calculated as m:r by using a group contribution method given in the same paper (Tables I1 and IV and eq 12). Equation 12 in the paper by Carrier et al. (1988) is given with a numerical error. The correct form is
m = 0.29942 + S - 0.21311S2 To summarize, method A1 determines the equation of state, eq 1,with eqs 2 and 3, which are calculated by using
6 = hg1,
m = mo or m = mi,
(4)
and is valid with hydrocarbons forming groups I11 and IV, the critical parameters of which are not known. Low Vapor Pressure: Method A2. Method A1 does not use the critical parameters, and consequently, the extrapolated saturation curve cannot pass through the critical point. When considering an extrapolation of this type, it is thus necessary to use a covolume value consistent with the critical specifications of eq 1:
6,
= Q&T,/P,
with
ob
= 0.045572
(5)
In this case, the parameter m zeeds t_o be modified. Differences between the values of b, and b, do not exceed 390,and it was-found that parameter m, consistent with the covolume (b,.), can be derived from m0 in the following way: m = mi,(l
+ A)
or m = mo(1 + A)
(6)
with 5. = 0.4084( 6, - gpr)/ 6,
(7)
It should be noted that aTbdepends on the covolume and should be recalculated as well. The results of the low vapor pressure estimations using method A2 are nearly the same as those obtained with method Al. With the components of group 111,both methods A1 and A2 can be used. The components of group IV, the critical parameters of which are unknown or not well-known, should be dealt with by using gethod Al. In short, method A2 determines eq 1 with b = b, and m given by eqs 6 and 7 and should be used with the components of groups I11 and IV, where the vapor pressures can be taken to be also above 2 bar. High Vapor Pressure: Method B. When looking for the best representation of the vapor pressures of the hydrocarbons between Tband the critical temperature, we tested various forms of a ( T ) . The function developed by Gibbons and Laughton (1984) was found to be particularly interesting, since it is perfectly consistent with the ex-
pression for uCRP in our model, eq 2. Accurate representations were obtained only when a function with at least two parameters was used. In this case, it is practically impossible to generalize the parameters (partly due to uncertainties about the critical parameters). For this reason, we decided to use the Soave-type function (1972) as = a,(l
+ ms(l with a, = 0.45724RTC2/P,2 (8)
ms =
[(aT,/ac)”2 -
11/P - (Tb/Tc)1/21
(9)
This function, though far from being perfect, is always fairly efficient with hydrocarbons, due to the specific error compensation. Parameter ms is established in a like manner, eq 9, to obtain an exact value of the vapor pressure a t the normal boiling temperature. Thus, ms is defined in terms of Tb (and not of the acentric factor), a parameter that is usually precisely known. According to P6neloux et al.’s (1982) study on the volumetric translations of cubic equations of state, identical vapor pressure calculation results can be obtained either with eq 1 or with the Peng-Robinson equation (1976) if both are used with the same a ( T ) (parameter b for the latter equation is given as 0.07780RTc/P,). This is due to the fact that the Peng-Robinson equation is the volume translated form of eq 1. Method B can be used with the components of groups 1-111 above the normal boiling temperature. Full Vapor Pressure Range: Method A2-B. In order to accurately calculate the vapor pressures from the triple point up to the critical point with a single equation, it was necessary to switch continuously from method A1 Lo B. This can be done by calculating parameters a and b between Tb and the upper junction temperature TI (after preliminary studies, T, = 1.25Tb was chosen) as a temperature-weighted mean of the values resulting from the two methods. The following expression was proposed: a ( T ) = Xas + (1 - X)aCRp (10) with as and aCRPdetermined by methods B and A2, respectively. Function X is chosen in such a way as t~ obtain a tangential junction between the two models for a ( T )a t Tb and TI and is defined as follows: when T IT b ,
x=0 when
Tb I T ITi,
x = (T
-
Tb)2/[(T- Tbl2 -k (Ti - TI2]
when T 2 TI, X = l (11) Plots of the octane vapor preessure deviations between the data given by the TRC Tables (1987) and those obtained with methods A2-B and B are shown in Figure 1. The use of the Soave-type function leads to considerable
Ind. Eng. Chem. Res., Vol. 29, No. 4,1990 661 combined functions are sufficiently similar to yield a smooth combination in the case of enthalpy calculations. Considerable deviations can be observed, however, with compounds where the vapor pressures cannot be very accurately represented with the CRP model (benzene, cyclohexane, and 3,3-diethylpentane). The deviations obtained with heat capacities are greater, of course. The contribution introduced by temperature derivatives of X is physically meaningless and results from the application of a combination of two models. In practical calculations of thermal properties, it seems preferable to neglect terms F1 and F2. This introduces a thermodynamic inconsistency (equivalent to considering X as independent of the thermodynamic temperature) but makes it possible to accurately represent the enthalpies and heat capacities.
T6P %
+
!
Volumes of Liquid Hydrocarbons As shown by PBneloux et al. (1982), the pseudovolume (0) in eq 1 needs to be corrected u=E-c (13) to be brought near the real volume ( u ) . As long as the volume correction is pressure independent, it does not modify the vapor pressure calculated with eq 1. Thus, if volumetric properties are to be calculated, eq 1 is defined with a set of three parameters, a, b, and c. In the previous section, we proposed several methods of calculating vapor prgssures, each defined by the mode of determining a and b. These methods should be completed with appropriate procedures to determine the volume correction ( c ) . In practice, the calculation mode needs only to be specified with methods A1 and A2-B. The liquid volumes calculated as described by PBneloux et al. (1982) usually come to within 1-270 of the experimental values between 0.55Tcand 0.85Tc. This method has two disadvantages when used with the heavy components of petroleum mixtures. First, the volume representation is not satisfactory at low reduced temperatures. Second, the critical parameters, which are not well-known for this class of compounds, are necessary to calculate c. In this paper, we propose a new method for determining the volume correction. It is based on the group contribution method, which makes it possible to determine c at the normal boiling temperature. With a weakly temperature-dependent function, it is possible to calculate c down to 273 K. Two formulations of the volume correction, completing the methods proposed in the previous chapter, are described below. Volume_Correction for Method Al. The parameters u ( T )and b are given by eqs 2 and 4. At the normal boiling temperature, the volume correction (c”,) is as follows:
I TJT J 1.0 1.3 1.6 Figure 1. Octane. Deviations of calculated vapor pressures from TRC data. (0)Method A2-B, (+) method B.
deviations at low pressures. These can be reduced by using the combined model, method A2-B, thanks to the contribution of the function aCRP. A similar compensatory effect was observed with other hydrocarbons. The results obtained with method A2-B at subatmospheric pressures were nearly identical with those reported previously-method A1 (Carrier et al. (1988), Table I). The calculated deviations of vapor pressures of more than 1 bar from TRC Table data are listed in Table 11. The values of the critical parameters used in the calculations were taken from the same tables. The overall mean deviation with the 46 compounds studied was 0.35% when parameter m was determined with the experimental data and was 0.38% when m was calculated with our group contribution method. It is difficult to compare these results with those obtained by using the original PengRobinson equation, since the latter is strongly dependent on the values of the acentric factors used. The overall average deviation obtained with critical parameters and the acentric factor reported by Reid et al. (1977) was 0,5370,whereas the corresponding results was 0.33% when the critical parameters from the TRC Tables and the ms determined from vapor pressures given in the same tables were used. A combination of two expressions for a ( T )gives satisfactory vapor pressure calculation results but unavoidably perturbs thermal properties in the junction range. In fact, the function X makes it possible to obtain the tangential junction in two limiting points. This is necessary to avoid numerical problems in practical applications of this EOS. On the other hand, in the middle junction range, some irregularities related to differences between two combined functions can appear with enthalpy or heat capacity calculations involving the use of the temperature derivatives of a ( T ) . This is evident with the following expressions for the first and second derivatives of a ( T ) : a’(T) = (1 - X ) a ’ c ~ p XU’S + F1 ~ ” ( 5 ” )= (1 - X)u”CRp + X U ” + ~ F2 (12) with F1 = X’(as - UCRP) and F2 = X”(us - acRp)+ 2X’(ah - abRP) As can be seen with eq 12, the importance of the term F1 is negligible if aCRp is similar to as. In practice, the two
13
cTb = -21.531
+ 0.00938Tb3/2- 141.60Am - j=1 ECiG,
(14)
The sum in eq 14 is calculated with all the groups forming a given compound. Cj and G, are a characteristic parameter of group j and a number of groups of this type in the molecule, respectively. The number of groups and the values of the group parameters are given in Table 111. Correcting factors (Am) are used only with benzene, toluene, cyclohexane, 3,3-dimethylpentane, and 3,3-diethylpentane and were established to rectify parameter m, which could not be satisfactorily calculated with group contribution methods (values of Am have been previously reported by Carrier et al. (19881, Table IV). The exceptional character of these compounds was again observed with regards to their volumetric properties. The evolution of the volumetric correction with temperature is given as a general function determined for
662 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 Table 11. Saturated Density Deviations (Sp) up to 0.85T,and compound T,K (min-max) 2,2-dimethylpropane 273-363 2-methylbutane 273-383 pentane 273-393 cyclopentane 273-323 2,2-dimethylbutane 273-303 273-333 2,3-dimethylbutane 273-333 2-methylpentane 273-333 3-methylpentane 273-423 hexane 273-343 methylcyclopentane 273-353 2,2-dimethylpentane 283-473 benzene 273-323 2,4-dimethylpentane 273-353 cyclohexane 2,2,3-trimethylbutane 273-323 3,3-dimethylpentane 273-323 1,l-dimethylcyclopentane 293-298 2,3-dimethylpentane 273-323 2-methylhexane 273-323 3-methylhexane 283-313 3-ethylpentane 273-363 heptane 273-453 2,2,4-trimethylpentane 273-378 methylcyclohexane 273-378 ethylcyclopentane 273-378 1,1,3-trimethylcyclopentane 293-298 273-323 2,2-dimethylhexane 2,5-dimethylhexane 273-383 2,4-dimethylhexane 273-323 273-493 toluene 3,3-dimethylhexane 273-323 273-323 2,3,4-trimethylpentane 1,1,2-trimethylcyclopentane 293-298 2,3-dimethylhexane 3-ethyl-2-methylpentane 273-323 2-methylheptane 273-333 4-methylheptane 273-333 3,4-dimethylhexane 273-333 3-ethyl-3-methylpentane 273-323 3-ethylhexane 273-333 3-methylheptane 273-333 1,l-dimethylcyclohexane 293-298 octane 273-483 propylcyclopentane 273-383 ethylcyclohexane 273-383 ethylbenzene 273-523 1,1,3-trimethylcyclohexane 293-298 p-xylene 293-523 m-xylene 293-523 o-xylene 293-533 3,3-diethylpentane 293-298 nonane 273-503 273-533 isopropylbenzene propylcyclohexane 273-383 propylbenzene 273-473 273-378 1-ethyl-3-methylbenzene 1-ethyl-4-methylbenzene 273-473 1,3,5-trimethylbenzene 273-473 273-378 1-ethyl-2-methylbenzene 1,2,4-trimethylbenzene 273-473 decane 273-523 1,2,3-trimethylbenzene 273-473 butylcyclohexane 273-383 butylbenzene 273-453 undecane 273-533 dodecane 273-553 naphthalene 343-473 tridecane 273-300 2-methylnaphthalene 273-413 1-methylnaphthalene 273-423 tetradecane 273-553 2-ethylnaphthalene 273-413 octylbenzene 273-453 pentadecane 283-423 decylcyclopentane 273-383 nonylbenzene 273-453 hexadecane 293-393
Vapor Pressure Deviations (6P ) above T b 6p,b Yo T , K (min-max) 6P," 7 0 0.32 298-423 0.29 0.60 3 18-443 0.20 0.38 328-463 0.19 1.11 338-503 0.37 0.16 338-483 0.18 0.77 348-493 0.20 0.5; 348-493 0.36 0.34 358-503 0.65 0.54 358-503 0.49 0.07 0.10 368-513 0.27 0.78 368-543 0.46 0.39 368-513 0.30 0.38 0.37 378-553 0.28 368-513 0.25 0.94 378-523 0.28 2.32 378-523 0.30 0.32 0.56 378-523 0.44 0.21 378-523 0.31 378-523 0.54 0.38 0.64 388-533 0.53 0.19 388-543 0.30 0.82 0.23 0.18 0.23 398-543 0.38 398-543 0.35 0.36 0.18 398-543 0.37 0.78 408-310 0.30 0.60 408-553 0.24 408-563 0.03 0.39 0.40 408-553 0.44 0.79 0.40 408-563 0.55 408-553 0.25 408-553 0.10 0.28 0.32 0.35 408-563 0.07 0.51 408-563 0.83 408-553 0.38 0.10 0.44 408-553 0.95 0.68 418-563 0.31 0.20 0.63 0.55 428-603 0.32 0.77 1.00 428-603 0.49 1.00 438-613 0.39 1.18 438-613 0.36 1.19 0.74 448-593 0.74 448-623 0.36 0.44 0.28 0.22 0.14 458-633 0.29 0.53 0.49 458-633 0.24 0.49 0.60 468-643 0.15 0.52 0.85 468-643 0.17 0.37 0.18 0.56 0.55 0.76 1.03 0.35 0.58 0.54 1.29 0.33 0.49 0.18
0.61 0.30
8P,b %
0.32 0.30 0.26 0.38 0.24 0.24 0.36 0.72 0.52 0.31 0.46 0.35 0.38 0.30 0.28 0.32 0.48 0.31 0.40 0.55 0.30
0.43 0.47 0.37 0.30 0.24 0.39 0.47 0.45 0.34 0.36 0.35 0.51 0.38 0.49 0.35 0.33 0.53 0.40 0.36 0.79 0.48 0.12 0.40 0.15 0.14
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 663 Table I1 (Continued) compound decylcyclohexane decyl benzene heptad ecan e octadecane nonadecane dodecylbenzene phenanthrene eicosane
T,K (min-max)
dp,b
mean deviations
" m determined with experimental data.
%
T , K (min-max)
6P," %
6P,b %
0.35
0.38
0.05 0.45 0.87 0.33 0.47 1.00 0.21 0.42
273-383 273-453 293-613 303-300 313-383 283-453 378-378 313-423
0.56
m calculated by a group contribution method.
Table 111. Group Contribution Parameters (Cj)for Calculating the Volumetric Correction ( c ) groups CHI CH; CH C CH2 CH C CH2 CH C CH C substituted C polycyclic
alkanes
cyclopentanes cyclohexanes aromatics
i
c,
1 2
13.133 13.542 15.335 15.896 14.408 10.336 1.555 13.868 9.809 7.742 12.521 11.391 13.460
3 4 5 6 7 8 9 10 11 12 13
1
particular compounds with the characteristic parameter m: when T I Tb, C
when T
85.8m2(1- T/Tb)2
(15)
+ 85.8m2(1- Tb/T)2
(16)
= Cfb
> Tb, C
=
Cfb
This method can be used with the components of group IV (it is also valid with the components of group 111). Volume Correction for Method A2-B. Parameters a and b are given with eqs 11and 5. At the normal boiling temperature, the volume correction is given as and the temperature functions are as follows: when T I Tb, C
=
+ 85.8m2(1- T/Tb)2
C T ~
85.8m2(1- Tb/T)2 eXp[(Tc - T)/Tb]
300
500
400
600
Figure 2. Comparison between saturated liquid densities given by method A2-B and TRC data. (+) Octane, (0) octadecane, (A)decylbenzene, and (0) naphthalene. Table IV. Deviations (dp, % ) with Boelhouwer's (1960) Data on the Densities of Liquid n-Alkanes UD t o 1200 b a r
T,K compd heptane octane nonane dodecane hexadecane
273 0.62
303 0.46 0.64 0.59 0.77 0.25"
333 0.80 0.61 0.57 0.74 1.21
363 1.34 0.85 0.66 0.71 1.14
393 2.17 1.49 1.01 0.74 0.92
'Up to 400 bar.
(18)
when T > Tb, C = C T ~
0.5
(19)
Satisfactory results can be obtained up to 0.85Tcwith this method. At higher temperatures, differences between the pseudovolumes and real volumes change sharply with temperature in a complex way. No modification of the temperature functions in eq 19 was proposed to improve the restitution of near-critical volumes, since this seems to be impossible with the equation of state used in this study. Since eq 19 preserves the continuity of the derivative (du/dT),, it is possible to continuously establish the volume correction applicable in the supercritical range. Estimation of supercritical PVT properties will be further discussed in the next section. In Table 11, the saturated liquid volumes calculated between 273.15 K and 0.85Tcwith components of groups
I11 (method A2-B) and IV (method A l ) are compared with data given by the TRC Tables. The overall average deviation obtained with 84 compounds was 0.56%. In Figure 2, the experimental liquid densities of four hydrocarbons are compared with those calculated at temperatures ranging from 273 K up to 600 K. The large deviations observed with octane above 0.85Tc (480 K) are not surprising. In Table IV, the experimental densities of n-alkanes under high pressures reported by Boelhouwer (1960) are compared with calculated values. The agreement can also be said to be satisfactory, taking into account the inherent weaknesses of the cubic equations of state. Method A2-€3 can also be used with components of groups I and 11. Light Components, Supercritical States The physical properties of the eight components of groups I and I1 (nitrogen, methane, carbon dioxide, ethane,
664 Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 Table V. Characteristic Constants for Eight Light ComDounds Pc, bar Tc,K compound Tb, K 126.26 nitrogen 77.35 33.97 190.55 methane 111.64 45.99 304.21 carbon dioxide 184.45' 73.79 305.42 48.80 ethane 184.57 373.53 hydrogen sulfide 212.87 90.06 369.80 propane 231.08 42.40 407.85 isobutane 261.34 36.40 425.14 butane 272.65 37.84
m1
m2
1.0172 0.7274 1.3827 1.0901 1.2974 1.4200 1.6473 1.6306
0.1856 0.0869 0.2211 0.1729 0.2710 0.2732 0.3436 0.3284
Tb
-14.03 -15.45 -13.55 -21.18 -15.00 -28.25 -36.00 -34.80
Metastable liquid.
Table VI. Saturated Density Deviations ( 6 p ) up to 0.85T,and Vapor Pressure Deviations (6P ) for Eight Light Compounds T , K (min-max) ref" 6 p , 70 P, bar (min-max) rep dP, 70 nitrogen 63-115 6 0.34 0.12-2.2 10 0.05 1.09-33.7 6 0.19 methane 93-153 1 0.22 0.12-3.1 9 0.14 1.68-42.7 1 0.43 ? 2 0.22 carbon dioxide 216-256 0.29 5.18-73.5 ethane 103-253 1 0.46 0.02-2.2 8 0.47 2.00-46.4 1 0.51 hydrogen sulfide 272-316 I 0.33 0.20-1.3 1 0.08 2.09-89.6 1 2.45 103-313 1 0.63 0.01-1.1 7 0.17 propane 2.03-37.7 1 0.25 113-340 5 0.49 0.01-1.1 3 0.15 isobutane 1.86-33.6 1 0.35 153-353 1 0.3; 0.01-1.1 4 0.14 butane 2.43-36.7 1 0.51 I
c
References (1) TRC Tables, 1987. (2) Angus et al., 1976. (3) Aston et al., 1940. (4) Aston and Messerly, 1940. (5) Goodwin, 1979. (6) Jacobsen et al., 1986. (7) Kemp and Egan, 1938. (8) Loomis and Walters, 1929. (9) Prydz and Goodwin, 1972. (10) Wagner, 1973.
hydrogen sulfide, propane, isobutane, and butane) because of their very nature cannot be considered in the framework of the group contribution methods proposed for hydrocarbons of groups 111 and IV. For this reason, their integration into the calculation scheme presented above has to be considered separately. Method A3-B. In fact, this method is a modified version of method A2-B. The main difference is that it involves the use of particular parameter values. With light components, the linear relation between parameters m, and m2,eq 3, is no longer valid, and a C R P should be calculated using two particular parameters from eq 2. In this case, the function a ( T )is equivalent to that proposed by Gibbons and Laughton (1984) and is called in this paper method A3. Equation 1 i s defined for each component with the normal boiling temperature, critical parameters, and parameters ml, m2,and cTb. The corresponding values are listed in Table V. Parameter m, calculated in terms of m, and m2,
I
1 .o
i _ - _ _ _ _ _ _ - -- _ _ - e
0.8
~
m = m,/2 - m2
(20)
(this is valid strictly for hydrocarbons only) is again used in eqs 18 and 19 to determine the temperature dependence of the volume correction ( c ) . Mean pressure deviations (pressures from subatmospheric up to the critical pressure were considered) and mean deviations of saturated densities (up to 0.85TJ are listed in Table VI. The results are quite satisfactory except in the case of hydrogen sulfide. The present model does not seem to be suitable for this polar compound. Supercritical volumetric properties were calculated using the volume correction established with eq 19. In this case, method A3-B is reduced to method B. Thus, it is equivalent to a slightly modified Peng-Robinson equation (eqs 9 and 10) in the volume-corrected form (eq 19) and is not very original. However, its high efficiency when used with appropriate values of the volume correction are worthy of discussion.
I
0 6
I
0
I
100
I
I
1
P/bar __j
200
Figure 3. Methane. A plot of the compressibility factor ( 2 )along the isotherms 300 and 400 K calculated with (-) method A3-B, (---I the PHCT equation (Gmehling et al., 1979), and ( - - - ) the BACK equation (Machat and Boublik, 1985). Experimental data by Goodwin (1979): (0)300 K, (+) 400 K.
Figures 3 and 4 compare the experimental and calculated compressibility factor z = Pu/RT of methane and carbon dioxide along two isotherms. It can be seen that the volume correction value, estimated with eq 19, is close to the optimal value. At the same time, these figures reveal an inherent default of cubic equations that makes it impossible to correctly represent high-pressure volumes simultaneously with second virial coefficients (corresponding to the slope of the z ( P ) curve when P = 0). This limits the possibility of improving the volumetric correction approach. Consistent results can be obtained with more rigorous equations of state only. This can be observed in
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 665 T Z
1 0
4
1.1
0.9 -
0.8
1 1 0
I
,
100
,
,
I
1
P/bar
_j
200
Figure 4. Carbon dioxide. A plot of the compressibility factor ( 2 ) along 400 and 550 K calculated with (-) method A3-B, (- - -) the PHCT equation (Gmehling et al., 1979), and (---) the BACK equation (Machat and Boublik, 1985). Experimental data by Angus (1976): (0)400 K, (+) 550 K.
P/bar
---+ I I I I 1 0 100 200 Figure 6. Nitrogen. A plot of the compressibility factor (Z) along 300 and 500 K calculated with (-) method A3-B and (---) the BACK equation (Machat and Boublik, 1985). Experimental data by Jacobsen and Stewart (1973): (0)300 K, (+) 500 K. Table VII. Determination of the Parameters of Equation 1 According to Methods Proposed in This Paper
Tz
group
T > T, Tb 5 T < T , T 5 Tb
I
I1
I11
IV
B A3-B A3
B A3-B A3
A2-B A 1 orA2
A1
equation of state at high reduced temperatures. Within a medium-pressure range and beyond 2.5Tc, the shape of the isotherms becomes more regular and it is possible to improve the temperature dependence of the volume correction. With eq 19, which is based on these experimental results, it can be seen that the value of c tends toward cTb at high reduced temperatures. It can be concluded that, except for the critical region, the results obtained with a properly established volume corrected cubic equation of state compare favorably with those yielded by more rigorous and complex equations. 0
100
200
Figure 5. Butane. A plot of the compressibility factor ( 2 )along 450 and 550 K calculated with (-) method A3-B, ( - - - ) PHCT equation (Gmehling et al., 1979), and (---) the BACK equation (Machat and Boublik, 1985). Experimental data by Haynes and Goodwin (1982): (0) 450 K, (+) 550 K.
Figures 3 and 4 with the BACK equation by Machat and Boublik (1985). Equation PHCT should in principle follow a similar pattern. Disappointing results given in the same figures were obtained with a version of this equation presented by Ghmeling et al. (1979), which is apparently not suitable for light fluids at high reduced temperatures. Figure 5 gives the volumetric properties of butane with the isotherm temperature (450 K) corresponding to the reduced temperature (1.06 K). In such a close vicinity to the critical point, this once again reveals weaknesses of a cubic equation of state with deviations beyond 10% in the region corresponding to the minimum of the z ( P ) curve. Both the BACK and PHCT equations correctly represent volumetric properties in this range, however. With the PVT properties of nitrogen illustrated in Figure 6, it is possible to test the efficiency of a cubic
Conclusions The aim of this study was to improve the performances of the Peng-Robinson-type cubic equation of state for representing vapor pressures and P V T properties in a wide range of temperatures and pressures. This model is valid not only for hydrocarbons but also for nitrogen, carbon dioxide, and hydrogen sulfide. A system of correlations for calculating the parameters of eq 1 makes it possible to apply the present method to most components of petroleum fluids in a large range of temperatures and pressures. The use of group contribution methods to establish parameters is particularly valuable with heavy petroleum fractions, the exact compositions of which are not and cannot be known. A scheme for applying the method is summarized in Table VII. For VLE calculations, the present EOS should be used with the group contribution methods of PBneloux et al. (1989) and Abdoul et al. (1990), which were proposed in the framework of the same research program as the present study. These methods make it possible to predict the temperature-dependent kij for mixtures of all the components dealt with in this study. The present equation of state used with the Abdoul method can be useful for modeling the phase equilibria of
Ind. Eng. Chem. Res., Vol. 29, No. 4,1990
666
petroleum fluids. Carrier et al. (1989) h a v e used this method in their s t u d y on composition regrouping of complex mixtures. Lastly, two important features of the proposed equation of state should be pointed out. It ensures homogeneous vapor pressure and PVT calculation results within a wide range of temperatures and pressures, with all the components being characteristic of petroleum fluids. When used with the Abdoul method, it is particularly well adapted to representing properties of complex mixtures the properties of which are known in terms of group compositions r a t h e r than real compositions.
Acknowledgment
We thank G. Auxiette for helpful discussions during the work on this paper. This research was sponsored b y TOTAL-Compagnie FranCaise des PBtroles. We thank Dr. Jessica Blanc for help with the E n g l i s h translation. Nomenclature
a ( T ) ,b = cubic equation of state parameters UCRP = function a ( T )calculated with e q 2 as = function u ( T ) calculated with e q 9 UT,
= value of parameter a calculated from normal boiling
- conditions
b, = pseudocovolume calculated with the group contribution method described in the paper by Carrier et al. (1988) hC = pseudocovolume calculated with e q 5 c = volume correction = value of c at the normal boiling temperature when
COT
6=
% cT = value of c at the normal boiling temperature when 6 = S C
C = jth group parameters for calculation of c
d, number of the jth group in a given compound ml, m2 = parameters of e q 2 m = paragetej of eq 3 determined with mk or mgr by eqs 5 a n d 6, b = b, mo-= parameter of eq 3 determined with experimental data, b = b,, mir = paraFeter-of eq 3 determined by a group contribution method, b = b,, Am = factor correcting values of mk with five "exceptional" hydrocarbons
ms = parameter of the Soave function, eq 9 P = pressure, bar T = absolute temperature, K Tb = normal boiling absolute temperature, K T , = critical temperature, K 2'1 = upper junction temperature, 7'1 = 1.25Tb u =, molar volume, cm3 mol-' u', b noncorrected volume a n d pseudocovolume related t o a cubic equation of state, cm3 mol-' X = junction function given by e q 12 8x70 = average absolute deviation of X calculated from n experimental determinations,
X
=
CyIXyP - XTdcl/n
Literature Cited Abdoul, W.; PBneloux, A,; Rauzy, E. A group-contribution equation of state for correlating and predicting thermodynamic properties
of weakly polar and nonassociating fluids. Fluid Phase Equilib. 1990, in press. Angus, S.; Armstrong, B.; de Reuk, K. M. (IUPAC) Carbon Dioxide. International Thermodynamic Tables of the Fluid State-3: Pergamon Press: Oxford, 1976. Aston, J. G.; Messerly, G. A. The Heat Capacity and Entropy, Heats of Fusion and Vaporization and the Vapor Pressure of n-Butane. J . Am. Chem. SOC.1940, 62, 1917-1923. Aston, J. G.; Kennedy, R. M.; Schumann, S. C. The Heat Capacity and Entropy, Heats of Fusion and Vaporization and the Vapor Pressure of Isobutane. J. Am. Chem. SOC.1940, 62, 2059. Boelhouwer, J. W. M. PVT Relations of Five Liquids n-Alkanes. Physica 1960,26, 1021-1028. Carrier, B.; Rogalski, M.; PBneloux, A. Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified PengRobinson Equation of State. 1. Low and Medium Vapor Pressures. Ind. Eng. Chem. Res. 1988, 27, 1714-1721. Carrier, B.; Rogalski, M.; PBneloux, A. Choice of Pseudocomponents for Flash Calculations of Petroleum Fluids. In Aduances in Thermodynamics V d . I-C7+ Fraction Characterization;Taylor, Francis, Eds.; New York, 1989. Gibbons, R. M.; Laughton, A. P. An Equation of State for Polar and Nonpolar Substances and Mixtures. J. Chem. SOC.,Faraday Trans. 2 1984, 80, 1019-1038. Gmehling, J.; Liu, D. D.; Prausnitz, J. M. High-pressure VaporLiquid Equilibria for Mixtures Containing One or More Polar Components. Chem. Eng. Sci. 1979, 34,951-958. Goodwin, R. D. The thermophysical Properties of Methane, from 90 to 500 K at Pressure to 700 bar. NBS Technical Note 653; National Bureau of Standards: Boulder, CO, April 1974. Goodwin, R. D. Isobutane: Provisional Thermodynamic Functions from 114 to 700 K at Pressure to 700 bar. NBSIR 74-1612; National Bureau of Standards: Boulder, CO, July 1979. Haynes, W. M.; Goodwin, R. D. Thermophysical properties of normal butane. NBS Monograph 169; National Bureau of Standards: Boulder, CO, April 1982. Jacobsen, R. T.; Stewart, R. B. Thermodynamics Properties of Nitrogen. J . Phys. Chem. Ref. Data 1973,2, 757-922. Jacobsen, R. T.; Stewart, R. B.; Jahangiri, M. Thermodynamics Properties of Nitrogen from the Freezing Line to 2000 K at Pressures to 1000 MPa. J . Phys. Chem. Ref. Data 1986, 15, 735-909. Kemp, J. 0.;Egan, C. J. Hindered Rotation of the Methyl groups in Propane. the Heat Capacity, Vapor Pressure, Heats of Fusion and Vaporization of Propane. Entropy and Density of the Gas. J . Am. Chem. SOC.1938,60, 1521. Loomis, A. G.; Walters, J. E. The Vapor Pressure of Ethane near the normal Boiling Point. J . Am. Chem. SOC.1926, 48, 2051. Machat, V.; Boublik, T. Vapor-Liquid Equilibrium at elevated pressures from the BACK Equation of State I. One Component Systems. Fluid Phase Equilib. 1985, 21, 1-9. PBneloux, A.; Rauzy, E.; FrBze, R. "A Consistant Correction for Redlich-Kwong Volumes" Fluid Phase Equilib. 1982, 8, 7-23. PBneloux, A.; Abdoul, W.; Rauzy, E. Excess Functions and Equation of State. Fluid Phase Equilibria 1989,47, 115-132. Peng, D.-Y.; Robinson, D. B. A New two Constant Equation of State. Ind. Eng. Chem. Fundam. 1976,15, 59-64. Prydz, R.; Goodwin, R. D. Experimental Melting and Vapor Pressures of Methane. J . Chem. Thermodyn. 1972, 4, 127. Soave, G. Equilibrium constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972,27, 1197-1203. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. TRC Thermodynamical Tables, Hydrocarbons; The Texas A&M University System: College Station, 1987. Wagner, W. New Vapour Pressure Measurements for Argon and Nitrogen and a New Method for Establishing Rational Vapour Pressure Equation. Cryogenics 1973, 17, 470-482. Received f o r review March 21, 1989 Accepted October 31, 1989
