Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
represent the ultimate enhancement. The mechanism responsible is the rapid setting up of a fast vertical streaming motion whose intensity is governed by density differences between h o y a n t and settling particles and suspending fluid. Nomenclature
uwl = particle velocity relative to wall when only heavy particles are present uw2= particle velocity relative to wall when heavy and light particles are present
49
Literature Cited Crowley, J. M., J . Fluid Mech., 45, 151 (1971). Crowley, J. M., Phys. Fluids, 19, 1296 (1976). Crowley, J. M., Phys;, Fluids, 20, 339 (1977). Hawksley, P. G. W., Conference on Some Aspects of Fluid Flow, 1950", p 114 Edward Arnold and Co., London, 1951. Sohn, H. Y.. Morland, C., Can. J. Chem. Eng., 46, 162 (1968). Steinour, H. H., Ind. Eng. Chem., 36, 618 (1944). Thacker, W. C., Lavelle. J. W., Phys. Fluids. 21, 291 (1978). Timbrell, V., Brit. J. Appl. Phys. Sup. No. 3 , 5, SI2 (1954). Whitmore, R . L., Brit. J. Appl. Phys., 6, 239 (1955).
Received for review April 13, 1978 Accepted October 21, 1978
A Third Parameter for Use in Generalized Thermodynamic Correlations Michael G. Kesler and Byung I k Lee Mob/ Research and Development Corporation, Princeton, New Jersey 08540
Stanley I. Sandler Department of Chemical Engineering, University of Delaware, Newark, Delaware 1971 1
Thermodynamic correlations based on corresponding states generally use T,, P,, and w. For heavy hydrocarbons, which decompose at temperatures far below the critical, T,, P,, and w cannot be measured and have been correlated as a function of only T, and SG. This work introduces a third parameter, w,, similar to w but calculated from vapor pressures near T,, without knowledge of T, and P,. Correlations of T,, P,, w , heats of vaporization at the normal boiling point, and vapor pressures in terms of T,, SG, and w, of a number of hydrocarbons show significant improvement over existing correlations. The work is based on general concepts of perturbation theory and, using n-alkane as reference, correlates departures of the other hydrocarbons from n-alkane behavior.
Thermodynamic properties correlations, developed in the past 20 years and based on the principle of corresponding states, generally use three correlating parameters. A widely accepted set is the critical temperature T,, the critical pressure P,, and Pitzer's acentric factor w. The introduction of this :latter parameter has significantly improved the prediction of volumetric and thermodynamic properties of pure coimponents and mixtures as well as vapor-liquid equilibria (Pitzer et al., 1955; Pitzer and Curl, 1957; Curl and Pitzer, 1958; Pitzer and Hultgren, 1958; Barner et al., 1966; Wilson, 1964; Chao et al., 1971; Chao and Greenkorn, 1971; Starling and Han, 1972; Soave, 1972; Carruth and Kobayasihi, 1972; Lu et al., 1973; Lee and Kesler, 1975). Such correlation schemes rely on the availability of T , and P, values for com:ponents and mixtures. In fact, the main limitation of these correlations results from the difficulties of measuring T , and P,; many industrially important components, such as heavy hydrocarbons, decompose a t temperatures far below the critical. For such components and petroleum cuts, T,, P,, and w have been correlated as functions of normal boiling point (Tb),specific gravity (SG),and/or the Watson Characterization Factor (WCF) defined as
W C F = (Tb)'/3/SG
(1)
0019-7874/79/1018-0049$01.00/0
where Tb is in degrees Rankine. Thus, any thermodynamic properties correlations for such components, based on T,, P,, and w , are reduced to correlations in the two parameters Tb and SG. The main objective of this work is to introduce a third, easily measurable parameter which, together with Tb and SG can be used to develop accurate predictions for T,, P,, and w. Another, broader objective is to use the new parameter together with T b and SG to develop thermodynamic properties correlations for heavy components and petroleum cuts. This second objective has only in part been accomplished. D e f i n i t i o n o f a Third P a r a m e t e r The third parameter proposed here has, in analogy with Pitzer's acentric factor, been defined as follows w , = a[ln P.,,(T = 0.85Tb) + b ] (2) where P,(T = 0.85Tb) is the vapor pressure of the liquid in atmospheres a t a temperature equal to 0.85 of Tb in the absolute scale. The constants a and b were chosen to make the values of w, most similar to w for the n-alkanes from C4 to C16 and Czo,for which reliable vapor pressure data are available. Thus we find that w, = -1.46456[1n P,,(T = 0.85Tb) '- 1.74731 = -3.37227tlog pv,(T = 0. jTb) + 0.758841 (3) 0 1979 American Chemical Society
50
Ind. Eng. Chem. Fundam., Vol. 18, No. 1 , 1979
Table I. Values of w and Deviations of w , T,, P,, and AH,,, % deviation in
n-pentane n-hexane n-heptane n -octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane cyclopentane cis-2-pentene trans-2-pentene 2-methyl-1-butene 2-methyl-2-butene 2-methylbutane benzene cyclohexene cyclohexane methylcyclopentane 1-hexene cis-2-hexene trans-2-hexene cis-3-hexene trans-3-hexene 2-methylpentane 3-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane toluene eth ylcyclopentane meth ylcy clohexane 1-heptene 2-methylhexane 3-methylhexane styrene o-xylene m-xylene p-xylene ethylbenzene ethylc yclohexane n-propylcyclopentane 1-octene trans-2-octene 3,3-dimethylhexane 2-methylheptane 3-ethylhexane 2,2,34rimethylpentane 2,2,44rimethylpentane n-propylbenzene isopropylbenzene 1-methyl-4-ethylbenzene 1,2,4-trimethylbenzene 1,3,5-trimethylbenzene n-propylcyclohexane 1-nonene n-butylbenzene sec-butylbenzene trans-decalin 1-decene 1-methylnaphthalene 1-undecene 1-dodecene 1-tridecene 1-tetradecene 1-pentadecene 1-hexadecene
0.2449 0.2994 0.3496 0.4030 0.4487 0.4972 0.5400 0.5801 0.6256 0.6573 0.6923 0.2539 0.2741 0.2681 0.2561 0.2730 0,1989 0.3421 0.2862 0.2735 0.2643 0.2821 0.3017 0.3102 0.2993 0.3188 0.2599 0.2553 0.1844 0.2153 0.3428 0.2926 0.2430 0.3341 0.3163 0.2497 0.3964 0.3849 0.3855 0.3713 0.3719 0.2957 0.3418 0.3820 0.3177 0.2884 0.3669 0.3580 0.2629 0.2447 0.3938 0.3703 0.4601 0.4321 0.4507 0.3426 0.4312 0.4365 0.3912 0.2947 0.4778 0.4475 0.5214 0.5639 0.6042 0.6420 0.6797 0.7137
- 0.006
+ 0.003
0.0 0.007 0.007 0.007 0.005 0.016 0.009 - 0.024 -0.016 + 0.006 + 0.003 0.006 -0.003 - 0.050 - 0.028 - 0.003 0.017 0.020 -0.012 - 0.020 + 0,008 + 0.029 + 0.041 + 0.042
0.0 - 0.001 t 0.011 t 0.008 - 0.011 - 0.004 - 0.006 - 0.033 0.009 t0.018 + 0.036 - 0.008 - 0.022 - 0.021 - 0.005 - 0.005 - 0.005 - 0.003 - 0.002 0.007 +0.015 0.018 -0.010 - 0.005 - 0.007 - 0.031 + 0.033 - 0.008 - 0.005 - 0.001 + 0.006 + 0.003 + 0.033
+
---
- 0.007 - 0.002 + 0.009 + 0.008 +0.006 - 0.007 - 0.007 -0.021
w, is well defined for most fluids except for liquefied noble gases which freeze a t a temperature far above 0.85 of Tb. In this work we have used the vapor pressure correlations of Reid et al. (1977) to compute w,; the values of w, thus calculated are given in Table I. The experimental determination of w, will be discussed in a subsequent section.
+ 0.035 -0.010 -0.036 + 0.007 + 0.002 + 0.036 + 0.028 + 0.002 + 0.043 -0.210 +0.125 -0.075 - 0.399 -0.575 + 0.442 t 1.75 + 0.919 +0.106 -0.527 -0.719 t 0.326 +0.633 - 0.242 -0.502 - 0,890 -1.37 t 0.079 - 0.202 -0.491 - 0.504 +0.142 +0.107 + 0.295 + 0.553 -0.045 - 0.387 -1.32 +0.171 +0.575 + 0.798 +0.459 +0.450 -0.895 + 0.301 -0.276 + 0,099 +0.169 t 0.1.08 + 0.383 + 0.243 + 0.458 + 0.810 + 0.221 + 0.147 + 0.093 - 0.497 t0.271 - 0.699 - 1.06 +0.087 +0.227 +0.055 +0.089 + 0.060 + 0.062 + 0.289 t0.030 +0.612
- 0.396 + 1.14 - 0.619 + 0.098
-1.00 0.444 0.116 0.707 -0.143 - 0.363 0.042 - 1.77 - 2.20 -3.77 + 5.63 + 4.40 1.20 0.545 - 0.475 - 2.06 + 1.48 + 1.09 - 1.24 -2.74 - 1.82 - 1.97 + 1.84 -1.19 - 1.59 - 1.93 - 0.292 - 0.097 - 0.075 + 2.19 + 1.75 + 1.36 -1.90 - 2.37 t 0.603 + 1.36 +1.31 + 3.66 -0.155 +0.129 - 5.89 +0.341 + 2.43 + 1.92 +0.210 + 1.71 -0.787 +4.51 +6.10 -1.52 2.29 - 0.825 2.33 - 7.39 - 1.22 + 0.087 - 0.227 + 0.409 + 0.847 +0.120 + 0.78 + 1.97 + 1.34 +4.23
-0.019 + 0.015 + 0.066 - 0.058 - 0.034 -0.021 + 0.025 + 0.009 + 0.041 -0.032 - 0.002 t0.137 + 0.358 -0.073 + 0.295 +0.072 -0.154 -0.046 -0.259 + 0.068 -0.078 +0.135 -0.706 -0.208 -0.068 -0.481 + 0.440 + 0.148 +0.072 + 0.065 +0.135 -0.792 + 0.597 + 0.523 i0.476 + 0.784 + 1.29 -0.135 - 0.608 - 0.108 + 0.695 - 0.025 + 1.55 + 0.491 - 1.60 -0.117 -0.856 -0.119 - 0.099 - 0.490 -1.35 - 1.77 -0.918 -0.111 -0.043 + 1.23 + 0.304 + 0.868 + 1.23 - 0.204 - 0.226 - 0.015 0.072 - 0.001 -0.174 -0.360 -0.195 -0.236
Approach Used to Improve Correlations for T,, P,, and w A number of correlations for T,, P,, and w for heavy hydrocarbons and petroleum cuts, using Tb and SG as characterizing parameters, have been published (Cavett, 1962; Kesler and Lee, 1976). The average errors of the
Ind. Eng.
Chem. Fundam., Vol. 18, No. 1, 1979 51
Table 11. Correlation Parameters for Eq 5 a
b
C
d
1.2539 1.98943+2 -1.1435 3.3060 2.1239Et2
- 6.42923-4
4.5328E-7 - 3.3696E- 4 -8.0152E-7 - 2.4892E- 6 8.14643-3
-1.4461Et 2 -1.9968Et4 1.3079E+ 2 1.6203Et2 -2.7944E+4
property, e A
SGA TCA w SA
In P , A AH,
1.1855 3.3716E- 3 - 2.925 6E- 4 1.7006Et1
vaporization at the normal point, AHlb,have been correlated by
I03
$ og9 $
Bs
I 00
~~~~~~~~
$60
n o o%
:+
;
- - - - - -% _ D S . P ----
--%:,---L--n 8,
I
098
%:am:
ey"
J 097 096 095
a
b
8 0
0
0 94 160
240
320
480 160
400
SPECIES N U M B E R
240
320
400
480
In PvpA = (0.9665 + 2.3021
SPECIES NUMBER
Figure 1. Ratio of the predicted heat of vaporization at the normal boiling point t o the experimental value for the 68 hydrocarbons of this study: a, as predicted from our correlation; b, as predicted using the Pitzer-Curl correlation, eq 14.
correlation of Kesler and Lee are, for example, 9.470, 0.7170, and 3.1% for I J , T,, and P,, respectively for the components given in Table I. Initial attempts to improve the accuracy by introducing the third parameter, us,into the correlation scheme (in somewhat arbitrary ways) proved futile, largely because Tb, SG, and w, are not completely independent. Careful analysis of the data showed that T,, P,, and w for the n-alkanes could be correlated to a high degree of accuracy as a function of only the normal boiling point. Further, systematic deviations in T,, P,, and w betweten alkanes and non-alkanes of the same boiling point seerned to depend upon the differences in their specific gravities and a,. Based on the above observations, we have: (1)developed smooth correlations of T,, P,, w , and S G for the n-alkanes as a function of normal boiling point; (2) developed correlations for T,, P,, w , and several other properties for non-alkanes using the n-alkane correlations as the reference, and Tb and the differences ASG and Aw, as the correlating variables. The terms ASG and Aw, are the differences in the specific gravity and us,respectively, between the real liquid and as obtained from the n-alkane correlations using the real liquid normal boiling point. The form of the equations used in these empirical correlations is based on perturbation theory, as described in a subsequent section. The choice of Tb and the differences ASG and Aw, as correlating variables is novel and deserves some comment. The variables Tb, SG, and w , are, in general, highly correlated; indeed for the n-alkanes, they are completely correlated, as shown below. The difference variables ASG and Aws are, however, independent of the normal boiling point and are measures of departure from n-alkane behavior. They have, therefore, been chosen together with T b as the independent correlating parameters. Description of the ni -Alkane Reference System The specific gravities of the C5-C15 n-alkanes (the only n-alkanes which are liquid a t 60 "F) have been correlated as follows SGA = 1.25386 - 6.42922
X
+ C T b 2 -k d/Tb
(5)
where 0* represents T,, a,,In P,, and H , b of the n-alkanes. The constants for these equations are given in Table 11. The vapor pressure (P,,, in atm) for the n-alkanes as a function of temperature in the range of 0.013 to 2 atm was correlated by
I
a
bTb
OA = a
10-4Tb+ 4.5328 X 10-7Tb2- 144.605/Tb (4)
where the superscript A denotes correlations specific to the n-alkanes. Simila.rly, T,, w,, In P,, and the heat of
10-3Tb- 5.4728 X 10-7Tb2+ 89.301/Tb)F1 (6)
X
where
and
c = (0.0953 + 3.179 x
10-4Tb)(Tb,- 0.85)
or more simply, by In P,pA= (1.7473 + 0.6828w,*)F1
(8)
The average errors of the correlations for the C5-C15 n-alkanes are 0.03% for S G , 0.05% for T,, 0.46% for P,, 0.29% for w , 0.03% for the heats of vaporization at the normal boiling point, and 0.13% for the vapor pressure as a function of temperature. It is obvious from the accuracy of these correlations that the normal boiling point temperature is an excellent characterization factor for the n-alkanes. It is for this reason that the normal boiling point has also been used as the primary (but not sole) characterizing parameter for the other hydrocarbons. Perturbation Theory Basis for the Empirical Correlations The following equation forms the basis for the correlations which follow O(x1,
..., x,)
=
Oref(xl,
c
..., x,)
Oi(X1,
+
..., X , ) ( X i
- xl'ef)
+
1
Equation 9 is in the form of a Taylor series expansion of the property 0 about its value in some reference state. The superscript ref denotes the reference system properties and Oi and OLl are the first and second partial derivative with respect to the difference variables ( x , - x,Te? evaluated for the reference system. In this work, 0, and Or, have been evaluated by correlating experimental data. It is worth noting that the convergence of any perturbation expansion depends on the choice of the reference system. The closer the system of interest is to the reference system, the more rapidly convergent the expansion will be. In particular, it is important that the relevant physics of the actual and reference systems be as similar as possible. Hence, it is advantageous to choose the family of n-alkanes as the reference for correlating the properties of hydrocarbons rather than, for instance, spherical molecules, as is often done in statistical mechanics.
52
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
Table 111. Correlation Parameters for Eq 1 2 T C
aS1) a,(2) ad31 a2(1) a2(2) a2(3) a11(1) a1,(2) a11(3) a22(1)
an(2) a22(3) a12(1) a12(2) a12(3)
-0.81018Et4 0.44480Et 2 - 0,60394E- 1 -0.82508Et 4 0.47911Et 2 -0.66552E-1 -0.45512Et 5 0.23255Et 3 - 0.28993 0.54163Et 5 -0.30390Et 3 0.41965 0.59140Et 5 -0.31402Et 3 0.40959
In Pc -0.10903E+ 2 0.62305E-1 -0.8 5 7 01E-4 -0.43524Et 2 0.24904 -0.3 3 2353- 3 0.13691Et 2 -0.22322 0.48566E-3 0.53097Et 3 -0.28166Et 1 0.37 0 7 6E- 2 0.50728E+ 3 -0.28390Et 1 0.39003E- 2
For the purpose of empirically correlating the deviations between n-alkane and non-alkane hydrocarbons, various forms of eq 9 can be used. For physical properties such as T,, P,, w, and AHvb,eq 9 reduces to B(Tb,SG,w,) = BA(Tb) Ol(Tb)(w,- wSA) &(Tb)(SG - SGA) O,,(Tb)(O, - usA)' B,z(Tb)(SG - SGA)(w, - LO:) + Oz,(Tb)(SG - SGA), (10) while for functions of temperature and density (or pressure) we have B(Tbr,dr,Tb,SG,ws)= BA(Tbr,dr,Tb)+ B,(Tbr,dr,Tb)(SG - SGA) -t ol(Tbr,dr,Tb) (us oll(Tbr,dr,Tb)( U s ab)' 812(Tbr,dr,Tb)(SG- SGA)(w, - w:) + &2(Tbr,dr,Tb)(SG - SGA)' (11)
W
0.33968Et 2 -0.17673 0.225 81E- 3 0.10729Et 2 -0.7 1111E- 1 0.10439E- 3 0.22971Et 3 -0.10991E+ 1 0.1 2309E-2 0.37560Et 2 -0.1 1038 0.4 6 89 IE--4 -0.68044Et 2 0.35989 -0.48588E-3
AH,
0.42917Et 5 -0.22522Et 3 0.30080 -0.37924Ei-5 0.20550Ei-3 -0.27614 0.53815Et 6 -0.29851Et 4 0.40411Et 1 0.74258Et 6 - 0.39555Et 4 0.52700Et 1 0.12505E.t 7 -0,65727Et4 0.85858E+ 1
the vapor pressure range of 0.013 to 2 atm for the 68 hydrocarbons in this study. The improved accuracy in AH,, resulting from use of the w, is illustrated in Figure la. The figure gives the ratio of the predicted to experimental AHvbfor each of the components considered in this study (The species numbers in this figure are the identification numbers used in Reid et al., 1977). For comparison, Figure l b gives the results for the same ratio when AHvb is predicted from the equation (Reid et al., 1977)
based on the Pitzer-Curl correction, with T = Tb. These figures show that the error in mvbexceeds 1%for only seven of the 68 hydrocarbons when the present correlation is used, while the error in AH,, is less than 1% for only where dr is a reduced density and the notation @(Tbr,dr,Tb) seven of the 68 compounds if eq 14 is used. indicates that all the B quantities are functions of reduced Discussion a n d Use of Correlations temperature and density, as well as the reference system In writing the above equations, we have made the folfunctions which are characterized by Tb. lowing observation and assumptions. First, it is observed A second-order polynomial has been chosen for Oi and that Bref, Oi, and Bij are only functions of reduced temBij in eq 10 as follows perature, reduced density, and Tb. This is because the Bi = ai (1) + ai (2)Tb + ai (3)Tb' reference system is characterized by only its normal boiling point, and its properties depend only upon Tbrand reduced (12) 8ij = U i j ( 1 ) + Uij(2)Tb + Uij(3)Tb2 density (or reduced pressure). Secondly, it is assumed that the normal boiling point can be taken as the primary Equations 10 and 12 were used to fit T,, P,, w , and AH,, characterizing parameter, so that the appropriate choice for the 68 hydrocarbons selected from Reid et al. (1977) of reference system for a general hydrocarbon is the "nand listed in Table I. Only those hydrocarbons were alkane" with the same boiling point. The next assumption selected for which the normal boiling point, critical is that the departure of the general hydrocarbon behavior properties, and vapor pressure data are all available and from that of the n-alkane can be described by only the two which are liquids a t 60 O F so that the specific gravity could difference variables (SG - SGA)and (w, - cob) where SGA be computed. (The vapor pressure data were needed to and usAare the specific gravity and w, for the n-alkane of compute us.) The errors in the prediction of Tc,P,, w, and the same normal boiling point as the hydrocarbon of inare given in Table I for the 68 hydrocarbons studied terest. Finally, it has been assumed that only the firstin this work. The average errors in these correlations are and second-order terms need be retained in the expansions. 0.38% for T,, 1.67% for P,, 4.13% for w , and 0.37% for In order to determine the convergence of the empirical These errors are approximately half as large as those perturbation theory, several of the properties considered mentioned earlier for the Kesler-Lee (1976) correlation. here were correlated with the simpler, first-order perThe constants for the present correlations are given in turbation expansion Table 111. O(Tb,SG,w,) = OA(Tb)+ OI(Tb)(SG- SGA) -k The vapor pressure for the hydrocarbons was correlated as a function of temperature with the following simple &(Tb)(O,- ab) (15) expression where eq 12 was again used for B1 and 0'. In this case, the values of the constants differ from those in Table 111, and In Pvp= In PwA+ (w, - wsA)FlF2 (13) the average errors in the correlations were found to be where the function F1 was given previously and about twice as large as with correlations based on eq 10, even though considerably fewer constants are involved. Fz = 0.6828 + (1.2532 - 4.3574 X 10-3Tb)(l- 0.85/Tbr) This suggests that the perturbation correlation using the n-alkane reference system is rapidly convergent. The average error in this correlation was only 0.20% over
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
f . 5
=
2000
I
l h l
b
'OOol ,
1
O i -
300
400
T O K
I /
500
Figure 2. The predicted and experimental heats of vaporization for n-heptane, benzene, and cis-2-pentene as a function of temperature: solid lines, experimental dam; 0 , predictions using eq 16; 0,predictions from eq 14 with experimental critical temperatures and acentric factors; V, predictions from eq 14 with calculated T , and o.
The accuracy of the present correlations indicates that the normal boiling point, specific gravity, and w, are useful characterizing parameters for physical properties correlations. We envision two ways in which these parameters can be used in more general correlations. The first is, as presented in the previous section, to use these parameters to calculate T,, P,, arid u,which can then be used with presently available correlations based on these three parameters, to predict thermodynamic properties. The second procedure is to develop new thermodynamic properties correlations., perhaps in the form of eq 9-11, in which Tb, S G , and w, are the characterizing parameters. Since these measured parameters would be used directly (rather than indirectly as above) such correlations should be more accurate than using Pitzer type correlations with estimated parameters. Unfortunately, much work would be involved in developing these new correlations. To assess the likely advantages and difficulties of using the correlations presented here in other correlations in which T,, P,, and w are the characterizing parameters, we consider, as an example, the temperature dependence of the heat of vaporization. The Watson relation (Reynes and Thodos, 1962; Reid et, al., 1977)
and eq 14 are two frequently used relations for predicting the heat of vaporization as a function of temperature. Figure 2 shows the experimental heats of vaporization for n-heptane, benzene, and cis-2-pentene (solid lines) together with the points (filled circles) which result from eq 16 with values of AHvb and T , obtained from our correlations (method a), the points (unfilled circles) resulting from eq 14 with experimental values of T , and w (method b), and the points (filled triangles) which result from eq 14 with correlated values of T , and w (method c). (Note: The unfilled circles and filled triangles sometimes overlap; in these cases only the former have been shown.) I t is evident, from Figure 2, that all three calculations of AHv are in reasonable agreement with experiment. Method a is the most accurate near the normal boiling point but less accurate than method b near the critical point. This is to be expected, since experimental normal boiling point information is used in method a, while experimental critical point information is used in method b.
53
(Note that since AHvdrops off so rapidly as T approaches T,, the main error in methods a and c is from the use of correlated rather than actual values of T,.) Finally, method c is the least accurate, since it contains the errors in the Pitzer type correlations near the normal boiling point and the errors in using correlated values of T , and w near the critical point. The above shows that correlations based on properties a t or near the normal boiling point can be expected to be reasonably accurate there but less accurate in the critical region. The Pitzer-Curl and Lee-Kesler correlations, which require critical point information, will better identify the critical region but can be expected to be less accurate elsewhere, such as in the vicinity of the normal boiling point. Finally, the Pitzer type correlations with estimated values of T,, P,, and w contain the errors inherent in these correlations together with those that arise from the uncertainities in T,, P,, and W . This suggests that for the characterizing parameters Tb,SG, and usto be most useful in thermodynamic properties correlations, the correlations should be developed using these parameters as the primary variables. The above calculation also suggests that when critical property data are available, the three parameters T,, P,, and os (rather than w ) may be useful for correlating thermodynamic data, since this parameter set may lead to accurate predictions near the normal boiling point and correctly identify the critical region. However, we have not yet explored this idea. It is useful, then, to consider how the third parameter oswould be experimentally evaluated. To compute wSfrom its definition, eq 4, one would have to accurately determine the normal boiling point for the substance of interest and then determine its vapor pressure at a temperature equal to 0.85 of Th. This is a very inconvenient procedure; more likely, one would measure the vapor pressure of the substance at a number of fixed temperatures (at least two, and preferably three or more) and then fit these data to a vapor pressure equation, such as the Antoine equation. The normal boiling point and w, would then be computed from the fitted experimental data. Alternatively, one could also use the vapor pressure correlation developed here and experimental data to find Tb and 0,. Finally, we note that while this study has been limited to pure components, we expect the ideas and correlations developed here also to be useful for calculating pseudocritical constants for mixtures and, especially, mixtures of undefined composition. Nomenclature d, = reduced density AH, = heat of vaporization, cal/g-mol AH& = H , at normal boiling point, cal/g-mol P, = critical pressure, atm P, = vapor pressure, atm ~ 6= 'specific gravity T = temperature, K Th = nor-mal boiling point, K Tbr
T,
X
= =
critical temperature, K
= independent variable
Greek Letters w = Pitzer acentric factor w , = a new third parameter
0 = dependent variable in eq 10, 11, 12, and 14
Literature Cited Barner, H. E., Pigford, R. L., Schreiner, W. C., Proc. Am. Pet. Inst.,Sect. 3 , 46, 244 (1966). Carruth, G. F., Kobayashi, R., fnd. Eng. Chem. Fundam., 11, 509 (1972). Cavett, R. H., Proc. A m . Pet. Inst., Sect. 3 , 42, 351 (1962).
54
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
Chao, K. C., Greenkorn, R. A., Nat. Gas Proc. Assoc., Research Report RR-3, Tulsa, Okla., 1971. Chao, K. C., Greenkorn, R. A., Olabisi, O., Hensel, B. H., AIChE J., 17, 353 (1971). Chappelow, C. C., Snyder, P. S., Winnick, J., J. Chem. Enq. Data, 16, 440 (1971). Curl, R. F., Pitzer, K. S., Ind. Eng. Chem., 5 0 , 265 (1958). Kesler, M. G., Lee, B. I., Hydrocarbon Process., 3 , 153 (1976). Lee, 6. I, Kesler, M. G., AIChE J., 21, 510 (1975). Lu, B. C-Y, Hsi, C., Chang, S.-D., Tsang, A., AIChE J., 19, 748 (1973). Pltzer, K. S., Curl, R. F., Jr., J. Am. Chem. Soc., 79, 2369 (1957). Pltzer, K. S.,Hultgren, G. O., J. Am. Chem. SOC.,80, 4793 (1958).
Pitzer, K. S., Lippmann, D. Z., Curl, R. F., Jr., Huggins, C. M.,Peterson, D. E., J . Am. Chem. Soc., 77, 3433 (1955). Reid, R. C., Prausnitz, J. M., Sherwood, T. K., "The Properties of Gases and Liauids". 3rd ed. McGraw-Hill. New York. N.Y.. 1977. Revnes E. G.. Thodos. G.. AIChE J.. 8. 357 (1962)' Soave, G., Chem. Eng. Sci., 27, 1197 (1972). Starling, K. E., Han. M. S., Hydrocarbon Process., 5 , 129 (1972). Wilson, G. M., Adv. Cryog. Eng., 9, 168 (1964).
Receiued f o r review April 17, 1978 Accepted November 1, 1978
Vapor-Liquid Equilibria in the Benzene-Polybutadiene-Cyclohexane
System S. Dlnqer and D. C. Bonner" Chemical Engineering Department, Texas A&M University, College Station, Texas 77843
Robert A. Elefritr Central Research Laboratories, Firestone Tire & Rubber Company, Akron, Ohio
We report vapor-liquid equilibrium results for the benzene-polybutadiene-cyclohexane system at 98 O C and at very small solvent concentrations characteristic of industrial devolatilization conditions. We show that the data are modeled well by the corresponding-statestheory of polymer solutions using only binary interaction parameters. The experimental technique used to obtain the data is a relatively simple modification of gas-liquid chromatography and is a rapid and precise technique.
Introduction It is common industrial practice in polymer production facilities to process many polymers in mixed solvent systems and then to remove the mixed solvents in various types of drying equipment. In addition, many polymers which are not readily soluble in a single solvent can be processed in a mixture of solvents (Wessling, 1970, 1973). It has also been observed that the volatility of a single solvent in a polymer solution can be increased or decreased by addition of a selected second solvent (British Patent, 1965). The unusual and interesting characteristics of dual solvent systems and their practical importance to industry have led us to investigate experimentally the vapor-liquid equilibria of a three-component polymer solution: benzene (1)-polybutadiene (2)-cyclohexane (3). Such an investigation has only been reported previously by one group (Katayama et al., 1971; Matsumura and Katayama, 1974). The experimental technique used by Katayama e t al. is laborious and would be difficult to use a t the elevated temperatures characteristic of solvent removal processes. Enhanced solubility in mixed solvents is a common characteristic of amorphous polymers. The mixtures usually include one component with a higher cohesive energy density than the polymer and another component with a lower cohesive energy density than the polymer. From the practical and economically important industrial *To whom correspondence should be addressed at Shell Development Co., P.O. Box 1380, Houston, Texas 77001.
viewpoint, this characteristic also means that synthetic polymer plants can feed a less expensive mixed solvents stream to the process, rather than specify a pure-component solvent, without specifying complete separations within the plant. This advantage eliminates some very expensive equipment and expensive, high-purity raw materials. In solutions involving polar polymers, the enhanced solubility in mixed solvent systems is often due to the formation of specific interactions between one (or both) solvents and the polymer. Often one component forms a complex with the polymer, and the complexed polymer is then solvated by the other solvent component (Wessling, 1970). In this situation, process plants require the dual solvent system, and the economics favor higher processing costs. Apart from the practical aspects of enhanced polymer solubility in mixed solvents, studies of such mixed solvent and polymer systems can yield valuable information for theoretical predictions. For instance, Flory-Huggins theory can be applied to many single solvent-polymer systems. For ternary polymer solutions involving two solvents, Pouch19 and Patterson (1976) have shown that the corresponding-states polymer solution theory of Prigogine (1957) and Flory (1965) yields better results than does application of the Flory-Huggins theory. Selection of the benzene-polybutadiene-cyclohexane system for this study was made on the basis that PBD is a commercially important, nonpolar polymer and that literature vapor-liquid equilibria data are available for two
0019-7874/79/1018-0054$01.00/00 1979 American Chemical Society
