Ind. Eng. C h e m . Res. 1989,28, 1893-1898
1893
Critical Evaluation of Equation of State Mixing Rules for the Prediction of High-pressure Phase Equilibria Steven K. Shibata and Stanley I. Sandler* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Some recent data on the high-pressure phase compositions and densities for the asymmetric nitrogen n-butane mixture are used to test a number of different combining and mixing rules for use with the Peng-Robinson equation of state. We find that the van der Waals one-fluid mixing rules with two adjustable parameters is superior to the mixing rules of Panagiotopoulos and Reid, Huron and Vidal, and Luedecke and Prausnitz and is as good as the Lee-Sandler density-dependent mixing rule which is computationally more complicated. Further, by use of fluid-specific (rather than generalized) pure component parameters, very good predictions are obtained for the densities of both the coexisting vapor and liquid phases.
+
I. Introduction Equations of state and their extension to mixtures using mixing and combining rules are used for estimating the PVT properties of fluids, for correlating phase equilibria data, in the design of separations processes, and for simulating petroleum reservoir operations. Simple generalized cubic equations of state of the van der Waals class are perhaps the most commonly used, especially when speed of computation is important or when information about the pure components is limited. For illustration here, our discussion will use only the Peng-Robinson equation of state a(t) p = - -RT (1) V - b ( V + CYb)(V+ pb) though our discussion would be little changed if other cubic equations of state were used. [Shibata (1988) reports some results for the Soave-Redlich-Kwong and Patel-Teja equations of state.] Separate from the volumetric form of the equation of state is the determination of the value of its parameters. Here P is pressure, T is temperature, R is the gas constant, CY = 1 21f2,/3 = 1- 2lI2,and a and b are discussed below. By demanding that the first two derivatives of P with respect to V be zero at the critical point and following the procedure of Soave, Peng and Robinson proposed the following recipes for obtaining the equation of state parameters from only the critical temperature T,, critical pressure P,, and the acentric factor o:
+
RTC b = 0.07780-
pc
a = 0.45724,[1 K
R2T,2
PC
+ ~ (-1(T/Tc)'/2)]2
= 0.37464 + 1.54226~- 0 . 2 6 9 9 2 ~ ~
(2)
(3) (4)
Alternatively, by fitting experimental vapor pressure and density data, Xu and Sandler (1987a,b) have presented polynomial expansions for both the a and b parameters which are specific to each fluid. Not surprisingly, the fluid-specific parameter set yields more accurate vapor pressures and liquid densities. To expand the range of equations of state from pure fluids to mixtures, a combination of mixing and combining rules is generally used. For example, the most commonly used mixing rules are the one-fluid rules of van der Waals
and nc nc
b, = C C x i ~ j b i j i=lj=1
(6)
where aii and bii are the pure fluid parameters and nc is the number of components. The cross parameters aij and bij are obtained from a set of combining rules. The most commonly used combining rules are and
Note that if dij = 0 then i=l
The following two general comments can be made about the class of cubic equations of state with generalized parameters and the mixing and combining rules discussed above. First, the liquid densities are generally in error by 5% or more for both pure fluids and mixtures. Second, for mixtures of compounds similar in size and chemical nature (i.e., the hydrocarbons and inorganic gases), phase equilibrium compositions can be correlated quite well with the one-fluid mixing rules above with a single adjustable parameter kik However, for mixtures containing molecules dissimilar in size or chemical nature, the predicted or correlated phase boundaries may not be in good agreement with experiment. Since the first of these shortcomings, the inaccurate density predictions, can largely be eliminated using the fluid specific parameters of Xu and Sandler, we consider here the second problem, the difficulty in obtaining accurate phase boundaries for mixtures of dissimilar molecules. To overcome this problem, a number of alternate mixing rules have been proposed. A discussion of some of these new mixing rules appears in the next section. 11. Alternate Mixing Rules
Panagiotopoulos and Reid (1987) have proposed the use of a nonquadratic mixing rule for a, in phase equilibrium calculations. Their two-parameter model uses eq 9 for b, and a linear function of mole fraction in the combining rule for ail: aij = (aiaj)'/2[1 - kij + (k.. 11 - k .11. ) X i ] (10) If kij = kji, eq 7 and 8 are recovered. Panagiotopoulos and
0888-5885/89/2628-1893$01.50/0
0 1989 American Chemical Society
1894 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989
Reid successfully applied this mixing rule to polar ternary mixtures of carbon dioxide, water, and ethanol. Vidal (1978) and Huron and Vidal (1979) developed a mixing rule by equating the P = m fugacity coefficients from an equaton of state to a local composition model excess Gibbs free energy expression using nc
g E = RT[ln cp -
Cxi In pi*] i=l
(11)
where cp is the fugacity coefficient of the solution at T, P = m, and x i and pi* is the fugacity coefficient of pure component i at T and P = m. Using the Peng-Robinson equation of state in eq 11 yields =
-(;
-
with
forces of dissimilar components at high densities. This mixing rule satisfies the low-density boundary condition of being quadratic in composition; also the second term vanishes as the mixture becomes very rich in either component. [This condition is necessary since the model was developed to account for only noncentral forces between unlike components.] Since this mixing rule was formulated for the Helmholtz free energy, the expression for the equation of state is obtained from = -(%)T,n#
The Luedecke and Prausnitz mixing rule in the PengRobinson equation of state gives p = - - RT V-6,
am
V(V+b,)+b,(V-b,)
nc nc 1 Cxixj[~i~iF(T,i)~ij + xjaj?(Tcj)~ji]x 2(2)'I2RTV2b, i+j
Substitution of the NRTL local composition model of Renon and Prausnitz (1968) into eq 11 gives the following
where
Lee and Sandler (1987) developed another, simpler density-dependent mixing rule by analyzing Monte Carlo computer simulation data for mixtures of square-well molecules. The final result of their detailed theoretical analysis of the simulation data is the remarkably simple expression that
The nonquadratic mixing rule for a , is obtained by equating eq 12 and 14
When applied to the Peng-Robinson EOS, this mixing rule yields
p = - - RT V - b,
with aij, Cij,and Cji as the three adjustable parameters. The mixing rule for b, must be eq 9 to obtain a finite limit for geXat infinite pressure. With aij = 0, the Huron-Vidal mixing rule reduces to eq 7 and 8. This empirical local composition mixing rule has been applied successfully to a variety of polar and asymmetric systems (Adachi and Sugie, 1985; Gupte and Daubert, 1986; Heidemann and Rizvi, 1986). Recently, density-dependent mixing rules have been developed, which reduce to eq 7 and 8 only at low density and have a higher order composition dependence at high densities. A density-dependent mixing rule for polar and asymmetric mixtures has been proposed by Luedecke and Prausnitz (1985) for the Helmholtz free energy based on the use of a noncubic equation. If we adapt their analysis to the Peng-Robinson equation of state, we obtain
This mixing rule for a, is a combination of two terms: the first term is the quadratic density-independent van der Waals one-fluid mixing rule and the second is a densitydependent term cubic in mole fraction which has been proposed to account for the noncentral intermolecular
'(V+
i=ij=1
aij cybij)(V
+ pbij)
(21)
By rearrangement, eq 21 can be written in pseudocubic form as
p = - - RT V-b,
am f o
(V+ab,)(V+pb,)
(22)
with nc nc fn
(V + ab,)(V
+
+ @b,) +
aij -
= ~ ~ x i x j ( cybij)(V V /3bij)a ,
(23)
The method of solution for the compressibilities is to start the calculation assuming f a = 1, solve eq 22, use the value of V obtained in eq 23, and then repeat the cycle until convergence is achieved. With this method, only one additional iteration loop is required; it is a rapidly convergent loop since f a is always close to unity. Note that as V m, f a 1, so that at low densities the van der Waals one-fluid mixing rules are recovered. Lee and Sandler (1987) found their density-dependent mixing rule did not yield significantlyimproved results for T P x y predictions over the van der Waals one-fluid mixing rules for the systems they studied; however, generally a smaller interaction parameter was needed to obtain optimum fits of experimental data than if eq 5 and 6 were used. Thus, if the interaction parameter were set equal
- -
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1895 to zero, as might be the case in a completely predictive calculation, the density-dependent mixing rule would be more accurate. However, when a fitted interaction parameter was used, there was no clear advantage over the van der Waals one-fluid rules. Note that the two density-dependent mixing rules we have considered result in the equation of state no longer being cubic in volume. In particular, the Lee-Sandler mixing rule results in the number of terms in the equation, and thus the polynomial order, being dependent on the number of components in the mixture. The LuedeckePrausnitz mixing rule results in a nonpolymonial form of the equation, as a result of the logarithm term, and two additional terms in the equation of state if there are two or more components. While some of the mixing rules discussed above were developed specifically for chemically dissimilar substances, i.e., water + hydrocarbons or other mixtures containing polar compounds, it is nonetheless of interest to see how they perform for nonpolar mixtures since a single equation of state and its mixing rule must be useful for such simple mixtures also. Therefore, in following section, we consider the application of the Peng-Robinson equation of state and the mixing and combining rules discussed above to accurate data for both the compositions and densities in the system of nitrogen + n-butane. One of the reasons for choosing this system is that the traditional van der Waals one-fluid mixing rules lead to a relatively poor correlation of the vapor-liquid phase boundary for this mixture of species with very different sizes and critical temperatures. [We have similar data for the nitrogen cyclohexane, carbon dioxide + n-butane, and carbon dioxide + cyclohexane binary systems, and ternary mixtures of nitrogen and carbon dioxide with + n-butane or cyclohexane; see Shibata (1988). Since identical conclusions are obtained by using these other data sets, we consider only the N2 + n-C, system here.] The 14 models we tested, all based on the Peng-Robinson equation of state but with different choices for the equation of state parameters and mixing and combining rules, are listed in Table I.
ki]=-O 0 4 1 . X ~ - S a n d l e r A&B k1]=-0 164 Di]=-O 233 Gen A&B k 1 ~ = - 0359 DIJ=-O 253,X-S A&B Expt I Data From This Study
100
h
0
m
80
I W I
3 vl W (0
e a 60
40
120-
- ki]=
__ _ _ -.
+
0
-
100
-
80
-
0 128 Generalized A&B klj=-O 041 Xu-Sandler A&B ki]=-O 164,Di]=-O 233 Gen A&B ki]=-O 359 D i j = - O 253 X-S A&B Expt'l Data From This Study
h
0
m
W I I
3
cn w m a Q
60 -
111. Results
The procedure used in fitting the models to the data was as follows. Since the experimental errors in the compositions of both phases were approximately equal and larger than the errors in temperature and pressure, we took the latter two as the independent variables. The adjustable parameters were chosen to minimize the objective function i=ljs1
-
i
b 90
I
I
I
I
01
02
03
04
05
PHASE DENSITY (gm/cm')
n p b nc
OBJ = C C ( I X- ,jEXPI ~ ~ +~ lyjcALc ~ ~ - yjEXPl)i
40
(24)
though we reached essentially the same conclusions with other objective functions we tested. It should be emphasized that we fit only the compositional data, not the phase density data. Therefore, the density results are predictions and not part of the fitting process, though clearly a better fit of the compositional phase envelope also results in more accurate predictions of the densities of the coexisting phases. The results of the calculations are displayed in Table I1 and the collection of largely self-explanatory figures. Parts a and b of Figure 1show the composition and density results, respectively, for the VDWlG, VDWlFS, VDW2G, and VDW2FS models. We see from Figure l a that there is marked improvement, especially in the location of the critical point, in going from one to two binary parameters, and for a fixed number of parameters, there is always an
Figure 1. (a, top) TPxy data of nitrogen + n-butane at T = 410.9 K. Fit with the PR equation of state and different variations of the van der Waals mixing rules. (b, bottom) Phase density data of nitrogen + n-butane at T = 410.9 K. Calculated results from the PR equation of state and different variations of the van der Waals mixing rules.
advantage in fitting the compositional data to using the fluid-specific rather than generalized equation of state parameters. Figure l b shows a much more striking difference between the generalized and fluid-specific parameters, in that only the latter set are capable of adequately describing the liquid density. In fact, we see that with the exception of the near-critical region (where no classical model is expected to perform well) model VDW2FS yields satisfactory phase compositions and densities but with quite large values of the two binary parameters. Parts a and b of Figures 2 and 3 present similar results for the Panagiotopoulos and Reid mixing rule (Models
1896 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989
';c
-1 I
--
e
k 1 2 = 0 376 k 2 1 = - 3 6 2 Gen A&B k 1 2 = 0 240 k 2 1 = - 3 Z 6 X - S A&B Expt I Data From This S t u d y
100
-
1
- -
1
-.
4
alphaij=O 3 3 , k i j = O 63 Cen A&B alphai]=O 3 1 . k i j = 0 21.X-S A&B a=-lOb12=0Ob21=01GenA&B a = 0 1 b I 2 = 0 2 b 2 1 = 0 0 X - S A&B Expt I Data From This S t u d y
h
o
m
W
80
W E
LL
3
3
u! u! W
u)
a IT
a (L
W cn
6'3
40
I i'
cc
I 03
I
0;
OC
c 4
01
- k 1 2 = 0 376 k 2 1 = - 3
'20~
0 3
02
c 4
MOLE FRACTION OF N t
MOLE FRACTION OF N2 8 2 Gen A&B k 1 2 = 0 240 k 2 1 = - 3 26 X-S A&B Expt I Data From This S t u d y
120
-:
- alphaij=O 3 3 k i j = 0 63 Gen
A&B alphai)=O 31 k i j = 0 21 X-Lc A&B a = - l O b 1 2 = 0 O b : ! l = O 1 GenA&B _ . a = 0 1 b 1 2 = 0 2 b 2 1 = 0 0 X-S A & B e Expt 1 Data From This S t u d y ..
__ 103
h
h
-
-
m k~
0
m
6C
83
Lz W
3 u! u! W
a IT 50
i
I
l 3C
e
01
e
02
03
0 4
05
PHASE DENSITY (gm/cm')
I
A
30
0'
i 02
a3
:a
-- -3
PHASE DENSITY ( g m / c m ' )
Figure 2. (a, top) TPxy data for nitrogen + n-butane at 2' = 410.9 K. Fit with the PR equation of state and different variations of the Panagiotopoulos-Reid mixing rule. (b, bottom) Phase density data of nitrogen + n-butane at T = 410.9 K. Calculated results from the PR equation of state and different variations of the Panagiotopou10s-Reid mixing rule.
Figure 3. (a, top) TPxy data of nitrogen + n-butane at T = 410.9 K. Fit with the PR equation of state and different variations of the Huron-Vidal mixing rule. (b, bottom) Phase density data of nitrogen + n-butane at ?' = 410.9 K. Calculated results from the PR equation of state and different variations of the Huron-Vidal mixing rule.
PRSG and PRSFS) and the Huron-Vidal mixing rule (models HV3G and HV3FS). We see that the Panagiotopoulos and Reid models lead to an approximately correct critical point but with a exaggeration of the dew point curve, which is also reflected in the shape of the density envelope. The Huron-Vidal model with three binary parameters, like the one-parameter van der Waals model, results in a good correlation of the bubble and dew point curves at low pressure but a large overestimation of the critical pressure and composition. As shown in parts a and b of Figure 4, the computationally complex three-parameter Luedecke-Prausnitz
models (LP3G and LP3FS) satisfactorily correlate the N2 + n-butane data to about 60 bar, but above approximately 70 bar we had difficulty in getting a converged solution, a problem not experienced with the other models at such low pressures. It appears, however, that these models will lead to a critical point that occurs at a pressure and nitrogen mole fraction that is much above the experimental results. Also, as with the other models, the density predictions are much better if the fluid-specific rather than the generalized equation of state parameters are used. The Lee-Sandler density-dependent mixing rule with one binary parameter (models LSlG and LSlFS), shown
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1897 120
Expt I Data From This Study a&b parameters . . Xu-Sandler a&b parameters G P a r a m ki]= 0 099 S 1 2 = 0 690. P 2 1 = - 0 005 X-S P a r a m k i ] = - 0 0 7 1 , E l Z = l 9 6 ? 2 1 = - 0 0 0 8 0
- kij=-O _ _ kij= -0
131 Gen A&B 234,Xu-Sandler A&B . _kij=-O 286 D i j = - O 191 Cen A&B -. k 1 ~ = - 0 447 D I J = - O 239 X-S A&B 0 Expt I Data From This S t u d y
- Gen
100
-
100
W
-
(L
Lz
m
h
O
m
80
80
w
3 v)
m E W L
60
6C
40
4C
00
0 2
0.1
03
0 4
00
01
MOLE FRACTION OF N t
120
Expt'l Data From This Study a&b parameters . . Xu-Sandler a&b parameters G P a r a m ki]= 0 0 9 9 E12=0 690, P 2 1 = - 0 005 X-S P a r a m k i j = - 0 0 7 1 , E'12=1 96. FZI=-0008 0
- Gen
lo@
-
03
02
04
MOLE FRACTION OF N,
-'
I
- kij=-O
A&B
131 Cen
- - k i j = -0 2 3 4 . X ~ - S a n d l e r A&B
'
..
j
0
-.
kij=-O 286 D i j = - O 191 Gen A&B kij=-O 447 D i j = - 0 239 X-S A&B Expt'l Data From This Study
1
h
m
80
I
W
E
l
I 00
I
4
01
02
03
0 4
05
PHASE DENSITY (gm/cm')
i 00
I
I
I
I
01
02
03
0 4
I 5
PHASE DENSITY (gm/cm')
Figure 4. (a, top) TPxy data of nitrogen + n-butane at T = 410.9 K. Fit with the PR equation of state and different variations of the Luedecke-Prausnitz mixing rule. (b, bottom) Phase density data of nitrogen + n-butane at T = 410.9 K. Calculated results from the PR equation of state and different variations of the Luedecke-Prausnitz mixing rule.
Figure 5. (a, top) TPxy data of nitrogen + n-butane a t T = 410.9 K. Fit with the PR equation of state and different variations of the Lee-Sandler mixing rule applied to the attractive pressure term. (b, bottom) Phase density data for nitrogen + n-butane a t T = 410.9 K. Calculated results from the PR equation of state and different variations of the Lee-Sandler mixing rule applied to the attractive pressure term.
in Figure 5a, leads to somewhat less of an overprediction of the mixture critical point than the van der Waals oneparameter models (VDWlG and VDWlFS). A similar conclusion applies when comparing the LS2G and VDW2G models with the LS2FS and VDW2FS models. Finally, the LS2FS and VDW2FS models lead to comparable and quite accurate results for both the compositions and densities of the coexisting phases.
n-butane, the van der Waals one-fluid and Lee-Sandler mixing rules lead to the most accurate correlations of the experimental composition data, while the two-constant Panagiotopoulos-Reid and three-constant Huron-Vidal models are the least accurate. The three-constant Luedecke-Prausnitz model appears to offer no advantage over the van der Waals or Lee-Sandler models, is difficult to use, and in our calculations failed to converge to a solution at lower pressures than the other models. In all fairness, however, we should point out that this model was developed for a completely different type of equation of state. Consequently, the results reported here are for our adap-
IV. Conclusions The work here shows that, for the asymmetric (by size and critical temperature) nonpolar mixture of nitrogen +
1898 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 Table I. Models Used for t h e Correlation of the Nitrogen n-Butane Data designation characteristics VDWlG van der Waals 1-fluid mixing rules, 1 binary parameter ( k , ),. generalized EOS parameters VDW2G as VDWlG but with 2 binary parameters (kl,, d , ) VDWlFS as VDWlG but with fluid-specific EOS parameters VDW2FS as VDWlFS but with 2 binary parameters (k,,, d,,) PR2G van der Waals 1-fluid mixing rules, Panagiotopoulos-Reid combining rule with 2 binary parameters (kll, k J, generalized EOS parameters as PR2G but with fluid-specific EOS parameters PR2FS Huron-Vidal mixing rule with 3 binary parameters HV3G and generalized EOS parameters as HV3G but with fluid-specific EOS parameters HV3FS Luedecke-Prausnitz mixing rule with 3 binary LP3G parameters and generalized EOS parameters as LP3G but with fluid-specific EOS parameters LP3FS Lee-Sandler mixing rule, 1 binary parameter (k,, ), LSlG generalized EOS parameters as LSlG but with 2 binary parameters (kl,, d , 1 LS2G as LSlG but with fluid-specific EOS parameters LSlFS as LSlFS but with 2 binary parameters (kl,, d,, ) LS2FS
+
+
Table 11. Results for t h e Correlation of t h e Nitrogen n -Butane Data for t h e Models Considered Herea model AADx AADv AADl AADv 0.0873 VDWlG 0.0080 0.0220 0.0636 0.0298 0.0812 VDW2G 0.0027 0.0067 0.0870 VDWlFS 0.0075 0.0303 0.0456 0.0124 0.0030 0.0149 VDWSFS 0.0016 0.1137 0.0604 0.0010 PR2G 0.0172 0.1146 0.0229 0.0776 0.0027 PR2FS 0.0862 0.0215 0.0636 0.0082 HV3G 0.0921 0.0307 0.0479 0.0071 HV3FS 0.0492 0.0618 0.0071 0.0111 LP3G 0.0349 0.0159 LP3FS 0.0594 0.0056 0.0122 0.0550 0.0633 0.0068 LSlG 0.0298 0.0788 0.0032 LS2G 0.0073 0.0582 0.0067 0.0219 0.0407 LSlFS 0.0133 0.0157 LSSFS 0.0019 0.0035
a Note: AAD = average absolute deviation are for experimental data points only and do not include calculations to identify the critical region of each model. We estimate an experimental uncertainty of f0.003 in mole fraction and *0.001 g/cm3 in density.
tation of their ideas to the Peng-Robinson equation of state. In all cases, the use of the fluid-specificrather than the generalized Peng-Robinson equation of state parameters led to more accurate liquid densities and a small improvement in the composition correlations. Also, the introduction of a binary parameter in the combining rule for the size parameter b in the van der Waals one-fluid and Lee-Sandler models resulted in a significant improvement in the correlation of the phase envelope.
Of the models considered here, the two binary interaction parameter versions of the van der Waals one-fluid and the Lee-Sandler models with the fluid-specific equation of state parameters are the most accurate in correlating the compositional data and predicting the phase densities. Of the two, since the van der Waals one-fluid model is easier to use, it is the one we recommend based on the calculations here. Acknowledgment This work was supported, in part, by Grants CBT8612285 from the National Science Foundation and DE 85ER13436 from the United States Department of Energy and a grant from the Chevron Oil Field Research Corporation, all to the University of Delaware. Registry No. N, 7727-37-9; C4HI0,106-97-8.
Literature Cited Adachi, Y.; Sugie, H. Effects of Mixing Rules on Phase Equilibrium Calculations. Fluid Phase Equilib. 1985,24, 353-362. Gupte, P. A.; Daubert, T. E. Extension of UNIFAC to High Pressure VLE using Vidal Mixing Rules. Fluid Phase Equilib. 1986, 28, 155-170. Heidemann, R. A.; Rizvi, S. S. H. Correlation of Ammonia-Water Equilibrium Data with Various Peng-Robinson Equations of State. Fluid Phase Equilib. 1986, 29, 439-446. Huron, M.-J.; Vidal, J. New Mixing Rules in Simple Equations of State for Representing Vapor-Liquid Equilibria of Strongly Non-Ideal Mixtures. Fluid Phase Equilib. 1979,3, 255-271. Lee, K.-H.; Sandler, S. I. The Generalized van der Waals Partition Function. IV. Local Composition Models for Mixtures of Unequal Size Molecules. Fluid Phase Equilib. 1987, 34, 113-147. Luedecke, D.; Prausnitz, J. M. Phase Equilibria for Strongly Nonideal Mixtures from an Equation of State with Density-Dependent Mixing Rules. Fluid Phase Equilib. 1985,22, 1-19. Panagiotopoulos, A. Z.; Reid, R. L. High-pressure Phase Equilibria in Ternary Fluid Mixtures with Supercritical Component. In Supercritical Fluids. Chemical and Engineering Principles and Applications; Squire, T. G., Paulaitis, M. E., Ed.; ACS Symposium Series No. 329; American Chemical Society: Washington, DC, 1987; pp 115-129. Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions For Liquid Mixtures. AIChE J. 1968, 14, 135-144. Shibata, S. K. High Pressure Vapor-Liquid Equilibria Involving Mixtures of Nitrogen, Carbon Dioxide, and Hydrocarbons. Ph.D. Dissertation, University of Delaware, Newark, 1988. Vidal, J. Mixing Rules and Excess Properties in Cubic Equations of State. Chem. Eng. Sci. 1978, 33, 787-791. Xu, Z.; Sandler, S.I. Temperature-Dependent Parameters and the Peng-Robinson Equation of State Znd. Eng. Chem. Res. 1987a, 26, 601-606. Xu, Z.; Sandler, S. I. Application to Mixtures of the Peng-Robinson Equation of State with Fluid-Specific Parameters. Znd. Eng. Chem. Res. 1987b,26, 1234-1238.
Received for review December 12, 1988 Accepted September 6, 1989
