Ind. Eng. Chem. Res. 1993,
32, 2077-2087
2077
Simultaneous Correlation of Viscosity and Vapor-Liquid Equilibrium Data Weihong Cao,1 Kim Knudsen, Aage Fredenslund,* and Peter Rasmussen *
Engineering Research Center IVC-SEP, Institut for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark
Ind. Eng. Chem. Res. 1993.32:2077-2087. Downloaded from pubs.acs.org by TULANE UNIV on 01/21/19. For personal use only.
A new statistical thermodynamic model for liquids has been proposed. A viscosity equation for pure liquids, a viscosity equation for liquid mixtures, and an equation for activity coefficients are obtained from the model and are expressed by the same parameters. The model can be used to correlate viscosities of pure liquids at different temperatures, to simultaneously correlate viscosities and activity coefficients of binary liquid mixtures at different temperatures and compositions, and to predict viscosities and activity coefficients of multicomponent systems. Finally, for binary systems viscosities can be predicted from activity coefficient information and vice versa. The results show that both the correlations and the predictions with the new model are very good. Introduction
viscosities of pure liquids and binary systems and to predict viscosities of multicomponent systems. For most systems, the correlation and prediction results are very good.
This paper is a followup of our previous work (Cao et al., 1992). In the previous paper, a viscosity model was established that can be used for the viscosity correlations and predictions of pure liquids and liquid mixtures. In this paper, the molecular size is introduced and the new model can be developed. The expressions of both viscosity and activity coefficients of liquid mixtures can be obtained from the new model. The new model is a so-called
Another application of Eyring’s theory is combining
models for viscosity calculations with activity coefficient equations. Wei and Rowley (1985) developed a local composition model for viscosity. The model is based on the NRTL equation and can be used to predict viscosities of nonaqueous liquid mixtures. The data needed in the model are viscosities of pure liquids, interaction parameters
“viscosity-thermodynamics" model (UNIMOD). Viscosity and activity coefficients belong traditionally to two different kinds of properties, transport and thermodynamic properties. Many statistical thermodynamic models for activity coefficients can be found, and many activity coefficient equations have been developed, such as Wilson (Wilson, 1964), NRTL (Renon and Prausnitz, 1968, Cukor and Prausnitz, 1969), ASOG (Kojima and Tochigi, 1979), UNIQUAC (Abrams and Prausnitz, 1975), and UNIFAC (Fredenslund, et al., 1975). These models generally give satisfactory results for correlation and prediction for activity coefficients. The parameters needed in the prediction are normally obtained from vapor-liquid equilibrium data for binary systems. For viscosity of pure liquids and liquid mixtures, the Eyring’s absolute rate theory (Glasstone et al., 1941) has been used. McAllister’s equation (McAllister, 1960), developed from the Eyring’s theory, is a very good viscosity correlation equation. There are two adjustable parameters for binary systems and three adjustable parameters for ternary systems, based on three-body interaction. Dizechi and Marschall (1982) modified McAllister’s equation for ternary systems, and they correlated viscosity data of ternary systems using six binary parameters and one adjustable ternary parameter. On the basis of McAllister’s equation, Asfour et al. (1991) reported a prediction method. In this method, the parameters in McAllister’s equation are calculated from pure component properties. The results for eight n-alkane systems are good. The parameters in all McAllister’s type equations are strongly temperature dependent. Cao et al. (1992) developed a statistical thermodynamic model for viscosity of pure liquids and liquid mixtures. Local composition is introduced into the model. The model can be used to correlate
in NRTL (determined from vapor-liquid equilibrium (VLB) data of the respective binary systems), and excess enthalpy data, which may be experimental data or calculated from the NRTL equation. Wu (1986) used a direct proportionality between the activation energy in Eyring’s theory and the Gibbs energy and developed a new viscosity model. The Gibbs energy may be expressed by the Wilson, NRTL, and UNIQUAC activity coefficient equations. This method can be used to correlate viscosities of liquid mixtures with good results, and to predict viscosities of liquid mixtures. To calculate viscosities of liquid mixtures with good accuracy, much information about the mixtures are needed. The information can of course be obtained directly from viscosity data but also from thermodynamic properties of the liquid mixtures. It is interesting to utilize thermodynamic rather than transport properties of liquid mixtures, because there are many thermodynamic models and there are more experimental thermodynamic data than viscosities. The purpose of this paper is to develop a new statistical thermodynamic model, from which both viscosity and activity coefficients of liquid mixtures can be obtained and expressed by the same parameters. The parameters in the new model can be easily determined by information of either viscosity or thermodynamic properties and predictions or correlations between viscosity and activity coefficients of liquid mixtures can be done.
Model For Viscosity and Activity Coefficient The UNIQUAC and UNIFAC models are based on Guggenheim’s lattice theory. Local composition is introduced and Staverman’s model is taken as the hightemperature boundary condition. The potential energy partition function for a liquid mixture may be expressed
f Present address: Department of Chemistry, Tsinghua Uni100084, People’s Republic of China. versity, Beijing * To whom correspondence should be addressed.
0888-5885/93/2632-2077$04.00/0
as ©
1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No.
2078
9, 1993
II
Following the idea (as in UNIQUAC) that a molecule in the mixture is represented by a set of segments, we express the kinetic energy partition function for the mixture as
Introducing eq
kT »ta ,
/ -?¡ /I =
(3)
The proportionality constant, called the dynamic viscosity of the liquid, is dependent on the molecular size, shape, and interaction and is possibly expressed by the partition function which is also dependent on the same molecular properties. Eyring developed an absolute rate theory (Glasstone et al., 1941) and described the molecular movement in the way of “chemical reaction”. A molecule, moving from one position to another in the liquid, should possess higher energy than other molecules in order to produce a hole and move to an available position. The extra energy is called activation energy and the energy barrier is symmetric between the nearest positions. If there is no shear force, the activation energy, Ua, in different directions is the same and the molecular movement frequencies in the directions are also the same:
If there is a shear force on the liquid, the activation energy barriers in the forward and backward directions of the shear force are different, Ua =f ea, and the molecular movement frequencies in the two directions are also different:
e0, done by the shear force is added to moving molecule. Subscript a denotes activation state. Here we represent a molecule in the mixture by a set of segments. Every segment has the same size: l\ X h X i3 as l x / x l (height X width X length). There is also an activation energy barrier between two nearest segment positions, and the shear force influences the segment activation energy barrier because work is done by the shear force to the segments. The net velocity of the segment movement and the work done by the shear force to a segment will be
eq 6 can be
rewritten
,
(kT®a
"ß
Ug\
GO)
where Avr is the net velocity of the segment movement, Ua is the activation energy without the shear force on the mixture, f is the shear force, e„ is the work done by the shear force to a segment, ka is the frequency of segment movement without the shear force, 0 is the partition function at activation state, and = *. Following the Eyring’s idea of molecule reaction process, the segment reaction process in the mixture is assumed as following: n
Nftisegi)
=
VNftisegi)a
7=1
the frequency of the segment movement for the mixture can
be expressed as
where Ua¡ is the activation energy barrier of segment i in the mixture. Eyring et al. (see Glasstone et al., 1941) have for a number of systems compared the deviations of the observed energy of activation for flow in mixtures from the linear additive law with the deviation from Raoult’s law. The comparison shows a close relationship between the energy of activation and the free energy of mixing. Eyring et al. determined the energy of activation from the internal energy of vaporization at the normal boiling point multiplied by a general constant. Wu (1986) determined the energy of activation from the free energy of mixing multiplied by a general constant. Cao et al. (1992) used a directly proportional relationship between the activation energy barrier and the potential energy of molecule i in the liquid mixture. Here we also assume that the activation energy is directly proportional to the potential energy of molecule i in the liquid mixture and the difference between ground state and activation state is that there are two degrees of freedom of molecular movement at the activation state because of the minimum energy limitation: Uai n
0
=
12
(12)
n,.[/0¡
\ /2
-= (—— / „ t=|V h2
1T¥expV
Ug + tg
kT
))
(6)
(13)
vfNin/3 ‘
n,· is the proportionality constant of segment i. Combining eqs 3 and 9-13, the dynamic viscosity of the mixture is written as
where
fh A>Vr
(
(9)
fk//kT
=
r2kT
r2
»
{MM/2
=
kT ) \( UaZS) kT®a
(
as
AvT
a
=
kT^a
(8)
T"o”pVHV
=
because the work,
^
(7)
and
7
6*
In the two equations, is the partition function, ¡ is the number of distinguishable configurations of lattice i in the mixture, N¡ is the number of molecules i, r¡ is the number of segments in a molecule i, mn is the average segment mass of molecule i, U0i is the potential energy of lattice i in the mixture, V¡, is the free volume of lattice i, k is Boltzmann’s constant, h is Planck’s constant, T is the temperature, n is the number of components in the mixture, superscript s denotes Staverman’s model, subscript p denotes potential energy, subscript k denotes kinetic energy, and subscript i denotes component i. If there is a shear force, f, on a Newtonian liquid, the molecules in different molecular layers move at different velocities, v, and the local velocity gradient is directly proportional to the force:
II
.
kav
J
vf*‘/s
x
(14)
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 where i, the average segment number of a molecule, and Uoi, the potential energy of lattice i in the mixture, are given by r
-
tNr&i
Uoi
*
=
·1
·1
(15)
kinematic viscosity of pure liquid i. Af and V are calculated = by Zx¡M¡ and V = Based on statistical thermodynamics, the following expressions for the excess Gibbs energy and the activity coefficients can be obtained, neglecting free-volume effects: z
7=1
where z is the coordination number of the lattice, q¿ is the area parameter of molecule i, Uji is the interaction potential energy between sites j and i, Mi is the molecular weight of component i, R is the gas constant, v is the volume of a molecule in the mixture, and V is the molar volume of
Ge
Xi
In 7/
=
In
-
x,
»,·
In —h -q¡ In--q,· ln(
=
+ -Qi In ¿
j,
+ L¡
-
-'¿XjLj xíj=i
M:\1/2
i
Z zq/itZA
...
(16,
For a liquid mixture, the dynamic viscosity equation, combining eqs 14 and 16, is written as n
ln(j?V)
=
,
ln(í7iVf) + 2
1 ( ;/0;)
-
=1
=1
^
/ );=
-
t*i
*
ifi*
+
v Hn
400.00
Number of Pure Liquids Figure
2.
MRSD of viscosity correlations for 433 pure liquids.
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 Table II.
Correlation Results of Viscosity for Binary Systems1 binary system
no.
Un.
carbon tetrachloride(l)-benzene(2) carbon tetrachloride(l)-cyclohexane(2)
1
2 3
n-octane(l)-n-nonane(2) n-octane(l)-n-decane(2) n-octane(l)-n-dodecane(2) n-octane(l)-n-hexadecane(2) n-nonane(l)-n-decane(2) n-nonane(l)-n-dodecane(2) n-nonane (l)-n-tetradecane(2) n-nonane(l)-n-hexadecane(2)
4 5
6 7
8 9 10
2081
-
Uu
U12-
U22
-46.1612 -183.6734
59.2637 218.2679
83.4446 10.5426
-73.2594 -12.5887
-93.1929 -175.1037
109.9741 255.9483
33.9821
-32.0643
-84.7322 -128.1115 -154.5588
101.1825 172.2859 214.0421
points 9 7
9
18 3 3 9 3 3 3
MRSD 0.0011 0.0015 0.0020 0.0003 0.0006 0.0038 0.0000 0.0008 0.0024 0.0032
temp, °C 298.15 298.15 298.15 293.15 298.15 298.15 298.15 298.15 298.15 298.15
298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15
reí* 12,574 29,332 35,206 35,206; 36, 285 35,206 35,206 35,206 35,206 35,206 35, 206
Total number of viscosity points 3900. Total number of binary systems 288. Average MRSD of viscosity correlations 0.0083. Journal of Chemical Engineering Data, volume and page. 0
1****6
0.10
CHL0R0F0RM(1)----T0LUENE(2) •
0.08
0.06
MRSD of each system Average MRSD
-
in or 0.04
0.02
-
-
*
vZ 0.00 0.00
200.00
100.00
300.00
Number of Systems Figure
3.
MRSD of viscosity correlations for 288 binary systems.
4. Correlation results of dynamic viscosity of chloroformtoluene at 298.15 K.
Figure
Parameters for Liquid Mixtures from Viscosity Data of Binary Systems 1. Viscosity Correlations for Binary Systems. In the new model, the interaction potential energy differences, Uji Uu, for a liquid mixture represent the adjustable parameters. These parameters are determined from experimental data for binary liquid mixtures. On the basis of eqs 17 and 21, they can be determined from the viscosity data of liquid mixtures. On the basis of eqs 22 or 23, they can also be determined from thermodynamic properties, such as vapor-liquid equilibrium data, liquid-liquid equilibrium data, or excess enthalpy data of the liquid mixtures. In this section, the viscosity data of binary system are used to determine the parameters, Uji Uu- For each binary system, the experimental viscosity data are fitted using eqs 17 or 21, using Fi of eq 29 as the objective function. The experimental viscosity data for binary systems have been found from the Journal of Chemical and Engineering Data. A total of 3990 sets of experimental viscosity data for 288 binary systems is collected here and correlated by the new model. The correlation results and the parameters of interaction potential energy, Uji Uu, for some systems are listed in Table II. A more extensive table for many systems is given in the supplementary material. The average MRSD of the correlation for all binary systems is 0.83%. Figure 3 shows MRSD for each binary system and the average MRSD. Figure 4 shows the correlation of a dynamic viscosity of a system with a maximum. Figure 5 shows correlation of the kinematic viscosity of a system with a minimum. Figure 6 shows correlation of the dynamic viscosity at different temperatures, in which the
1,2-DICHL0R0RETHANE(l)----BENZENE(2)
-
-
-
5. Correlation results of kinematic viscosity of 1,2-dichloroethane-benzene at 303.15 K.
Figure
parameters of interaction potential energy, Uji Uu, are temperature independent. The correlation results show that the new viscosity equation of liquid mixtures can be used, with very good accuracy, to correlate viscosity data for binary systems, (1) having a maximum or a minimum in viscosities, (2) at different temperatures, (3) for systems which include mixtures with polar-polar, nonpolarnonpolar, nonpolar-polar components, and (4) with a big difference of molecular size between the two components. 2. Viscosity Predictions for Multicomponent Systems. Predictions of viscosity for multicomponent systems -
2082
Ind. Eng. Chem. Res., Vol. 32, No.
9, 1993 0.20 ·* -
0.15
Q (/)
MRSD of each system by New Model Average MRSD of New Model MRSD of each system by UNIVISC --Average MRSD of UNIVISC
·
'
+
·
·
·
0.10
Q2
0.05
0.00 0.00
10.00
20.00
30.00
Number of Systems Figure
6. Correlation results of dynamic viscosity of 1,4-dioxaneethanol at 288.15, 293.15, 298.15, 303.15, and 308.15 K.
Figure
MRSD of viscosity predictions for 28 multicomponent
used to predict viscosity data of multicomponent systems
be done if the parameters of interaction potential energy, U¡¡ Uu, are known for all the relevant binary systems (listed in Table II). A total of 451 sets of viscosity data for 28 multicomponent systems is predicted and can
with parameters estimated from viscosity data of binary
-
mixtures.
3. VLE Predictions for Binary Systems. The interaction potential energy parameters, Uj¡- Uu, obtained from viscosity data of binary systems (listed in Table II), can be used to predict VLE data based on the activity coefficient equation, eq 23. Here the vapor phase is assumed as an ideal gas mixture, and the following relationship is used to calculate VLE data:
compared with the experimental data, which are found from the Journal of Chemical and Engineering Data. The prediction results are listed in Table III. The average MRSD for 28 multicomponent systems is 3.30%. Figure 7 shows MRSD for each multicomponent system and the average MRSD. More detailed results are given in the supplementary material. UNIVISC (Wu, 1986) is a group-contribution viscosity model. The parameters in the model can determined from viscosity data or VLE of liquid mixtures. If the parameters are determined from VLE data, the prediction is normally poor. If the parameters are determined from viscosity data of liquid mixtures, the prediction is good. A total of 331 sets of viscosity data for 20 multicomponent systems has been predicted by the UNIVISC model, where the parameters have been determined from viscosity data of binary liquid mixtures. The results are also listed in Table III. The average MRSD for 20 multicomponent systems is 5.88%. Figure 7 gives MRSD for each multicomponent system and the average MRSD. Other 120 sets of viscosity data for eight multicomponent systems cannot be predicted by UNIVISC model because there are no parameters. The average MRSD of the predictions of the new model for the 20 multicomponent systems is 3.35%. The predicted results of the new model, compared with the experimental data and UNIVISC predictions, show that the new viscosity equation of liquid mixture can be Table III.
7.
systems.
=
(3D
PyJP0^
(32) lnPpi = Ai + Bi/T+Cln(T) + DiTEi where P is the pressure of the system, P°¿ is the saturated vapor pressure of pure liquid i, y¡ is the mole fraction of component i in the vapor phase, and the parameters, A,·, Bu C„ Di, and E, are found from the DIPPR data bank. The activity coefficients for all components in a liquid mixture can be predicted from the activity coefficient equation at given temperature and mole fractions in liquid phase. Then pressure and mole fractions in the vapor phase at the given temperature and liquid phase composition can be calculated from eq 31. Here the predictions for 134 binary systems have been done. All experimental VLE data of the binary systems are found from the Dortmund data bank (Gmehling et al., 1977). All parameters Uji Uu, are obtained from the viscosity correlations of the binary systems (listed in Table II). The viscosity correlation results and VLE prediction results of the 134 binary systems are listed in Table IV, where MRSD -
Prediction Results of Viscosity for Multicomponent Systems*1 MRSD
no. 1
2 3
4 5
6 7
8 9 10
multicomponent system
points
temp, °C
ethanol(l)-benzene(2)-n-heptane(3) n-heptane(l)-2-methylheptane(2)-toluene(3) carbon tetrachloride(l)-n-hexane(2)-benzene(3) ethanol(l)-acetone(2)-cyclohexane(3) acetone(l)-n-hexane(2)-ethanol(3) acetone(l)-ethanol(2)-methanol(3) acetone(l)-n-hexane(2)-cyclohexane(3) n-hexane(l)-cyclohexane(2)-ethanol(3) methanol(l)-ethanol(2)-isopropyl alcohol(3) acetone(l)-ethanol(2)-isopropylalcohol(3)
24 15 14 12 12
298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15
12 12 12 12 12
New Model
UNIVISC
ref”
0.0234 0.0187 0.0038 0.0758 0.0864 0.0894 0.0173 0.0352 0.0106 0.0398
0.0316 0.0995 0.0378 0.0325 0.0336 0.0789 0.0495 0.0175 0.0166 0.0565
34,200 20,46 14,55 29,336 29,336 29,336 29,336 29, 336 29, 336
29,336
Total number of viscosity points 451. Total number of multicomponent systems 28. Average MRSD of viscosity predictions by new model b Total number of viscosity points 331. Total number of multicomponent systems 20. Average MRSD of viscosity predictions by new model 0.0335. Average MRSD of viscosity predictions by UNIVISC 0.0588.c Journal of Chemical Engineering Data, volume and page. 0
0.0330.
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993
2083
Table IV. Prediction Results of VLE from Viscosity for Binary Systems*"®
Un-Uu
binary system
no.
-46.1620
carbon tetrachloride(l)-benzene(2)
1
Ul2
-
U22
59.2648
points
MRSD viscosity VLE P y 0.0012 0.0104 0.0015 0.0139 0.0035 0.0050 0.0010 0.0337 0.0031 0.0242 0.0162 0.0514 0.0022 0.0260 0.0012 0.0617 0.0017 0.0004 0.0017 0.0007
9
68
-183.6727
218.2664
-73.3940
62.4229
-9.5561
8.9153
n-hexane(l)-n-decane(2)
-120.6709
151.4583
6
ethyl acetate(l)-carbon tetrachloride(2)
-124.1177
141.4610
7
n-hexane(l)-n-octane(2)
67.9488
-62.2806
8
n-hexane(l)-n-dodecane(2)
-157.1308
219.9935
11 3 12
9
p-xylene(l)-m-xylene(2)
-3.6407
3.8577
36
10
p-xylene(l)-m-xylene(2)
-3.6407
3.8577
36
2
carbon tetrachloride(l)-cyclohexane(2)
3
benzene(l)-ethyl acetate(2)
4
n-heptane(l)-n-octane(2)
5
7 41 14
59 18 12 18 19 19 52 18
9 9
0.0040 0.0089 0.0026 0.0096
0.0165 0.0132
0.0001
temp, 298.15 311.65 298.15 283.15 293.15 323.15 293.15 328.15 293.15 308.15 293.15 346.10 293.15 328.15 298.15 308.15 288.15 411.58 288.15 411.56
pressure, Pa
®C
298.15 353.05 298.15 353.75 318.15 353.16 298.15 328.15 298.15 417.95 318.15 349.71 298.15 328.15 298.15 308.15 303.15 412.18 303.15 412.17
24 767.27
101 577.70
6 220.82
101 557.70
36 879.93
102 030.70
9 743.80
21 811.50
2 827.77
102 384.20
89 833.45
101 325.00
10 839.10
61 594.98
2 209.15
28 071.11
101 276.60
101 325.00
101 231.60
101 325.00
Viscosity: total number of viscosity points 1991; total number of viscosity systems 134; average MRSD of viscosity correlations 0.0086. VLE(pressure): total number of pressure points 3988; total number of pressure sysetms 131; average MRSD of pressure predictions 0.1027. ® VLE(molar fraction): total number of molar fraction points 3569; total number of molar fraction systems 114; average MRSD of molar fraction predictions 0.0452. *
6
1.00
0.10
• •
0.08
0.06
-
MRSD of each system Average MRSD
0.80
MRSD of eoch system Average MRSD
0.60
-
if)
00
CC
a:
0.04
0.02
0.40
0.20
-
^-7V 0.00 0.00
I
50.00
I
100.00
8.
-
0.00 0.00
V
150.00
50.00
100.00
150.00
Number of the Systems
Number of the Systems Figure
-
-
MRSD of viscosity correlations for 134 binary systems.
Figure
9.
MRSD of VLE predictions (pressure) for 131 binary
systems.
is defined by
/ /^ ,-^ ^
)
Parameters for Liquid Mixtures from VLE Data of Binary Systems 1331
1. VLE Correlations and Predictions for Liquid Mixtures. The parameters Uji Uu can also be deter-
in
(1
\2 ¡ average segment fraction of component i = partition function 0 = partition function at activation state ; = number of distinguishable configurations y¡
.
ea
=
.
.
=
Superscript s
=
Staverman’s model
Subscripts a = activation state cal = calculated data exp = experimental data i = component i k = kinetic energy p = potential energy
P
=
pressure
=
segment
molar fraction in vapor phase
S upplementary Material
Available: List of detailed
current masthead page.
Literature Cited Abrams, D. S.; and Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116-128.
Asfour, A.-F. A.; Cooper, E. F.; Wu, J.; Zahran, R. R. Prediction of McAllister Model Parameters from Pure Component Properties for Binary n-Alkane Systems. Ind. Eng. Chem. 1991, 30, 1666-
-
=
y
2087
information corresponding to the examples given in Tables I-VI (19 pages). Ordering information is given on any
-
and i U„i
r
=
j
1669. Cao, W.; Fredenslund, A.; Rasmussen, P. Statistical Thermodynamic
Model for Viscosity of Pure Liquids and Liquid Mixtures. IEC
Res. 1992, 31, 2603-2619. Cukor, P. M.; Prausnitz, J. M. Proceedings of the International Symposium on Distillation; Inst. Chem. Eng.: London, 1969; Vol. 3, p 88. Daubert, T. E; Danner, R. P. Physical and Thermodynamic
Properties of Pure Chemicals: Data Compilation; Hemisphere Publishing Corp.: New York, 1989. Dizechi, M.; Marschall, E. Correlation for Viscosity Data of Liquid Mixtures. Ind. Eng. Chem. Process Des. Dev. 1982,21, 282-289. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21,1086-1099. Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Process; McGraw-Hill: New York, 1941; Chapter 9, p 477. Gmehling, J.; Arlt, W.; Onken, U. DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1977; Vol, I. Hansen, . K.; Rasmussen, P.; Fredenslund, A. Vapor-Liquid Equilibria by UNIFAC Group Contribution. 5. Revision and Extension. IEC Res. 1991, 30, 2352-2355. Kojima, K.; Tochigi, K. Prediction of Vapor-Liquid Equilibrium by ASOG Method; Elsevier, Amsterdam, 1979. McAllister, R. A. The Viscosity of Liquid Mixtures. AIChE J. 1960, 6, 427-431. Renon, H.; Prausnitz, J. M. Local Composition in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968,14,135. Skjold-Jorgensen, S.; Rasmussen, P.; Fredenslund, A. On the Temperature Dependence of UNIQUAC/UNIFAC Models. Chem. Eng. Sci. 1980, 35, 2389-2403. Wei, I. C.; Rowley, R. L. A Local Composition Model for Multicomponent Liquid Mixtures Shear Viscosity. Chem. Eng. Sci. 1985, 40, 401-408. Wilson, G. M. A New Expression for the Excess Gibbs Energy. J. Am. Chem. Soc. 1964, 86,127. Wu, D. T., Prediction of Viscosities of Liquid Mixtures by a Group Contribution Method. Fluid Phase Equilibria 1986,30,149-156.
Received for review November 17,1992 Revised manuscript received May 12,1993 Accepted May 18, 1993· •
Abstract published in Advance ACS Abstracts, August 15,
1993.
