Ind. Eng. Chem. Res. 1993,32, 2017-2087
2077
Simultaneous Correlation of Viscosity and Vapor-Liquid Equilibrium Data Weihong Cao,t Kim Knudsen, Aage Fredenslund,’ and Peter Rasmussen Engineering Research Center IVC-SEP, Institut for Kemiteknik, The Technical University of Denmark, DK-2800 Lyngby, Denmark
A new statistical thermodynamic model for liquids has been proposed. A viscosity equation for pure liquids, a viscosity equation for liquid mixtures, and a n equation for activity coefficients are obtained from the model and are expressed by the same parameters. The model can be used t o 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 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 “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, 19691, 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’stheory, 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 + Present address: Department of Chemistry, Tsinghua University, Beijing 1OOO84,People’s Republic of China. To whom correspondence should be addressed.
viscosities of pure liquids and binary systems and to predict viscositiesof multicomponent systems. For most systems, the correlation and prediction results are very good. 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 in NRTL (determined from vapor-liquid equilibrium (VLE) 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 as
0888-5885/93/2632-2071~0~.00f 0 0 1993 American Chemical Society
2078 Ind. Eng. Chem. Res., Vol. 32, No. 9 , 1993 t,
13 =fl 1 -
232
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
2
(7)
Introducing eq 7 and kT
>> to
(8)
eq 6 can be rewritten as In the two equations, Q is the partition function, wi is the number of distinguishable configurations of lattice i in the mixture, Ni is the number of molecules i, ri is the number of segments in a molecule i, rn, is the average segment mass of molecule i, Uoi is the potential energy of lattice i in the mixture, Vfiis the free volume of lattice i, k is Boltzmann's constant, h is Planck's constant, Tis 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, u, and the local velocity gradient is directly proportional to the force: f = -qAv/l (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: =7 kT 2 Qa exp(-
2)
(4)
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 7 e,, and the molecular movement frequencies in the two directions are also different: (5)
because the work, to, done by the shear force is added to a 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: 11 X 12 X 13 = 1 X 1 X 1 (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 A v r = l ( k--exp T%
h Q
(--uiita)
-
Avr = fka14/kT (10) where Avr is the net velocity of the segment movement, U, 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, Q, is the partition function at activation state, and Q = Q$k. Following the Eyring's idea of molecule reaction process, the segment reaction process in the mixture is assumed as following: n
n
the frequency of the segment movement for the mixture can be expressed as
where Uaj 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 = niUoi
where ni is the proportionality constant of segment i. Combining eqs 3 and 9-13, the dynamic viscosity of the mixture is written as
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2079 kinematic viscosity of pure liquid i. M and Vare calculated by M = CziMi and V = &Vi. Based on statistical thermodynamics, the following expressions for the excess Gibbs energy and the activity coefficientscan be obtained, neglectingfree-volumeeffects:
where i,the average segment number of a molecule, and Uoi,the potential energy of lattice i in the mixture, are given by t =~
n
n
n
N i r i / ~ N iUoi= ( z q i N i / 2 ) x U j i 6 j i(15) J=1
r=l
1=1
where z is the coordination number of the lattice, qi is the area parameter of molecule i, Ujiis the interaction potential energy between sites j and i, Mi is the molecular weight of component i, R is the gas constant, u is the volume of a molecule in the mixture, and V is the molar volume of the mixture. For a pure liquid, the dynamic viscosity equation, from eq 14, can be expressed as
j-1
'-
p -1 ekTkj
where GE is the excess Gibbs energy of the mixture, yi is the activity coefficient of component i, and Li is a parameter given by the following equation: L, = (z/2)(ri- si)- (ri- 1) (24) This completes the development of the new model. The equations for viscosity of pure liquids, the viscosity for liquid mixtures, and the activity coefficients for liquid mixtures are obtained from the same model and are expressed using the same parameters in the model. The parameters in the final equations are Vi (molar volume of component i), Mi (molecular weight of component i), ni (proportional constant of segment i), qi (area parameter of molecule i), ri (number of segments in a molecule i), and Uji - Uii (interaction potential energy).
For a liquid mixture, the dynamic viscosity equation, combining eqs 14 and 16, is written as n
n
Here it is assumed that the excess value of the free volume, V f , is zero. The definitions of different composition variables are in this work given by
Comparison with Other Well-Known Thermodynamic Models and Viscosity Models n
n
where 6;i is the local composition, 8i is the average area fraction of component i, 4i is the average segment fraction of component i, x i is the mole fraction of component i, ~ j is the interaction parameter between sites j and i in the mixture:
uji
z - uii ( 2 RT The kinematic viscosity equations are also obtained from eqs 16 and 17 by the definition of kinematic viscosity, qV = YM, T~~
= exp - --
for a pure liquid:
for a liquid mixture:
i
The new model can be reduced to or compared with some well-known viscosity equations and activity coefficient equations separately. The equations for the excess Gibbs energy and activity Coefficientsare exactly the same as those of UNIQUAC. If we let ri be equal to the same constant, e.g. 1,for all components in the mixture, the viscosity equations of the new model are reduced to our original equations (Cao et al., 1992). The viscosity equations of the model can be reduced to or compared with the de Guzman equation and its modified equations, the McAllister equation, the Dymond-Brawn equation, the Grunberg-Nissan equation, and the Wei-Rowley equation, among others. In this work,the equations of viscosity, the excess Gibbs energy,and the activity coefficientsare given by one model. The model can be used to calculate viscosities and activity coefficients not only separately, like the above well-known equations, but also together using the same parameters. It is possible, using the new model, to correlate viscosities of binary systems and to predict viscosities of multicomponent systems, to correlate viscosities and to predict activity coefficientsof liquid mixtures, to correlate viscosity and activity coefficients of liquid mixtures together, and to correlate activity coefficients and to predict viscosities of liquid mixtures.
Parameters for Pure Liquids
where M is the molecular weight of the mixture, Vi is the molar volume of component i, v is the dynamic viscosity of the mixture, vi is the dynamic viscosity of pure liquid i, Y is the kinematic viscosity of the mixture, vi is the
Most parameters in the model are those of pure liquids and are determined from the properties of pure liquids. and qi are calculated by UNIFAC group volume and surface area (Hansen et al., 19911, Mi is obtained from DIPPR data bank (Daubert and Danner, 1989), Vi is calculated by the following equation:
2080 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
where parameters Ai, Bi, Ci, and Di are obtained from DIPPR data bank, z is calculated by the Skjold-Jorgensen equation (Skjold-Jorgensen et al., 1980): z = 35.2 - 0.1272T
+ 0.00014P
(26) Uii is calculated from the molar heat of vaporization of the pure liquid i: (zqi/2)uii = RT - AuapHm(i,
(27)
where AuapHm(i,is the molar heat of vaporization of the pure liquid i and calculated by
where parameters Ai, Bi, Ci, and Di are obtained from DIPPR data bank and Triis the reduced temperature of pure liquid i (Tri= Tci/T,Tciis the critical temperature of component i and is found from DIPPR data bank). We choose to incorporate the value of Vfi into the parameter ni and set arbitrarily Vfi= 1 unit volume. ni is temperature dependent and is obtained by fitting eqs 16 and 20 to experimental viscosities of pure liquid i at different temperatures. The temperature function and the objective function of the correlation are j-0
e(
viscositycal- viscositye,
Fl =
1
“viscosity
)
2
(29)
where A, (j= 0, 1, 2 , ...I are the parameters in eq 29 for
pure liquid i , m is the number of experimental viscosity data points, subscripts cal and exp denote calculated and experimental viscosities, a a i t y is the standard deviation of the correlation (a,,= 10-7 Pa s, a” = 10-lo m2/s). A total of 5952 sets of viscosity data for 433 pure liquids has been collected from DIPPR data bank and used to check the correlation ability of the new model for pure liquid viscosity. The pure liquids include almost all kinds of organic chemicals. For each pure liquid, the viscosity data are fitted by the new viscosity equation for pure liquid, eq 16 or eq 20, using F1 as the objective function, and the parameters in the ni equation for each pure liquid are determined simultaneously. The correlation results and the parameters of ni for some pure liquids are listed in Table I, where MRSD is the mean relative standard deviation and is defined by
MRsDuiscosity =
(:$(
uiscositycalviscosity,,,
)2Yi2
(30) One example of the correlation is shown in Figure 1. Similar results are shown for 433 pure liquids in Table I of the supplementary material (see paragraph at end of paper). The average MRSD for 433 the pure liquids is 0.95%. Figure 2 shows MRSD for each pure liquid and the average MRSD. The correlation results show that the new viscosity equation for pure liquids can be used to correlate viscosity data of pure liquids over a large temperature range and with very good accuracy. Note that the calculated viscosities using the parameters Aj in Table I are in units of Pa s.
Table I. Correlation Results of Viscosity for Pure Liquids. pure liquid n-pentane formaldehyde acetaldehyde n-propionaldehyde n-butyraldehyde isobutyraldehyde 1-heptanal l-hexmal cyclopropane
no. 1 2 3 4 5 6 7 8 9 10 0
1-octanal
points 12 11
19 9 8 10 12 11 12 11
temo. OC 213.10 181.10 273.20 288.50 291.50 108.10 230.10 217.10 145.60 246.00
308.10 251.10 293.50 321.00 348.00 336.10 420.10 397.10 318.30 446.00
MRSD 0.0174 0.0123 0.0143 0.0017 0.0073 0.0018 0.0107 0.0057 0.0139 0.0035
102Ao -4.388 338 -136.307 892 -80.359 711 -13.319 610 -3.130 864 -129.080 505 -92.345 510 -98.569 557 -142.490 692 -79.384 211
103A1 8.548 564 11.821 600 7.994 473 5.987 621 5.852 275 18.219 950 15.399 941 15.878 320 22.569 290 -146.229 904
lO’A2 -34.477 192 -100.072 052
WAS
-386.665 990 -278.166 290 -297.082 700 -623.873 780 2.105 100
4.035 928 2.479 976 2.727 134 8.665 390
Total number of viscosity pointa 5952. Total number of pure liquids 433. Average MRSD of viscosity correlations 0.0095.
-
0.0020
cn
2
0.0015
* + + + *MRSD of each pure liquid Average MRSD
I
A
viscosity - experimental correlation results 4 4
data
5 .* cn
0.0010
.-07 >
0.03
m
I 0.02
0.01
0.00 C
Figure 1. Correlation results of dynamic viscosity of carbon tetrachloride at different temperatures.
Figure 2. MRSD of viscosity correlations for 433 pure liquids.
Ind. Eng. Chem. Res., Vol. 32,No. 9,1993 2081 Table 11. Correlation Results of Viscosity for Binary Systems. temp,"C points MRSD rep 1 -46.1612 9 0.0011 59.2637 298.15 298.15 12,574 -183.6734 2 218.2679 7 0.0015 298.15 298.15 29,332 83.4446 3 9 0.0020 -73.2594 298.15 298.15 35,206 4 10.5426 18 0.0003 -12.5887 293.15 298.15 35,206,36,285 -93.1929 5 109.9741 3 298.15 o.ooo6 298.15 35,206 -175.1037 6 255.9483 3 0.0038 298.15 298.15 35,206 7 33.9821 -32.0643 0 . m 9 298.15 298.15 35,206 -84.7322 8 101.1825 3 0.0008 298.15 298.15 35,206 9 -128.1115 172.2859 3 0.0024 298.15 298.15 35,206 -154.5588 10 214.0421 3 0.0032 298.15 298.15 35,206 a 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. no. -
U21- 4
binary system carbon tetrachloride(1)-benene(2) carbon tetrachloride(l)-cyclohexane(2) n-octane(l)-n-nonane(2) n-octane(l)-n-decane(2) n-odane(l)-n-dodecane(2) n-octane(l)-n-hexadecane(a) n-nonane(l)-n-decane(2) n-nonane(l)-n-dodecane(2) n-nonane(l)-n-tetradecane(2) n-nonane(l)-n-hexadecane(2)
O.1°
1
U12- U22
CHLOROFORM( 1 )----TOLUENE( 2) I 0.00057
i -
* * + * * MRSD of each system
0.08
= i
Average MRSD
0.04
*
*
I I
._
E x
0.00054
* * * * experiment01 viscosity doto __ correlation results ot 298.15K
0.00053 0.00
0.20
0.40
0.60
0.80
1 0
X1
Figure 3. MRSD of viscosity correlations for 288 binary systems.
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 - Uii, 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- Uij.For each binary system, the experimental viscosity data are fitted using eqs 17 or 21,using F1 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 - Uii,for some systems are listed in Table 11. 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 viscosityof 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
Figure 4. Correlation results of dynamic viscosity of chloroformtoluene at 298.15 K. 1,2-DICHLORORETHANE( 1 ) - - - - B E N Z E N E ( 2 ) 6.5E-007
6 3E-007
experimental viscosity dota results at 303.15K
- correlation
+
+
+ +
5 5E-007
0.00
0.20
0.60
0.40
0.80
1.00
X1
Figure 5. Correlation results of kinematic viscosity of 1,2-dichloroethane-benzene at 303.15 K.
parameters of interaction potential energy, Uji - Uii, 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)a t 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
1 ,LF-DIOXANE( 1 )----ETHANOL(Z)
,
0.0015 *
+ +
+
__
,. m
-
I + + + +
MRSD of each system b
New Model
-Average MRSD of New &del MRSD of each s stem b UNIVISC - -* Average MRSD or UNIVIS6
*
experimental viscosity data correlation results
0.15
ot 288.15K
0.0013
a
2
5 .-
m :00011 .m -
0.10
H
1
*
>
.-U
E
: 0.0009 x
0
4
0 0007
0 00
0 20
0.60
0.40
0.80
Number of S y s t e m s
1.00
X1
Figure 6. Correlation results of dynamic viscosity of 1,a-dioxaneethanol at 288.15,293.15,298.15,303.15,and 308.15 K.
can be done if the parameters of interaction potential energy, Uji - Uii, are known for all the relevant binary systems (listed in Table 11). A total of 451 sets of viscosity data for 28 multicomponent systems is predicted and compared with the experimental data, which are found from the Journal of Chemical and Engineering Data. The prediction results are listed in Table 111. 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 111. The average MRSD for 20 multicomponent systems is 5.88%. Figure 7 gives MRSD for each multicomponent system and the average MRSD. Other 120sets 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
Figure 7. MRSD of viscosity predictions for 28 multicomponent systems.
used to predict viscosity data of multicomponent systems with parameters estimated from viscosity data of binary mixtures. 3. VLE Predictions for Binary Systems. The interaction potential energy parameters, Uji- Uii,obtained from viscosity data of binary systems (listed in Table 111, 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: yi
= Pyi/poixi
(31)
+
In P i = Ai + Bi/T C ln(7') + D i p i (32) where P is the pressure of the system, Pi is the saturated vapor pressure of pure liquid i, yi is the mole fraction of component i in the vapor phase, and the parameters, Ai, Bi, Ci, Di, and Ei 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 - Uii, are obtained from the viscosity correlations of the binary systems (listed in Table 11).The viscosity correlation results and VLE prediction results of the 134binary systems are listed in Table IV, where MRSD
Table 111. Prediction Results of Viscosity for Multicomponent Systems*b no. 1 2 3 4 5 6 7 8 9 10
multicomponent system 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)-methano1(3) acetone(l)-n-hexane(2)-cyclohexane(3) n-hexane(l)-cyclohexane(2)-ethanol(3) methanol(l)-ethanol(2)-isopropyl dcohol(3) acetone(l)-ethanol(2)-isopropyl dcohol(3)
points 24 15 14 12 12 12 12 12 12 12
temp, OC 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15
MRSD New Model UNIVISC 0.0234 0.0316 0.0995 0.0187 0.0378 0.0038 0.0758 0.0325 0.0864 0.0336 0.0789 0.0894 0.0495 0.0173 0.0352 0.0175 0.0106 0.0166 0.0398 0.0565
reF 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 0.0330. 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. Journal of Chemical Engineering Data, volume and page.
Ind. Eng. Chem. Res., Vol. 32,No. 9,1993 2083 Table IV. Prediction Results of VLE from Viscosity for Binary Systems-0 MRSD viscosity UIZ-UB points VLEP Y
uzl-Ull
binary system no. 1 carbon tetrachloride(l)-benzene(2)
59.2648
-183.6727
218.2664
-73.3940
62.4229
-9.5561
8.9153
-120.6709
151.4583
6
ethyl acetate(1)-carbon tetrachloride(2) -124.1177
141.4610
7
n-hexane(l)-n-octane(2)
2
carbon tetrachloride(l)-cyclohexane(2)
3
benzene(l)-ethyl acetate(2)
4
n-heptane(l)-n-octane(2)
5 n-hexane(l)-n-decane(2)
67.9488 -62.2806
8 n-hexane(l)-n-dodecane(2)
-157.1308
219.9935
p-xylene(l)-m-xylene(2)
-3.6407
3.8577
10 p-xylene(l)-m-xylene(2)
-3.6407
3.8577
a
pressure, Pa ~~
-46.1620
9
temp, O C
9 68 7 41 14 59 18 12 18 19 19 52 18 11 3 12 36 9 36 9
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
0.0040 0.0089 0.0026 0.0096
0.0165 0.0132
0.0001
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
298.15 353.05 24 767.27 101 577.70 298.15 353.75 6 220.82 101 557.70 318.15 353.16 36879.93 102030.70 298.15 328.15 9743.80 21 811.50 298.15 2 827.77 102 384.20 417.95 318.15 349.71 89 833.45 101 325.00 298.15 328.15 10 839.10 61 594.98 298.15 308.15 2 209.15 28 071.11 303.15 412.18 101276.60 101325.00 303.15 412.17 101 231.60 101 325.00
Viscosity: total number of viscosity pointa 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 syeetms 131; average MRSD of pressure predictions 0.1027.
VLE(mo1ar fraction): total number of molar fraction points 3569 total number of molar fraction systems 114; average MRSD of molar fraction predictions 0.0452. 1.00
* * * + * MRSD of each system
0.08
0.80
Average MRSD
-
".r"
0.00 1.8 0.00
+
'+l' *' +*.*T' *-;*-*++ * ,r5
I # - ,
4
*#0*
I
I
I
I
I
50.00
.
I * & !
-* * I
I
*+' !
I
,
+ .
*+*+ I
I
f f
I
I
?'I*#
I
i
MRSO of each system
+ t i + +
-Average
+
++
100.00
,
I
150.00
0.00 0.00
is defined by
More detailed information is given in the supplementary material. The viscosity correlation results are shown in Figure 8. The average MRSD of the correlations for 1991 viscosity data of 134binary systems is 0.865%. The VLE prediction results are shown in Figure 9 for pressure and Figure 10 for mole fraction in vapor phase. The average MRSD of the predictions for 3988pressure data of 131binary systems is 10.27% and for 3569 mole fraction data of 114 binary systems is 0.0452. For most systems, the VLE predictions are qualitatively good, the MRSD for pressure is around 2% , and the MRSD for composition in vapor phase is below 0.02.
r h +
50.00
100.00
15 00
Number of the Systems
N u m b e r of the Systems Figure 8. MRSD of viscosity correlations for 134 binary systems.
MRSD
Figure 9. MRSD of VLE predictions (pressure) for 131 binary systems.
Parameters for Liquid Mixtures from VLE Data of Binary Systems 1. VLE Correlations and Predictions for Liquid Mixtures. The parameters Uji - Uii can also be determined from phase equilibrium or/and excessenthalpy data via the Gibbs-Helmholtz equation based on the eqs 23 and 31. Since the expressions for activity coefficients are the same as those for UNIQUAC, the accuracies of the correlations and the predictions of the above properties with the model are as same as the accuracies of UNIQUAC. Here the parameters Uji - Uii are determined from VLE data for binary systems. For each binary system, the experimental VLE data are fitted using the following objective function:
where u p is the standard deviation of the correlation for pressure ( u p = 10 Pa) and uy is that for composition in
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
2084
f
+ + 4 +
0 40
*
MRSD of each system Average MRSO
+
+
+
MRSO of each system Average MRSO
I I
020/
.
* +
++
Number of the Systems
Number of the Systems
Figure 10. MRSD of VLE predictions (molar fraction in vapor phase) for 114 binary systems.
Figure 11. MRSD of VLE correlations (pressure) for 131 binary systems. 0.10
vapor phase (cy= 0.001). The experimental VLE data of 134 binary systems are found from Dortmund data bank and correlated for obtaining the parameters of interaction potential energy, Uji - Uii of the binary systems. Correlation results are listed in Table V. More detailed results are given in the Supplement. The MRSD for each system and the average MRSD are shown in Figure 11for pressure and Figure 12 for mole fraction in the vapor phase. The average MRSD of the correlations for 3988 pressure data of 131 binary systems is 1.03% and for 3569 mole fraction data of 114 binary systems is 0.0123. 2. Viscosity Predictions for Binary Systems. If the parameters Uji - Uii,are obtained from VLE data, the viscosity predictions can be carried out. Here 1991 viscosity data of 134 binary systems have been predicted using the parameters Uji - Uii obtained from VLE data and listed in Table V (see also the supplementary material). The prediction results are compared with the experimental viscosity data which are found from the Journal of Chemical and Engineering Data. The prediction results are listed in Table V. The MRSD of the predictions for each binary system and the average MRSD are shown in
*
+
+
+
MRSO of each system MRSD
- Average
-1
0 00 0.00
50.00
100.00
15( 00
Number of the Systems Figure 12. MRSD of VLE correlations (molar fraction in vapor phase) for 114 binary systems.
Figure 13. The average MRSD of the predictions is 22.64% and the predictions are thus relatively poor.
Table V. Prediction Results of Viscosity from VLE for Binary SystemseC
no. binary system 1 carbon tetrachloride(l)-benzene(2)
Uzl- U11 243.3878
2
carbon tetrachloride(1)-cyclohexane(2)
-230.2583
3
benzene( 1)-ethyl acetate(2)
-62.8271
4
n-heptane(l)-n-octane(2)
-234.0954
5
n-hexane(l)-n-decane(2)
-244.6624
6
ethyl acetate(1)-carbon tetrachloride(2)
126.9049
7
n-hexane(l)-n-octane(2)
326.6270
8
n-hexane-n-dodecane(2)
250.0960
9 p-xylene(l)-m-xylene(2)
-3.5910
10 p-xylene(l)-m-xylene(2)
-26.6547
MRSD viscosity U ~ ZUZZ points VLE P y 9 0.0552 -201.2416 68 0.0065 0.0042 7 0.0307 302.1576 41 0.0090 0.0080 14 0.0045 47.9581 59 0.0048 0.0021 18 0.1773 310.9013 12 0.0023 0.0025 18 0.1912 317.3956 19 0.0174 -52.3391 19 0.1295 52 0.0058 0.0033 -269.8750 18 0.2639 11 0.0053 0.0050 3 0.0198 -194.3233 12 0.0076 3.8064 36 0.0017 9 0.0004 0.0001 36 0.0019 28.1410 9 0.0004
temp, OC 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
298.15 353.05 24697.00 101547.40 298.15 353.75 6220.82 101 688.50 318.15 353.16 36814.51 101775.10 298.15 328.15 9938.33 21 811.50 298.15 417.95 2 702.19 102 576.50 318.15 349.71 95 455.51 101 872.50 298.15 328.15 10839.10 61 526.77 298.15 308.15 2 209.15 27 826.30 303.15 412.18 101276.50 101325.00 303.15 412.17 101 259.90 101 377.20
Viscosity: total number of viscosity points 1991; total number of viscosity systems 134; average MRSD of viscosity predictions 0.2264. total number of pressure points 3988; total number of pressure systems 131; average MRSD of pressure correlations 0.0103. c VLE(mo1ar fraction): total number of molar fraction points 3569; total number of molar fraction systems 114;average MRSD of molar fraction correlations 0.0097. a
* VLE(pressure):
Ind. Eng. Chem. Res., Vol. 32, NO. 9,1993 2085
+ + + + *
1.50
,
0.20
2.00
-
1
MRSD of each system MRSD
MRSD of each system MRSD
*+**I
-Average
-Average
30.15
.-m 0 0
0 w 1.00 2 "I
.-m
' ;
1 1
*'
0.10
I
+
It
0
*
n
v, 0.05
+
*
*.
%
*
*
+
*+
* *
5 *
+
' i t , .
+
,+
+ ,*
%+* '
I
Figure 13. MRSD of viscosity predictions for 134 binary systems.
+
h'*
:*:@+%+ *"* +
*+I* + + + t
o,oo
,
+
I
*+
,
I
'
* +*
+
't
4
+*
"t T ' I
, V I I ! I
+4$
.
t.
f
f
'+
f
***+ '
,
'
**
.A+
,
3
'
Figure 14. MRSD of viscosity + VLE correlations (viscosity) for 134 binary systems.
Parameters for Liquid Mixtures from Both Viscosity and VLE Data of Binary Systems
0.10
As already stated, it is possible to determine the parameters Uji - Uii from both viscosity data and phase equilibrium data of liquid mixture simultaneously. For each binary system, the experimental viscosity data and VLE data are fitted by eqs 17 and 23 using F I +FZas the objective function. The experimental viscosity data of all binary systems are found from the Journal of Chemical and Engineering Data and the VLE data are from Dortmund data bank. Correlations for 134binary systems have been carried out and the results are listed in Table VI (the detailed results are given in the supplementary material). The MRSD for each binary system and the average MRSD are shown in Figures 14for viscosity,Figure 15 for pressure, and Figure 16 for mole fraction in the vapor phase. The average MRSD for 1991 viscosity data of 134 binary systems is 2.77%, for 3988 pressure data of 131 binary systems is 1.96%, and for 3569 composition data of 114 binary systems is 0.0123. One example of the correlations for carbon tetrachloride + bromobenzene binary systems is shown in Figures 17-19. Viscosity data at two temperatures and VLE data at five temperatures
I + + + +
MRSD of each system MRSD
- Average
Ln
'c
0
n
fn [r
I
0.00 0.00
100.00
50.00
151 00
Number of the Systems Figure 15. MRSD of viscosity 131 binary systems.
+ VLE correlations (pressure) for
are correlated together by one pair of the parameters of interaction potential energy Uji - Uii, which are temper-
Table VI. Correlation Results of Both Viscosity and VLE for Binary Systemsea
no. binary system 1 carbon tetrachloride( l)-benzene(2)
U21- UII -31.2242
2
carbon tetrachloride(l)-cyclohexane(2)
139.6494
3
benzene(1)-ethyl acetate(2)
-49.3151
4
n-heptane(l)-n-octane(2)
-92.3581
5
n-hexane(l)-n-decane(2)
6
ethyl acetate(1)-carbon tetrachloride(2)
7
n-hexane(l)-n-octane(2)
8
n-hexane(l)-n-dodecane(2)
-131.5515 275.8278 5.3309 -128.7926
9 p-xylene(l)-rn-xylene(2)
-15.6335
10 p-xylene(l)-rn-xylene(2)
-15.6335
MRSD viscosity U12- UZZ points VLEP y 39.2353 9 0.0013 68 0.0085 0.0035 -106.9238 7 0.0046 41 0.0181 0.0069 33.7419 14 0.0036 59 0.0048 0.0020 114.1249 18 0.0027 12 0.0109 0.0028 172.0565 18 0.0038 19 0.0359 -173.4638 19 0.0251 52 0.0069 0.0029 18 0.0031 -7.7631 11 0.0173 0.0102 157.4837 3 0.0064 12 0.0124 16.4779 36 0.0017 9 O.oo00 O.oo00 16.4779 36 0.0017 9 0.0003
temp, O C 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
298.15 353.05 24738.68 101411.80 298.15 353.75 6 220.82 101 743.30 318.15 353.16 36826.65 101 793.70 298.15 328.15 9923.50 21 811.55 298.15 2 827.77 102 507.90 417.95 318.15 349.71 95184.88 102071.10 298.15 328.15 10 839.11 61 575.81 298.15 308.15 2 154.01 27 896.96 303.15 412.18 101300.00 101342.40 303.15 412.17 101 247.40 101 342.40
*
Viscosity: total number of viscosity points 1991;total number of viscosity systems 134;average MRSD of viscosity0.0277. VLE(pressure): total number of pressure points 3988; total number of pressure systems 131; average MRSD of pressure 0.0196. VLE(mo1ar fraction): total number of molar fraction points 3569; total number of molar fraction systems 114; average MRSD of molar fraction 0.0123.
2086 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 .
CARBON TETRACHLORIDE( 1)-
- - -BROMOBENZENE(2)
^ ^
1 .vu
* * * * * MRSD of each system Average MRSD
5
I
L
0.06
4
0.80
-
0.04
v-
x
0 ,
f
I
r+*
/#/
4 0.60
*
+ +
*
+
experimental data results at 313.15K 323.15K 333.15K 343.15K 353.15K
- calculation
.oo 0.00
50.00
100.00
150.00
Number of the Systems
,
CARBON TETRACHLORIOE(l)----BROMOBENZENE(2)
0.001 1
0.40 0.00
I ' " ~ ' ' ' I' I ~ I ,
I
Figure 16. MRSD of viscosity + VLE correlations (molar fraction in vapor phase) for 114 binary systems. I
0.20
0.60
0.40
/ ' ' , '
0.dO
1. 0
X1
Figure 19. Viscosity + VLE correlation results (molar fraction in vapor phase) of carbon tetrachloride-bromobenzeneat 313.15,323.15, 333.15, 343.15,and 353.15 K.
Conclusions
a" 0.0010
5 .; + m
fi- 0.0009
v) ._
at 298.15K and 318.15K
>
.-0
E 0.0008 5 C
x
n
0.0007 ! a 8 3 t 0.00
I I I b I , I
r I r I r r I , I I ~ I I I I , I
I
0.20
0.40
0.60
1.00
0.80
X1
Figure 17. Viscosity + VLE correlation results (dynamic viscosity) of carbon tetrachloride-bromobnzene at 298.15 and 318.15 K. CARBON TETRACHLORIDE( l)----BROMOBENZENE(2)
looooo
*
experimental data
__ calculation results at 313.15K 323.15K 333.15K
80000
5
a
\ 60000
A new statistical thermodynamic model has been proposed and checked. A viscosity equation for pure liquids, a viscosity equation for liquid mixtures, and an equation for activity coefficients are obtained from the model and expressed using the same model parameters. The new viscosity equations of pure liquid has been used to correlate experimental viscosity data with excellent results. The viscosity equation for liquid mixture has been used to correlate experimental viscosity data for binary systems in order to obtain the parameters of interaction potential energy. The correlation results are very good. The parameters can be used (1)to predict the viscosity data of multicomponent systems with good results and (2) to predict VLE data of binary systems with qualitatively good results. The activity coefficient equation for liquid mixture has been used to correlate and predict phase equilibrium data of liquid mixtures. Here the equation has been used to correlate experimental VLE data of binary systems in order to obtain parameters for the interaction potential energy. Then viscosity predictions of binary systems have been performed. The prediction results are relatively poor. The viscosity and activity coefficientequations for liquid mixture have been used together to correlate experimental viscosity data and VLE data of binary systems. The correlation results for both viscosity and VLE are very good.
Nomenclature
0 0.00
I I I I I I I I I I I I I I I I I I I I I I
0.20
I I I I I I I , I I I I I I I I
0.60
0.40
I
(
I
0.80
I
I
7
-
m
1.00
X1
Figure 18. Viscosity + VLE correlation results (pressure) of carbon tetrachloride-bromobenzene at 313.15, 323.15, 333.15, 343.15, and 353.15 K.
ature independent. Figure 17 shows the viscosity results, Figure 18 shows the pressure results and Figure 19 shows the mole fraction results of the correlations. The correlation results for both viscosity and VLE are very good.
A j = constant in ni equation for pure liquid i f = shear force F = objective function GE = excess Gibbs energy of the mixture h = Planck's constant
Ava,,Hm(i)= molar heat of vaporization of the pure liquid i k = Boltzmann's constant m = number of experimental data M = molecular weight of the mixture Mi = molecular weight of component i m, = average segment weight of molecule i
Ind. Eng.Chem. Res., Vol. 32, No. 9,1993 2087
MRSD = mean relative standard deviation n = number of components ni = proportional constant of segment i Ni = number of molecules i P = pressure of system Poi = saturate vapor pressure of liquid i qi = area parameter of molecule i ri = number of segments in a molecule i P = average segment number of a molecule R = gas constant T = temperature Tci= critical temperature of component i Tri = reduced temperature of pure liquid i U, = activation energy barrier Uai = activation energy barrier of segments i Uji = interaction potential energy between sites j and i Uoi = potential energy of lattice i u = velocity u = volume of a molecule V = molar volume of the mixture Vi = molar volume of component i V f= free volume xi molar fraction of component i in liquid phase yi = molar fraction of component i in vapor phase z = coordination number of the lattice is
Greek Symbols activity coefficient of component i t, = work done by the shear force q = dynamic viscosity of the mixture qi = dynamic viscosity of component i tli average area fraction of component i Oji local composition v = kinematic viscosity of the mixture vi = kinematic viscosity of component i u = standard deviation 7 j i = interaction parameter between sites j and i di = average segment fraction of component i Q = partition function Q, = partition function at activation state wi = number of distinguishable configurations yi
I
I
I
Superscript s = Staverman’s model Subscripts a = activation state cal = calculated data exp = experimental data t = component i k = kinetic energy p = potential energy P = pressure
r = segment y = molar fraction in vapor phase SupplementaryMaterial Available: List of detailed information corresponding to the examples given in Tables I-VI (19 pages). Ordering information is given on any 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. 1976, 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. Znd. Eng. Chem. 1991,30, 16661669. Cao, W.; Fredenslund, A.; Rasmussen,P. Statistical Thermodynamic Model for Viscosity of Pure Liquids and Liquid Mixtures. ZEC Res. 1992,31, 2603-2619. Cukor, P. M.; Prausnitz, J. M. Proceedings of the Znternutionul Symposium on Distillation; Inet. Chem. Eng.: London, 1969;Vol. 3, P 88. Daubert, T. E; Danner, R. P. Physical and Thermodynumic Properties of Pure Chemicals: Data Compilation; Hemisphere Publishing Corp.: New York, 1989. Dizechi, M.; Marschall, E. Correlation for Viscosity Data of Liquid Mixtures. Znd. Eng. Chem. Process Des. Dev. 1982,21, 282-289. Fredenslund, A,; Jones, R. L.; Prausnitz, J. M. Group-Contribution Estimation of Activity Coefficienta in Nonideal Liquid Mixtures. AIChE J. 1975,21, 1086-1099. Glasstone. S.: Laidler. K. J.: Evrine. H. The Theow of Rate Process: McGraw-Hilk New York, i94c’Chapter 9, p 477. Gmehling,J.; Arlt, W.; Onken,U. DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1977; Vol. I. Hansen, H. K.; Rasmussen, P.; Fredenalund, A. Vapor-Liquid Equilibria by UNIFAC Group Contribution. 5. Revision and Extension. ZEC 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. AZChE J. 1960, 6,427-431. Renon, H.; Prauanitz, 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 Dependenceof UNIQUAC/UNIFAC Models. Chem. Eng.‘Sci. 1980, %, 2389-2403. Wei. I. C.: Rowlev. R. L. A Local ComDosition Model for MulticomponentLiqGd Mixtures Shear Viscbeity. Chem.Eng. Sci. 1985, 40, 401-408. Wilson, G. M. A New Expression for the Excess Gibba 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.
Receiued for review November 17, 1992 Revised manuscript received May 12, 1993 Accepted May 18, 1993. Abstract published in Aduance ACS Abstracts, August 15, 1993. @
