Ind. Eng. Chem. Fundam. 1981, 20, 77-81
77
Generalized Corresponding States Method for the Viscosities of Liquid Mixtures A. S. Teja' and P. Rice Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire LE 1 1 3TU, England
The three-parameter corresponding states principle (CSP) has been used with considerable success for the correlation and predition of the thermodynamic and transport properties of fluids and fluid mixtures. All published methods, however, are based on the properties of a spherical reference fluid such as argon or methane. An extension of the principle using two nonspherical reference fluids has recently been proposed for thermodynamic properties. This work uses a similar extension for the viscosity of liquids and liquid mixtures. For binary mixtures, the two pure components are used as the reference fluids and the van der Waals one-fluid model is used to extend the pure-component relationships to mixtures. A single binary interaction coefficient is required for the mixture calculations. Over the range of conditions studied, this coefficient is independent of temperature and composition. The average absolute deviations between calculated and experimental viscosities ranged from 0.7 % for nonpolar nonpolar mixtures to 2.5% for systems containing polar components and 9.2% for systems containing water.
+
Introduction The viscosities of liquid mixtures are required in many chemical engineering calculations involving heat transfer, mass transfer, and fluid flow. However, the theoretical derivation of the viscosity has only been possible for certain special cases. A method for the prediction of this property is therefore of great theoretical and practical interest. Continued interest in the viscosity of liquid mixtures is shown by the large number of investigations carried out since the work of Arhhenius in 1877 (see, for example, the compilation of Irving, 1977a) as well as by the appearance of a t least 50 empirical or semiempirical equations to describe such mixtures which have appeared in the literature. (These equations have been reviewed by Irving (1977b) and Reid et al. (1977).) In contrast, although similar theoretical problems are encountered in the correlation and prediction of thermodynamic properties, considerable success has been achieved in this area by the application of the three-parameter corresponding states principle of Pitzer et al. (1955). The original work of Pitzer et al. and the many later extensions (in particular, the analytical representation by Lee and Kesler, 1975) are based on the use of two reference fluids, one spherical and the other nonspherical, and the use of a Talyor series expansion of a thermodynamic property about its value for a spherical reference fluid. Pitzer's acentric factor is then used for a linear interpolation/extrapolation of the reference fluid properties. An extension of the three-parameter corresponding states principle using two real nonspherical reference fluids has recently been proposed (Teja, 1980; Teja and Sandler, 1980; Teja et al., 1980). If the reference fluids chosen are similar to the pure component of interest or, in the case of mixtures, to the key components of interest, very accurate predictions of thermodynamic properties can be made. In this work, we extend the generalized corresponding states principle (GCSP) presented previously for thermodynamic properties to the viscosity of liquid mixtures. The method is similar in principle to the procedures used by Hanley (1976) and Mo and Gubbins (1976).
Generalized Corresponding States Principle (GCSP) A pure fluid (with critical parameters T,,P,, V,, and molecular weight M) is defined to be in corresponding states with a reference fluid, 0,if the compressibility 2 and the reduced viscosity ( q f ) of the two substances a t t h e same reduced temperature TRand reduced pressure PR are given by (1)
(75) = (t15)'"'
(2)
where
Vc2/3Tc-1/2M-1/2
(3) Equations 1and 2 are strictly valid only for pairs of substances (such as the noble gases) in which the molecules interact with spherically symmetric two-parameter potentials. The resulting statements then describe the two-parameter corresponding states principle and the superscript o denotes the properties of a spherical reference substance. In the more general case of nonspherical molecules, Pitzer et al. (1955) showed that eq 1 can be written as a Taylor series expansion in the acentric factor f
=
z = Z(0) +
(4)
where Z ( O ) is the compressibility of a simple fluid with zero acentric factor (i.e., a spherical reference substance) and 2'l)is a complicated deviation function. Letsou and Stiel (1973) later extended this approach to viscosities of liquids by rewritting eq 2 in the form In (75) = In (75)(0)+ w In ( q f P (5) More recently, Lee and Kesler (1975) provided an analytical framework for the three-parameter corresponding states principle by writing eq 4 as
where the compressibility of any fluid of acentric factor w is expressed in terms of a simple fluid (analytical) equation of state and a (heavy) reference fluid equation
*School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Ga. 30332. 0196-4313/81/ 1020-0077$0 1.OO/O
z = z(0) and
0
1981 American Chemical Society
78
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981
Table I. Comparison of Experimental and Calculated Viscosities. AAD % = 100 lqcdc._ . . .
___.____
benzene
Nonpolar
s .ox
+ ethylbenzene
1.00
t
+ ethylbenzene
0.93
benzene t carbon tetrachloride
0.99
benzene
+
+
0.98
+ cyclohexane
benzene
T, K
ii
I _
n-dodecane
n-hexane toluene
rc
system
___-
..
chloroform
0.99
-
Nonpolar 298.15 313.15 328.15 253.15 263.15 273.15 283.15 293.15 303.15 313.15 323.15 333.15 343.15 353.15 298.15 318.15 328.15 283.15 297.15 298.15 313.15 293.15 303.15 313.15 285.15 288.15 293.15 303.15 313.15
no. of data
9 9 9 4 4 4 4 4 4
4 4 4 4 4 11
11 11 9 9 9 9 5 5 5 5 6 6 6 6
7jexp1/qeXp
AAD %
max dev, %
0.80 0.62 1.12 0.27 0.97 1.08 0.80 0.19 0.49 0.70 1.17 1.35 1.42 1.56 0.29 0.40 0.85 2.03 0.74 1..32 2.23 0.35 0.51 1.31 1.88 0.86 0.92 0.98 0.71
--2.52 -1.45 2.01 0.66 -1.22 --1.18 -1.01 0.49 1.01 1.43 -1.58 1.69 2.09 2.03 -0.75 1.58 2.21 4.02 1.47 2.48 3.79 -0.84 1.27 1.90 2.43 -1.29 -1.49 -2.29 -1.19
2.81 2.24 2.47 2.21 0.62 0.78 0.85 0.56 2.51 0.67 1.94 2.41 2.70 2.63 3.73 3.32 3.54 2.83 2.83 5.52 4.55 4.17 4.43 3.78 3.77 3.00 2.48 3.25 3.34 2.76 2.85 2.70 1.44 0.61 0.48 0.88 3.17 3.26 3.15 2.85 2.80 2.46 2.51 3.64
-3.72 -3.96 -5.32 -4.93 -1.28 -2.37 1.80 1.39 4.96 1.49 3.08 -5.08 -6.01 -5.69 5.88 5.15 - 6.06 -5.74 6.21 -9.49 -6.47 -5.39 -6.45 -5.96 -6.38 -4.61 5.02 6.43 -5.82 4.58 5.36 6.02 -2.76 - 2.05 1.63 1.75 5.18 5.02 -4.46 - 4.06 --4.46 -4.70 - 7.36
overall AAD for nonpolar-nonpolar mixtures = 0.69% benzene
+ methanol
Nonpolar 1.00
benzene
+ acetone
0.99
benzene
+ aniline
1.02
1.2-dichloroethane -methanol i
0.97
1,2-dichloroethane + ethanol
0.90
1,2-dichloroethane + 1-propanol
0.89
1.2-dichloroethane + 1-butanol
0.92
1,2-dichloroet,hane t 7 -pentan01
0.94
1,&dioxane + methanol
0.95
toluene
+ methanol
trichloroethylene
0.99
+ methanol
oG6rall AAD for nonpolar
1.00
+ polar mixtures = 2.54%
+ Polar 293.15 303.15 313.15 323.15 293.15 298.15 310.95 323.25 303.15 313.15 323.15 303.15 313.15 323.15 333.15 303.15 313.15 323.15 333.15 303.15 313.15 323.15 333.15 303.15 313.15 323.15 333.15 303.15 313.15 323.15 333.15 283.15 293.15 303.15 313.15 323.15 293.15 298.15 311.15 288.15 293.15 303.15 313.15 .___ 323.15
6 6 6 6 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 6 6 6 6 7 n
7 7 8 8 8 8 9 9 9 9 9 9 9 9 9
so
10 10
10
-6.9.9
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981 79 Table I (Continued) system
LZ
acetic acid
+ methyl propyl ketone
Polar 1.02
acetic acid
+ methyl ethyl ketone
1.02
Aqueous 298.15 303.15 308.15 313.15 323.15 1.36 288.15 298.15 308.15 1.37 298.15 303.15 308.15 313.15 1.34 293.15 298.15 303.15 1.37 293.15 298.15 303.15 313.15 333.15 353.15 1.36 283.15 288.15 293.15 298.15 303.15 308.15 328.15
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
4.43 2.19 1.65 1.85 3.97 15.58 11.87 10.39 14.91 13.25 12.71 12.32 22.79 21.10 19.58 9.78 8.27 7.03 6.27 5.71 6.14 7.62 6.49 6.35 8.37 6.03 5.53 5.93
9.08 5.59 3.21 -4.78 - 8.81 35.16 23.90 20.00 26.20 23.40 19.88 16.92 44.29 39.10 35.00 -20.22 -18.76 -19.90 -19.50 -17.50 -17.60 30.47 27.02 24.69 21.74 17.81 17.70 10.85
+ methanol
1.00
acetone
+ acetic acid
1.06
acetone
+ phenol
1.14
+ ethanol
water
+ 1-propanol
water
+
2-propanol
water
+
1,4-dioxane
water
+ acetic acid
+ Polar -0.74 1.02 -1.61 1.90 1.73 -0.76 1.16 1.33 -1.96 4.70 3.76 -2.92 -4.70 2.80 2.66 -1.57 -1.17 15.80 -8.90 10.70 13.8 17.0
acetone
water
max dev, %
0.30 0.52 0.85 0.81 0.69 0.39 0.63 0.73 0.81 2.03 2.00 2.07 2.00 0.78 1.07 0.64 0.45 6.43 3.61 5.49 7.37 10.29
1.09
+ methanol
AAD %
9 9 9 9 9 9 9 9 9 7 7 7 7 9 9 9 9 9 9 9 9 9
+ n-butyric acid
water
no. of data
298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 293.15 303.15 313.15 323.15 293.15 298.15 310.95 323.25 283.15 293.15 302.95 313.25 322.95
acetone
overall ADD % for polar
T,K
+ polar mixtures = 2.06% 1.34
overall ADD for aqueous mixutres = 9.22% average for 1010 data points = 3.77%
of state .@I). Lee and Kesler, however, retained Pitzer's original proposal of a Taylor series expansion of a thermodynamic property about that property of a simple spherical reference fluid. We have recently proposed a generalized corresponding states principle (GCSP) for thermodynamic properties which no longer retains the simple spherical fluid as one of the references. Equation 6 is written as
then eq 7 of course, reduces to eq 6. Equation 7 provides a method for generalizing equations of state using the known equations of two pure components. Thermodynamic properties can then be predicted with considerable success, as has been shown elsewhere (Teja, 1980; Teja and Sandler, 1980). In an analogous manner, we extend eq 5 to viscosities in this work as follows In (at) = In ( q t ) ( ~+ l)
- ,(rl) ,(rZ)
-
,(Ill
[In (q[)(r2) - In (qt)(rl)] (8)
where the superscripts r l and r2 refer to two (nonspherical) reference fluids which are chosen so that they are similar to the pure component of interest or, in the case of mixtures, to the key components of interest. If one of the reference fluids is a simple fluid of zero acentric factor,
where the superscripts (rl) and (r2) again refer to two (nonspherical) reference fluids. Equation 8 can be extended to mixtures using the van der Waals one-fluid model to replace T,, V,, w , and M of a pure fluid by the pseudocritical properties T,,, V,, w,,
80
Ind. Eng. Chem. Fundam., Vol. 20, No. 1, 1981
and M , of a hypothetical equivalent substance as follows
TcmV c m = C C X i X j Tcij Vctj Vcm
=
CxLxI Vcij
(9) (10)
I 1
w, = ,&w,
(11)
M , = CX,M,
(12)
Equations 9-11 have been used successfully for thermodynamic properties (see, for example, Teja, 1978). Equation 12 arises naturally in thermodynamic calculations and although Mo and Gubbins (1976) suggested a more complicated relationship using the Enskog dense gas theory, we have preferred to retain the simpler relationship given by eq 12. We have found that the simpler relationship works better in many cases. The one-fluid model can be used to obtain the properties of mixtures provided values can be assigned to the crossparameters Tc, and Veil.(i # 1). For nonpolar substances, the most successful mixing rules are (Reid and Leland, 1960; Teja, 1978)
Tc,,V,,, = ~l~(TcrrVcllTc~jV,,)'~2; 1 # j Vcr,= (VC,.l'J+
V$3)3
+8
(13)
(14)
where +L,is a binary interaction coefficient which must be calculated from experimental data. A second binary interaction coefficient is often inserted in eq 14. However, one of the advantages of using appropriate reference substances (and essentially using small perturbations) is that a single binary interaction coefficient is sufficient to characterize each binary mixture. In principle, no additional coefficients are required to predict the properties of ternary and higher mixtures. Reference Fluid Calculations The bulk of the viscosity measurements of liquid mixtures are at atmospheric pressure, or in terms of the equations above, at low reduced pressures. In this region, pressure has little effect on liquid viscosity, and in applying eq 8, we have chosen to ignore the effects of pressure altogether. Thus all evaluations are carried out at the same reduced temperature and we require the reference fluid viscosities as functions of the reduced temperature only. (It should be added here that in the more general case, we would require equations for the reference fluid viscosities as functions of reduced pressure as well as reduced temperature. One consequence of this is that an additional mixing rule for Z,, = C1x,Z,would also be required in the one-fluid model. This is not. however, a difficulty in principle.) We have chosen to correlate the properties of our reference substances as functions of the reduced temperature by means of the Andrade equation In ( q t ) = A + B T f l (15) The choice of the reference equation is a matter of accuracy and convenience. We feel that the Andrade equation is one of the best two constant equations for the temperature variation of the viscosity of liquid mixtures and, certainly, we found that it fitted the data to within the accuracy of the experimental measurements (for the range of temperatures studied). The choice of reference substances is also arbitrary. For each binary mixture studied, we chose the two pure components as our reference fluids. This ensures that a good fit is obtained at the pure component end of the viscosity
vs. composition relationship and, since the pure component viscosities were available, the constants A , B in eq 15 could easily be calculated by least-squares methods. Experimental data for the pure component references were taken from the compilation by Irving (1977a). Should experimental viscosities of any pure fluid not be available, then eq 8 and the properties of two similar fluids can easily be used to predict the (unknown) pure component data. Method of Calculation Given T,, V,, M , and w for each component and the constants of eq 15 plus the acentric factor for each reference fluid, the calculation of the viscosity of any mixture at a given temperature T and composition x i may be carried out as follows. (i) Calculate the pseudocritical quantities T,,, V,,, M,, and w, using eq 9-12, 13, and 14. The value of the binary interaction coefficient #;. may be set initially equal to 1.0. (ii) Calculate T~ = T/+~,,,. (iii) Calculate the quantities In (q,$*') and In (T&(") at the given T , using eq 15 and the appropriate constants for each reference fluid. (iv) Calculate the quantity In (74) for the given mixture using eq 8. (v) For the given mixture 5 = Mm-1/2Tcm-'i2Vc,2/3, and since all these quantities are known, the viscosity 7 of the mixture can be calculated. (vi) If some experimental data are available, the quantity can be varied until the difference between calculated and experimental viscosities is minimized. It will be shown below that, to a good approximation, +i, is independent of temperature and composition. It should be noted that none of the evaluations require any iterations, so that the whole procedure may be carried out on a simple hand calculator. Results a n d Discussions Irving (1977a) has compiled viscosity data for a large number of binary liquid mixtures. Althouth there is disagreement of as much as 15% between various investigators for certain mixtures, we selected 123 binary mixtures for study (in most cases, where a single investigator had measured viscosities at three or more different temperatures). The range of mixtures selected is, we feel, representive of systems with nonpolar + nonpolar, nonpolar + polar, polar + polar and water + other interactions. Aqueous systems were treated separately because such mixtures usually exhibit large deviations from ideal behavior and because water is strongly associated in solution. In all, a total of 1010 viscosity data points were tested. The results of our calculations are shown in Table I. The single adjustable constant + , . used in our calculations was, to a good approximation, inJeependent of temperature and composition. We therefore evaluated this constant using the data at the lowest temperature for each binary pair. As can be seen from the results, extrapolation over a 100° range of temperature did not lead to any significant deterioration in accuracy. As expected, the method works best for nonpolar nonpolar mixtures. The errors increase from 0.7% for these systems to about 2.5% for systems containing polar components and to about 9% for aqueous mixtures. Irving (197713) also compared 25 published equations for the estimation of mixture viscosities and concluded that the Grunberg and Nissan (1949) equation with one temperature dependent constant per mixture was the best. The Grunberg-Nissan equation may be written as In q = x1 In q l + x 2 In q2 + 2x1x2G (16) or in terms of volume fractions In = 6 In v1 + 42 In 772 + 24142G' (17)
+
The constants G and G'were optimized for each data set
Ind. Eng. Chem. Fundam., Vol. 20, No.
1, 1981 81
Table 11. Comparison with Grunberg-Nissan Equations rms error, eq 16
type of system nonpolar t nonpolar nonpolar + polar polar + polar aqueous a
.5
1.0
X
Figure 1. The viscosity vs. composition relationship of n-hexane + n-dodecane mixtures. The full lines are the predicted curves for qI2 = 1.08. The experimental points are denoted by A at 298.15 K and A at 328.15 K; x is the mole fraction of n-dodecane. I
I
2
rms error = (100/n)
%a
eq 1 7 thiswork
2.3 3.0 8.9
1.9 2.9 10.3
24.0
17.4
1.0 2.9 2.5 12.0
( v c d c- qexp)2/qex
