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Prediction of the McAllister Model Parameters by Using the Group-Contribution Method: n-Alkane Liquid Systems Abdulghanni H. Nhaesi, Walid Al-Gherwi, and Abdul-Fattah A. Asfour* Environmental Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4
The McAllister model is considered to be the best correlating technique for viscosity-composition data. In a series of publications, Asfour et al. and Nhaesi and Asfour successfully converted the McAllister model into a predictive model which requires only the viscosities of the pure components and the molecular parameters of the constituents of a liquid mixture. They validated the model for the cases of n-alkane and regular liquid systems by using data on a large number of liquid mixtures at different temperatures. In this paper, we propose a novel technique for predicting the McAllister model parameters, for n-alkane systems, by the group-contribution method. The predictive capability of the McAllister model in this case is shown, for n-alkane binary and multicomponent systems, to be better or at least as good as the techniques reported earlier. The main advantage of the technique we are proposing here is that we expect it to be able to successfully and reliably predict more classes of liquid solutions than earlier methods. Introduction The viscosities of liquid mixtures are required in most engineering calculations, such as in the determination of flow in pipelines and heat transfer and mass transfer operations. Moreover, the knowledge of the dependence of viscosities of liquid mixtures on composition is of paramount importance from a theoretical standpoint, since it may lead to a clearer insight into the behavior of liquid mixtures. Models for estimating the viscosity of liquid mixtures can be classified into two categories, namely, semiempirical models and empirical models. The latter may further be classified as either correlative, where experimental mixture data are needed, or predictive, where the properties of the pure components constituting a mixture are utilized. The literature on predictive viscosity models has revealed that the idea of having a global viscosity model for viscosity prediction has not been very successful, since, in the opinion of the present authors, global models fail to properly recognize the molecular interactions which differ in different classes of liquids. The present unsatisfactory state of predictive models, with respect to the structure of liquids, led Asfour1 to break liquid solutions down into three classes: n-alkane mixtures, regular solutions, and associated systems. Such a classification led to success in predicting the dependence of mutual diffusivities for various liquid systems.2 Asfour et al.3 successfully extended that classification to the prediction of the dependence of viscosity on composition for n-alkane liquid systems. McAllister4 utilized Eyring’s absolute rate theory to develop a cubic equation for the kinematic viscosity of binary liquid mixtures. Chandramouli and Laddha5 extended the McAllister model to ternary liquid mixtures. Kalidas and Laddha6 verified the Chandramouli and Laddha model by using experimental data on ternary liquid systems. It should be pointed out that the McAllister model is regarded by many investigators * To whom correspondence should be addressed. Fax: (519) 735-6112. E-mail:
[email protected].
to be the best correlating technique available for liquid binary and ternary systems.7 Asfour et al.3 pointed out that the major drawback of the McAllister model is the presence of adjustable parameters, which makes it correlative in nature and drastically limits its practicality and usefulness since costly and time-consuming experimental data would be needed for the determination of the values of such parameters. The group-contribution concept has been extended by many investigators8-10 and was employed in predicting the viscosities of liquid mixtures. The basic idea of a group-contribution model is that a solution may be considered to be a mixture of the individual groups making up the molecules in the solution. Consequently, the properties of the solution are assumed to be due to interactions among those groups. The objectives of the present study include the following: (i) to develop a technique for the calculation of the McAllister model parameters for n-alkane systems by using the group-contribution method and (ii) to compare the predictive capability of the developed technique with other models which were derived on the basis of the group-contribution concept. Furthermore, the proposed method is compared with the Allan and Teja correlation.11 Development of the Technique for Predicting the McAllister Model Parameters. Nhaesi and Asfour12 developed and reported a generalized form of the McAllister model suitable for multicomponent liquid systems. The reported model is as follows: n
ln νm )
n
n
xi3 ln(νiΜi) + 3∑∑xi2xj ln(νijΜij) + ∑ i)1 i)1 j)1
i*j n
6
n
n
∑ ∑ ∑ xixjxk ln(νijkMijk) - ln(Mavg) i)1 j)1k)1
(1)
i*j*k Equation 1 is the generalized McAllister three-body interaction model. Due to the assumption of three-body
10.1021/ie050461z CCC: $30.25 © 2005 American Chemical Society Published on Web 11/10/2005
Ind. Eng. Chem. Res., Vol. 44, No. 26, 2005 9963
interactions, only two types of interaction parameters are involved: the binary interaction parameters, νij, and the ternary interaction parameters, νijk. Consequently, the kinematic viscosity of an n-component liquid mixture can be calculated from eq 1 if the properties of the pure components as well as the binary and ternary interaction parameters are known at a given temperature. The number of binary and ternary interaction parameters depends on the number of components in a liquid mixture. The number of the binary interaction parameters, N2, in an n-component mixture can be determined from the following relation:
N2 )
n! (n - 2)!
(2)
whereas the number of the ternary interaction parameters, N3, in an n-component mixture is given by
N3 )
n! 3!(n - 3)!
(3)
where n is the number of pure components in a liquid mixture. In an earlier communication, Asfour et al.3 reported a technique for predicting the McAllister model parameters from pure component and molecular parameters for liquid n-alkane binary systems. The reported technique is based on the carbon numbers. Utilizing the concept of the effective carbon number, Nhaesi and Asfour13 extended the technique to include regular binary solutions. Nhaesi and Asfour14 developed a method to predict the McAllister ternary interaction parameters for n-alkane as well as regular liquid systems. The method is based on the effective carbon number. In the present study, we report a new technique for predicting the numerical values of the McAllister model parameters by using the group-contribution method for n-alkane liquid solutions. The Basic Equations. Numerical values of νij (i < j) at 298.15 K for binary n-alkanes were chosen as reference values for each binary mixtures. This temperature was selected as a reference temperature since viscosity data are most frequently measured and reported at that temperature. The values of the νij were obtained by fitting eq 1, for the case where n ) 2, to experimental data on some binary n-alkane systems (cf. the caption of Figure 1 for the systems employed). The resulting values of νij were found to be related, in an approximately linear relationship, to the average carbon number, as shown in Figure 1. One may deduce from Figure 1 that the CH3 and CH2 groups make a simple additive contribution to the magnitude of νij. To estimate the contribution of each group, two sets of binary liquid mixtures with close carbon numbers were selected. In this case, we specifically used data on the binary systems hexane + heptane and heptane + octane. Two linear independent equations were formed; the solution of the formed equations yielded the numerical values of the CH3 and CH2 contributions, as reported in Table 1. Since the obtained parameters were calculated by using systems with close carbon numbers, a correction is necessary for systems where the difference in the carbon number is more pronounced. An explanation of how such corrections are made is in order.
Figure 1. Variation of ν12 with the average carbon number: (+) hexane (1) + heptane (2); (]) hexane (1) + octane (2); ([) heptane (1) + octane (2); (.) octane (1) + decane (2); (b) decane (1) + tridecane (2); (2) undecane (1) + tridecane (2). Table 1. McAllister Group-Contribution Parameters group
contribution
CH3 CH2
-0.0263 0.0663
Prediction of the McAllister Model Binary Parameters from the Group-Contribution Method. To estimate the values of νij, the following formula is suggested:
νij ) Rij +
m (Sm ∑ i + Sj )Gm m
(4)
where Sm i ) the number of groups of type m in component i; Gm ) the group-contribution parameter, as reported in Table 1; and Rij ) the correction term. For the case of binary systems, this term can be obtained from the following correlation:
Rij ) 2.3658 - 0.5661Cavg + 0.0339Cavg2
(5)
The Cavg term, in eq 5, denotes the mean carbon numbers in a binary liquid n-alkane system. In developing eq 5, the correction terms, Rij, represent the difference between the value of the McAllister binary interaction parameter calculated from fitting experimental data to eq 1 for the case of n ) 2 and the value of the same parameter calculated on the basis of the second term contained in eq 4, namely, m (Sm ∑ i + Sj )Gm m
The correction terms and the corresponding mean concentrations for the following binary n-alkane systems at 298.15 K were correlated: C6 (1) + C10 (2), C7 (1) + C12 (2), C8 (1) + C14 (2), C11 (1) + C15 (2), and C13 (1) + C15 (2). Figure 2 depicts a plot of the correction terms Rij versus their corresponding Cavg values. The curve shown can be fitted to a second degree parabola. It is clear from Figure 2 that the curve has a flat minimum.
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Figure 2. Change in the correction terms Rij with the average carbon number, Cavg, for the case of the binary McAllister parameters.
Therefore, one can see from that diagram that Rij is practically equal to zero between Cavg values of 7 and 9. Therefore, in this work, the values of Rij, for the binary parameters, at Cavg values e 8 were set equal to zero. Correction terms, Rij, for values of Cavg higher than 8 were calculated with the help of eq 5. It should be pointed out here that the use of eq 4 requires that the index “i” is assigned to the component with the lower number of carbon atoms in a binary mixture whereas the index “j” is assigned to the component with the higher number of carbon atoms. Equation 4 allows the calculations of the values of the McAllister parameters, νij, from group-contribution at 298.15 K. Asfour et al. 3 showed that
()
νji ) νij
νj νi
1/3
(6)
Equation 6 permits the calculation of the McAllister parameters, νji, from the value of νij obtained from eq 4 and the pure component kinematic viscosities. Prediction of the McAllister Model Ternary Parameters from the Group-Contribution Method. The treatment here is similar to that of the binary parameters, which was explained in detail, except that experimental viscosity data on ternary n-alkane systems were employed. Values of the McAllister ternary interaction parameter for n-alkane liquid mixtures may be calculated from the following equation:
νijk ) Rijk +
m m (Sm ∑ i + Sj + Sk )Gm m
(7)
where Rijk is a correction for large differences in the carbon numbers. The correction terms for the following ternary n-alkane systems at 298.15 K were fitted versus the average carbon number: C8 (1) + C11 (2) + C13 (3), C8 (1) + C11 (2) + C15 (3), C11 (1) + C13 (2) + C15 (3), and C8 (1) + C13 (2) + C15 (3). This is shown in Figure 3. The curve shown in Figure 3 could be fitted to a second
Figure 3. Change in the correction terms Rij with the average carbon number, Cavg, for the case of the ternary McAllister parameters.
degree polynomial given by the following equation:
Rijk ) - 7.569 + 1.1254Cavg - 0.0402Cavg2
(8)
Again, it should be noted here that the index “i” is assigned to the component with the lowest carbon number in the mixture, whereas the index “k” is assigned to the component having the highest carbon number value. The roots of the quadratic equation given by eq 8 were found to be 11.22 and 16.79, respectively. For the systems tested in this study, the values of Rijk were set equal to zero for systems having a Cavg ) 11.22. For any systems having a Cavg value between 11.22 and 16.79, Rijk is calculated from eq 8, and the calculated values of Rijk are incorporated, for those systems, into eq 7 for the calculation of the values of νijk. It should be noted here that in searching for data in the literature we could not find any systems with a Cavg value equal to, higher than, or even approaching 16.79. Therefore, this condition was not applied in this work. Correction for Temperature. Since the McAllister model parameters strongly depend on temperature, the values of the binary parameters at temperatures other than 298.15 K must be corrected by using the following equation:
( ) νi2νj
νij ) ν°ij
1/3
(9)
ν°i2ν°j
where ν°ij is the McAllister model three-body interaction parameter at the reference temperature of 298.15 K. A similar equation for the ternary parameters was derived. The equation has the following form:
( )
νijk ) ν°ijk
νiνjνk ν°iν°jν°k
1/3
(10)
Testing the Predictive Capability of the McAllister Three-Body Interaction Model in the Case of Binary and Ternary n-Alkane Liquid Systems. Experimental kinematic viscosity data gathered from
Ind. Eng. Chem. Res., Vol. 44, No. 26, 2005 9965 Table 2. Summary of the Results from Binary Systems: Comparison with the GC-UNIMOD Method, the Grunberg-Nissan Equation, the Allan and Teja Correlation, and the Wu Model Using n-Alkane Data McAllister
GC-UNIMOD
Grunberg-Nissan
system
T (K)
% AAD
% max
% AAD
% max
% AAD
% max
dodecane (1) + hexadecane (2)24 dodecane (1) + tetradecane (2)24 decane (1) + hexadecane (2)24 decane (1) + tetradecane (2)24 decane (1) + dodecane (2)24 nonane (1) + hexadecane (2)24 nonane (1) + tetradecane (2)24 nonane (1) + dodecane (2)24 nonane (1) + decane (2)24 octane (1) + hexadecane (2)24 octane (1) + dodecane (2)24 octane (1) + nonane (2)24 hexane (1) + tetradecane (2)24 heptane (1) + nonane (2)24 hexane (1) + dodecane (2)24 hexane (1) + nonane (2)24 hexane (1) + heptane (2)25 hexane (1) + octane (2)25 hexane (1) + decane (2)25 heptane (1) + octane (2)25 heptane (1) + decane (2)25 heptane (1) + dodecane (2)25 heptane (1) + tetradecane (2)25 octane (1) + decane (2)25 octane (1) + tetradecane (2)25 tetradecane (1) + hexadecane (2)25 octane (1) + undecane (2)15 octane (1) + tridecane (2)15 octane (1) + pentadecane (2)15 decane (1) + pentadecane (2)15 undecane (1) + pentadecane (2)15 tridecane (1) + pentadecane (2)15 decane (1) + tridecane (2)15 undecane (1) + tridecane (2)15 overall %
298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 293.15-308.15 293.15-308.15 293.15-308.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15 293.15-313.15
2.06 0.16 2.51 1.41 0.86 2.58 1.50 0.29 2.90 2.24 0.21 1.66 0.92 1.04 0.65 3.80 0.04 6.47 1.66 1.92 0.13 0.40 1.05 1.59 1.36 0.14 0.95 0.66 1.98 1.94 1.37 0.29 0.26 1.07 1.31
-4.57 -0.52 -5.25 -3.06 1.96 -5.61 -3.16 0.68 4.72 -5.03 -0.58 2.74 2.29 -1.75 0.69 3.80 -0.09 -10.9 -2.84 -3.20 -0.29 -0.88 -2.39 2.59 -2.67 0.25 1.74 1.36 -3.90 -3.52 -2.54 -0.51 0.33 1.58
1.75 0.66 3.09 2.11 0.50 4.71 2.98 1.10 0.09 6.68 2.26 0.25 9.17 0.71 5.28 1.12 0.55 0.34 2.16 0.17 1.00 2.11 5.31 0.52 3.66 0.23 1.17 2.68 4.58 1.93 0.78 0.26 0.88 0.25 1.71
-2.33 -0.84 -3.70 -2.51 -0.57 -5.91 -3.71 -1.60 -0.16 -8.48 -3.04 -0.89 -11.04 -0.90 -6.93 -1.61 -2.04 -0.93 -3.66 -0.33 -2.12 -5.17 -9.08 -0.92 -6.01 -0.56 -2.12 -4.87 -8.50 -3.44 -1.81 -0.3 -1.47 -0.71
0.10 0.33 0.66 0.73 0.50 0.90 0.48 0.61 0.37 1.58 0.36 0.36 2.77 0.45 0.99 1.05 1.99 1.98 1.84 2.78 2.66 2.68 2.84 2.74 2.73 2.76 2.65 2.57 2.66 2.85 2.81 2.76 2.78 2.81 2.59
0.21 0.71 0.78 -0.98 0.76 -1.49 -0.59 0.77 0.48 -2.47 0.54 0.60 5.00 0.69 1.50 1.54 -3.45 -3.45 -3.52 -5.50 -5.10 -5.89 -6.75 -5.43 -6.08 -5.34 -5.49 -5.65 -6.36 -5.55 -5.40 -5.35 -5.42 -5.43
the literature for liquid n-alkane solutions at different temperature levels were used to validate the proposed technique. The percent average absolute deviation (% AAD) was calculated by the following equation:
AAD )
1
n
∑
ni)1
|νexp - νcal i i | νexp i
× 100
(11)
To predict the kinematic viscosities of n-alkane liquid mixtures, the kinematic viscosities and molecular properties of the pure components must be known. The McAllister parameters for binary n-alkane solutions at 298.15 K are calculated with the help of eqs 4, 6, and 9. The average carbon number of a binary subsystem is substituted into eq 5 to obtain the correction term for the difference in the carbon number. The correction term and the proper number of groups along with the values of the group parameters for CH2 and CH3, as reported earlier in Table 1, are substituted into eq 4. This results in the value of the parameters νij. To obtain the value of this parameter at a different temperature, eq 9 must be employed. The value of the other parameters νji can be obtained by using eq 6. The values of the above parameters were substituted into the McAllister three-body model for binary systems, eq 1, for n ) 2 to obtain the predicted viscosity. It should be noted here that eqs 4 and 5 were tested by using experimental data which were not used in their development. Equations 7 and 8 were tested by using the viscosity data of ternary n-alkane systems from the literature. First, the values of νijk for each ternary subsystem at
Allan & Teja
Wu
% AAD % max % AAD % max 0.77 1.59 1.46 0.78 1.36 2.55 0.75 0.66 1.44 4.10 0.34 1.62 7.31 1.06 3.93 0.63 1.42 0.40 2.16 1.29 0.65 2.28 4.33 0.39 2.56 1.12 0.50 1.82 3.45 1.29 0.79 1.08 0.46 0.76 1.57
-1.19 -1.80 2.15 -1.83 -1.64 3.39 1.08 -0.99 -1.47 5.33 0.66 -2.35 9.73 -1.57 5.44 1.08 -2.07 -1.5 3.5 -1.95 1.28 3.82 6.77 -0.97 4.21 2.90 0.93 2.67 5.50 2.45 2.18 -1.89 -1.60 -1.80
0.82 0.18 2.78 1.04 0.33 3.82 2.00 0.85 0.17 5.13 1.49 0.19 6.65 0.57 3.87 1.86 0.20 0.44 2.54 0.18 1.39 3.34 5.97 0.59 4.24 0.34 1.19 3.11 5.51 2.64 1.71 0.60 0.98 0.40 2.01
1.74 0.64 6.28 2.85 0.82 8.37 4.49 1.70 0.29 11.08 3.14 - 0.47 13.12 1.01 8.20 2.91 0.36 1.26 4.25 0.33 2.30 5.58 9.91 0.99 7.00 0.58 2.13 5.46 9.00 4.40 2.99 0.99 1.64 0.68
298.15 K were calculated by eqs 7 and 8, and the binary interaction parameters, for the binary subsystems of the ternary system under consideration, were predicted by following the calculation procedure outlined earlier; then, the values of the calculated parameters were substituted into the McAllister three-body model for ternary systems, eq 1, for n ) 3 to obtain the predicted viscosity. Again, it should be noted here that the value of the ternary interaction parameter at a temperature other than 298.15 K should be corrected when necessary by using eq 10. It should be indicated here that systems utilized in obtaining eq 8 were excluded from the test. The results of testing the predictive capability of the proposed technique, for the case of binary and ternary n-alkane systems at different temperature levels, are reported in Tables 2 and 3, respectively. The overall AADs are 1.31% and 0.66%, respectively. Testing the Predictive Capability of the McAllister Three-Body Interaction Model in the Case of Quaternary and Quinary n-Alkane Liquid Systems. The general expression of the extended n-component McAllister equation is given by eq 1. As it can be seen, the equation contains binary and ternary interaction parameters. The number of the binary and ternary interaction parameters is calculated by means of es 2 and 3. To predict the kinematic viscosities of n-alkane liquid mixtures, the kinematic viscosities and molecular properties of the pure components must be known. The calculation scheme described earlier must be followed in order to determine the binary and the ternary interaction parameters for the binary and the ternary subsystems. The predicted parameters, along
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Table 3. Summary of the Results for Ternary Systems: Comparison with the GC-UNIMOD Method, the Grunberg-Nissan Equation, the Allan and Teja Correlation, and the Wu Model Using n-Alkane Data McAllister
GC-UNIMOD
system
T (K)
octane (1) + tetradecane (2) + hexadecane (3)15 octane (1) + undecane (2) + tridecane (3)15 octane (1) + undecane (2) + pentadecane (3)15 undecane (1) + tridecane (2) + pentadecane (3)15 octane (1) + tridecane (2) + pentadecane (3)15 decane (1) + tridecane (2) + pentadecane(3)15 overall %
293.15-313.15
0.39
1.28
3.48
293.15-313.15
0.53
1.19
293.15-313.15
0.93
293.15-313.15
Grunberg-Nissan
% AAD % max % AAD % max
Allan & Teja
Wu
% AAD
% max
-7.04
2.90
-5.99
3.00
7.28
4.60
10.18
2.01
-3.90
2.57
-5.47
1.14
2.16
2.25
4.05
-2.71
3.49
-6.33
2.56
-5.86
2.41
4.25
4.11
0.40
-1.10
0.98
-1.90
2.74
-5.17
0.71
-1.72
1.17
2.09
293.15-313.15
0.97
-2.06
3.60
-6.77
2.55
-5.19
2.37
4.28
4.21
7.29
293.15-313.15
0.77
-1.92
1.49
-2.97
2.70
-4.88
0.74
1.68
1.89
3.41
0.66
2.51
% AAD % max % AAD % max
2.67
1.73
7,28
3.04
Table 4. Summary of the Results from Quaternary Systems: Comparison with the GC-UNIMOD Method, the Grunberg-Nissan Equation, the Allan and Teja Correlation, and the Wu Model Using n-Alkane Data McAllister
GC-UNIMOD
system
T (K)
heptane (1) + octane (2) + undecane (3) + tridecane (4)26 octane (1) + decane (2) + undecane (3) + pentadecane (4)26 octane (1) + undecane (2) + tridecane (3) + pentadecane (4)26 heptane (1) + decane (2) + tridecane (2) + hexadecane (4)26 heptane (1) + octane (2) + dodecane (3) + hexadecane (4)26 overall %
293.15-313.15
1.00
-2.90
4.24
308.15-313.15
1.44
1.49
308.15-313.15
0.72
308.15-313.15 308.15-313.15
Grunberg-Nissan
% AAD % max % AAD % max
Allan & Teja
Wu
% AAD
% max
-7.14
3.00
-7.21
2.37
4.79
1.75
4.31
3.85
-4.43
4.09
-5.95
2.07
2.66
2.21
4.89
-1.92
3.65
-4.71
4.90
-6.11
1.89
2.73
2.15
4.77
0.43
-1.75
5.73
-6.72
4.72
-6.93
4.13
5.65
3.90
8.41
0.48
-2.79
6.62
-7.91
4.08
-6.93
5.16
6.40
4.34
9.63
0.84
4.82
4.18
% AAD % max % AAD % max
3.12
2.68
Table 5. Summary of the Results from Quinary Systems: Comparison with the GC-UNIMOD Method, the Grunberg-Nissan Equation, the Allan and Teja Correlation, and the Wu Model Using n-Alkane Data McAllister
GC-UNIMOD
Grunberg-Nissan
Allan & Teja
Wu
system
T (K)
% AAD
% max
% AAD
% max
% AAD
% max
% AAD
% max
% AAD
% max
heptane (1) + octane (2) + undecane (3) + tridecane (4) + pentadecane (5)26
298.2
0.18
1.33
5.93
-6.99
0.72
-1.84
3.4
4.2
3.44
6.92
with the properties of the pure components, are substituted into the appropriate form of eq 1 to obtain the predicted kinematic viscosity of the n-alkane liquid mixture. The applicability of the McAllister equation for n-components (n > 3) was tested by using n-alkane solution quaternary and quinary liquid system viscosity data obtained from the literature. The results are listed in Tables 4 and 5 along with the necessary information. For the quaternary systems, the overall AAD is 0.84%. The proposed method was tested by using kinematic viscosity data of a quinary system. This is the heptane (1) + octane (2) + undecane (3) + tridecane (4) + pentadecane (5) liquid mixture at 298.15 K which was reported by Wu.15 This represents the only viscosity data on quinary n-alkane liquid systems reported so far in the literature. The AAD of the predicted data, as shown in Table 5, is 0.18%, and the maximum deviation is 1.29%. The excellent predictive capability of our proposed technique not only converts the McAllister model from a correlative to a predictive model, which enhances the usefulness of the McAllister model, but also provides a very reliable means for predicting the viscosities of multicomponent liquid systems at different temperature levels. Comparison with the Allan and Teja Correlation. Allan and Teja11 reported an Antoine-type equation for calculating absolute viscosities. The three constants of the equation are correlated with the
Figure 4. Summary of the results of testing the various viscosity models by using data of n-alkane systems.
number of carbon atoms in the n-alkane by simultaneously fitting the viscosities of liquid n-alkanes C2C20. Figure 4 shows a comparison between the results of applying the Allan and Teja correlation and the results obtained from the McAllister model for n-alkane viscosity data. According to that figure, the McAllister model predicts the viscosities of n-alkane liquid mixtures much better than the Allan and Teja correlation. Comparison with Group-Contribution Based Methods. Cao et al.9 reported the GC-UNIMOD (group-
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contributions-viscosity-thermodynamics model). They introduced group-contribution such that the residual parts of the viscosity model used the same group binary interaction parameters as the UNIFAC vapor-liquid equilibrium (VLE) model. According to Monnery et al.,16 in their critical review on the viscosity of liquid systems, the results of the test were very impressive for a generalized predictive model. The results of testing the GC-UNIMOD are reported in Tables 2-5 for n-alkane liquid solutions. Furthermore, Chevalier et al.8 and Gaston-Bonhomme et al.10 reported the UNIFAC-VISCO group-contribution method to predict the viscosities of liquid systems. The main difference between the GC-UNIMOD and the UNIFAC-VISCO methods is that, in the latter method, the kinematic viscosity data were utilized to obtain the group interaction parameters involved in the residual part, whereas in the GC-UNIMOD method the VLE data were employed. Grunberg and Nissan17 introduced a binary adjustable parameter to correct the deviation from the ideal solution from the viscosity point view. Isdale et al.18 reported a group-contribution method to predict the binary adjustable parameters. The procedures of calculation were outlined by Reid et al.19 A comparison of the predictive capabilities of all the equations tested is summarized in Figure 4. The overall % AAD of each of the investigated viscosity models is represented by the height of the corresponding bar. According to Figure 4, the lowest % ADDs of prediction were obtained when the McAllister model was employed. It can also be observed that, in the case of the McAllister model, the AADs decrease as the number of components in a liquid mixture increases. In other words, the capability of viscosity prediction is enhanced. This is not surprising since the number of parameters in the model increases with the increasing number of components in a liquid mixture. When GC-UNIMOD is employed, the opposite trend is observed; that is, the AADs increase as the number of components in a liquid mixture increases. The viscosity prediction by means of the UNIFACVISCO method is not included, since the outcome of this model in the case of the ternary, quaternary, and quinary n-alkanes is similar to that of GC-UNIMOD. Only insignificant differences in the case of binary n-alkane systems were observed. This is because both methods have the same combinatorial part; while the residual term for the GC-UNIMOD method in the case of n-alkane systems is equal to zero,20 the residual term in the case of the UNIFAC-VISCO method played an insignificant role in improving the viscosity predications.12,21 A comparison of the Grunberg-Nissan equation and the McAllister model (cf. Figure 4) for all systems under investigation is absolutely in favor of the later method, since the overall % AADs of the McAllister model with our proposed technique are significantly lower than those of the Grunberg-Nissan method. The predictive capability of our proposed modification of the McAllister model was also compared with the Wu model,22 which also is based on the group-contribution method and requires only the knowledge of the pure component viscosities. The results of the comparison reported in Tables 2-5 clearly indicate that our proposed modification of the McAllister model outperforms the Wu model.
Figure 4 shows the stark differences in the predictive capabilities of the models tested. It also shows the clear superiority of the McAllister equation, when our proposed technique was employed to predict its parameters, over GC-UNIMOD especially in the cases of quaternary and quinary systems. It may be argued that we have picked a very select type of liquid mixture (n-alkanes) to test our proposed technique. Such an argument cannot be farther from the truth. First of all, we have argued in a series of publications1,3,13,21,23 that the main reason behind the failure of earlier attempts in the literature to obtain reliable predictive models has been due to the lack of knowledge of the structure of liquids. This led one of the authors1 to suggest breaking liquid mixtures down into three classes, namely, n-alkane mixtures, regular solutions, and associated systems. Earlier attempts to develop “global” models for viscosity prediction have failed, since, in our opinion, they failed to properly recognize the molecular interactions which differ in different classes of liquids. Work is currently underway to develop similar techniques for calculating the McAllister model parameters by using our technique for the cases of regular solutions and mixtures of associated solutions. The initial results obtained so far are very promising. We believe that the only way to enhance the predictive capabilities of any of the group-contribution based methods is to follow our idea of breaking liquid systems down into classes and calculating the contribution of each group in each class of liquid systems. We are confident that this idea would improve the predictive capabilities of any model and, hence, would provide additional evidence to support our pioneering work in this field. We elected not to compare the predictive capabilities of our present technique with those of the earlier techniques proposed by Asfour et al.3 and Nhaesi and Asfour12,14 because many of the systems used earlier in those publications were utilized in developing the equations reported in this work. However, a cursory comparison of the results reported in this manuscript and the results reported in those earlier publications would undoubtedly show that the technique we are proposing herein is at least as good as the techniques we reported in those earlier publications. Conclusions A new technique for predicting the values of the McAllister model parameters by using the groupcontribution method is proposed. The values of the McAllister binary and ternary interaction parameters for n-alkane liquid mixtures were calculated by a groupcontribution method. Group parameters for CH3 and CH2 groups at 298.15 K are reported. The validity of the technique was validated at other temperature levels. Using a common experimental viscosity database of various n-alkane systems, the prediction capabilities of the McAllister model with the proposed technique, the GC-UNIMOD model, the Grunberg-Nissan equation, and the Allan and Teja correlation were subjected to critical testing. The results indicated that the McAllister model predicts the viscosity data better than the other predictive models.
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Acknowledgment We acknowledge with thanks financial support through a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Research Grant. Nomenclature AAD ) average absolute deviation, %, eq 115.2 Cavg ) average carbon number Gm ) group parameter i ) index number j ) index number k ) index number m ) number of groups M ) molecular weight, g/mol n ) number of components of the mixture and number of interaction parameters Sm i ) number of groups of kind m in component i Greek Letters ν ) kinematic viscosity, m2/s νij ) McAllister three-body binary model interaction parameter νji ) McAllister three-body binary model interaction parameter νijk ) McAllister three-body ternary model interaction parameter Subscripts avg ) average i,j ) refer toith and jth component in the mixture, respectively ij ) refer tointeraction of type i-j ijk ) refer tointeraction of type i-j-k m ) mixture n ) refer tonth component in the mixture Superscripts cal ) calculated viscosity value exp ) experimental viscosity value ° ) refers tostandard temperature
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Received for review April 18, 2005 Revised manuscript received September 15, 2005 Accepted October 4, 2005 IE050461Z
