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Ind. Eng. Chem. Res. 2010, 49, 6250–6254
Equivalence of the McAllister and Heric Equations for Correlating the Liquid Viscosity of Multicomponent Mixtures Gustavo A. Iglesias-Silva* Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya, 38010 Celaya, Guanajuato, Me´xico
Kenneth R. Hall Artie McFerrin Department of Chemical Engineering, Texas A&M UniVersity, College Station, Texas 77843-3122
This paper demonstrates the equivalence between two equations for correlating liquid kinematic viscosities: the Heric and McAllister equations. For ternary and multicomponent mixtures, the only difference is in the ternary interaction term. If this term is constant, both equations correlate the viscosity data identically. This approach can prove the equivalence of the Nissan and Grunberg equation and the Frenkel equation to correlate the dynamic viscosity of liquid mixtures. Examples are included for binary and ternary mixtures. Now, McAllister4 presents his model as
Introduction Viscosities of liquid mixtures are important in many engineering problems related to mass transfer, fluid flow, and heat transfer. Their representation as a function of the composition has been of interest for many years. The most successful equations follow the Eyring theory of absolute reaction rates.1 Arrhenius2 established a relation equivalent to an ideal solution: a composition average of the logarithm of the viscosity. Katti and Chaudhri3 found an expression for what they term regular solutions. McAllister,4 based upon Eyring’s theory and considering multibody interactions, developed a theory that provides expressions for the logarithm of the kinematic viscosity of binary mixtures. The composition functionality depends upon the number of body interactions. The most popular expression is the function based upon three-body interactions. Later, Kalidas and Laddha5 extended the McAllister binary model to ternary mixtures. Almost simultaneously, Heric6 extended the Katti and Chaudhri3 equation to multicomponent mixtures and presented the McAllister equation also for those mixtures. The paper shows that for binary mixtures the three-body model of McAllister is the same as the Heric equation for binary mixtures and for ternary mixtures both models are the same if the term that accounts for ternary interactions is a constant. Equivalence between the Models of McAllister and Heric The Heric model6 for binary mixtures is ln ν ) x1 ln ν1 + x2 ln ν2 + x1x2[R12 + R′12(x1 - x2)] ln Mmix + x1 ln M1 + x2 ln M2 (1) where ν is the kinematic viscosity of the mixture, ν1 and ν2 are the kinematic viscosities of the pure components, Mmix ) ∑xiMi and xi is the mole fraction composition of specie i. Now, if x1 + x2 ) 1 ln ν ) x1 ln ν1 + x2 ln ν2 + x1x2[R12(x1 + x2) + R′12(x1 - x2)] - ln Mmix + x1 ln M1 + x2 ln M2 ) x1 ln ν1 + x2 ln ν2 + x1x2[x1(R12 + R′12) + x2(R12 - R′12)] - ln Mmix + x1 ln M1 + x2 ln M2
(2)
* To whom correspondence should be addressed. E-mail: gais@ iqcelaya.itc.mx.
ln ν ) x13 ln ν1 + x23 ln ν2 + 3x12x2 ln ν12 + 3x1x22 ln ν21 ln[x1M1 + x2M2] + x13 ln M1 + 3x12x2 ln M12 + 3x1x22 ln M21 + x23 ln M2 (3) where M1 and M2 are the molecular weights of components 1 and 2, and M12 ) (2M1 + M2)/3 and M21 ) (M1 + 2M2)/3. It is possible to reformulate eq 3 by writing it as ln ν ) x13 ln ν1 + x23 ln ν2 + x1x2(x13 ln ν12 + x23 ln ν21) ln Mmix + x13 ln M1 + x1x2(3x1 ln M12 + 3x2 ln M21) + x23 ln M2 (4) and using x1 ) 1 - x2 and x2 ) 1 - x1 to reduce the cubic terms into quadratic terms ln ν ) x12 ln ν1 + x22 ln ν2 + x1x2{x1(3 ln ν12 - ln ν1) + x2(3 ln ν21 - ln ν2)} - ln Mmix + x12 ln M1 + x22 ln M2 + x1x2{x1(3 ln M12 - M1) + x2(3 ln M21 - ln M2)} (5) Repeating the same procedure to reduce the quadratic terms to linear terms, ln ν ) x1 ln ν1 + x1 ln ν2 + x1x2{x1(x13 ln ν12 - ln ν1) ln ν1 + x2(3 ln ν21 - ln ν2) - ln ν2} - ln Mmix + x1 ln M1 + x2 ln M2 + x1x2{x1(3 ln M12 - ln M1) - ln M1 + x2(3 ln M21 - ln M2) - ln M2} (6) Now, using x1 + x2 ) 1 in the noncomposition terms of the brackets ln ν ) x1 ln ν1 + x1 ln ν2 + x1x2{x1(3 ln ν12 - 2 ln ν1 ln ν2) + x2(3 ln ν21 - ln ν1 - 2 ln ν2)} - ln Mmix + x1 ln M1 + x2 ln M2 + x1x2{x1(3 ln M12 - 2 ln M1 ln M2) + x2(3 ln M21 - ln M1 - 2 ln M2)} (7) Grouping terms using the property of logarithms produces
10.1021/ie1002763 2010 American Chemical Society Published on Web 05/19/2010
{ ( )
Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010
ν12
ln ν ) x1 ln ν1 + x1 ln ν2 + x1x2 x1 ln
( )} ν21
x2 ln
3
ν1ν22
2
+
ν1 ν2
- ln Mmix + x1 ln M1 + x2 ln M2 +
{ ( ) ( )}
x1x2 x1 ln
M123
+ x2 ln
M12M2
M213
M1M22
(8)
Mij3
(9)
ln ν ) x13 ln ν1 + x23 ln ν2 + x33 ln ν3 + 3x12 x2 ln ν12 + 3x1x22 ln ν21 + 3x12x3 ln ν13 + 3x1x32 ln ν31 + 3x22x3 ln ν23 + 3x3x22 ln ν23 - ln Mmix + x13 ln M1 + x23 ln M2 + 3 x3 ln M3 + 3x12x2 ln M12 + 3x1x22 ln M21 + 3x12x3 ln M13 + 3x1x32 ln M31 + 3x22x3 ln M23 + 3x2x32 ln M32 + 6x1x2x3 ln ν123 + 6x1x2x3 ln M123 (17)
- ln Mmix + x1 ln M1 + x2 ln M (10)
where M123 ) (M1 + M2 + M3)/3. Following the same procedure used for binary mixtures but using xi ) 1 - xj - xk with i * j * k * i and eq 9
2
Mi Mj
and substituting eq 9 into eq 8 results in
{ (
ν123
ln ν ) x1 ln ν1 + x1 ln ν2 + x1x2 x1 ln
(
x2 ln
ν21
)}
3
ν1ν22
M*21
ν12ν2
)
M*12 +
Equations 2 and 10 are identical if
(
R12 + R12 ′ ) ln
ν123 ν12ν2
M*12
)
(11)
and
(
(R12 - R12 ′ ) ) ln
ν213 ν1ν22
M*21
)
(12)
Solving for the adjusting parameters, R12 and R′12,
(( )
)
(
ν12ν21 3 ν12ν21 M12M21 1 3 ln M*12M*21 ) ln 2 ν1ν2 2 ν1ν2 M1M2
R12 )
ln ν ) x1 ln ν1 + x2 ln ν2 + x3 ln ν3 - ln Mmix + x1 ln M1 + x2 ln M2 + x3 ln M3 + x1x2[x1(R12 + R12 ′ )+ x2(R12 - R12 ′ )] + x1x3[x1(R13 + R13 ′ ) + x3(R13 - R13 ′ )] + x2x3[x2(R23 + R13 ′ ) + x3(R23 - R23 ′ )] + x1x2x3(β + R12 + R13 + R23) (16) In this form, the term with three interactions includes parameters with two interactions. The extension to ternary mixtures of the McAllister equation suggested by Kalidas and Laddha5 is
Defining a dimensionless molecular weight as M*ij )
)
(13)
ln ν ) x1 ln ν1 + x2 ln ν2 + x3 ln ν3 - ln Mmix + x1 ln M1 + ν123 x2 ln M2 + x3 ln M3 + x1x2 x1 ln 2 M*12 + ν1 ν2 3 ν21 ν133 * M21 + x1x3 x1 ln 2 M*13 + x2 ln ν1ν22 ν1 ν3 ν313 ν233 * M + x x x ln M*23 + x3 ln 31 2 3 2 ν1ν32 ν22ν3 ν323 (ν123M123)6 * M + x x x ln (18) x3 ln 32 1 2 3 ν2ν32 (ν1M1ν2M2ν3M3)2
[ ( ) ( )] [ ( ) ( )] [ ( ) ( )] { }
Comparing eq 16 to eq 18 and solving explicitly for the Heric’s parameters
and R12 ′ )
{( )
1 ln 2
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Or using x1 + x2 + x3 ) 1 as before
3
}
{( ) ( ) } ( ) ( )
ν12 3 V2 M12 ν21 ν1 M21 ν12 M12 3 1 ln + ln 2 ν21 M21 2
ν12 3 ν2 M*12 1 ) ln ν21 ν1 M*21 2
3M
2
)
M1 ν2 M2 (14) ν1 M1
For a binary mixture the McAllister equation and Heric equation are the same if their parameters satisfy eqs 13 and 14. Using eq 1 or eq 10 to correlate the experimental data should produce the same minimum of the same objective function because the adjusted parameters are eqs 13 and 14. Multicomponent Mixtures For a ternary mixture, the Heric equation consists of three terms with i-j interactions ln ν ) x1 ln ν1 + x2 ln ν2 + x3 ln ν3 - ln Mmix + x1 ln M1 + x2 ln M2 + x3 ln M3 + x1x2[R12 + a12 ′ (x1 - x2)] + x1x3[R13 + a13 ′ (x1 - x3)] + x2x3[R23 + a23 ′ (x2 - x3)] + x1x2x3β (15) with β ) β123 + β123 ′ (x1 - x2)
R12 ) R12 ′ )
(
(19)
( ) ) ( ) ) ( )
(
(20)
ν13ν31 M13M31 3 ln 2 ν1ν3 M1M3
(21)
(
(22)
)
ν13 M13 ν3 M3 3 1 ln + ln 2 ν31 M31 2 ν1 M1
R23 ) R23 ′ )
)
)
ν12 M12 ν2 M2 1 3 ln + ln 2 ν21 M21 2 ν1 M1
R13 ) R13 ′ )
(
ν12ν21 M12M21 3 ln 2 ν1ν2 M1M2
(
ν23ν32 M23M32 3 ln 2 ν2ν3 M2M3
(23)
(
(24)
)
ν23 M23 ν3 M3 3 1 ln + ln 2 ν32 M32 2 ν2 M2
For the 1-2-3 interaction, if β ) β123 is not a function of the composition, then β123 + R12 + R13 + R23 ) ln
{
(ν123M123)6 (ν1M1ν2M2ν3M3)2
}
(25)
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Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010
Again, If the Heric and McAllister equations satisfy eqs 19-25, they are identical. If β ) β123 + β′123(x1 - x2), the only term different from the McAllister equation is the compositiondependent term. For multicomponent mixtures, as apparent from the previous analysis, the Heric equation has exactly the same i-j interaction terms as the McAllister equation with each having an adjustable parameter. The only difference between the equations is in the i-j-k term. The Heric equation in the original paper appeared for a ternary mixture, but for n-components the equation would be n
∑ x ln ν
ln ν )
i
i
i
i)1 n-2 n-1
n
Rij )
i)1
i)1 j>i
Rij′ ) xixjxk[βijk +
∑ x ln ν
ln ν )
i
- ln Mmix +
i
∑ x ln M i
+
i
i)1
n
∑ x x [x (R i j
i
i)1 j>i n-2 n-1
+ Rij′ ) + xj(Rij - Rij′ )] +
xixjxk(βijk + Rij + Rik + Rjk) + n-1
n
∑ ∑ ∑ i)1
xixjxkβijk ′ (xi - xj) (27)
j)i+1 k)j+1
Also, the McAllister4 equation for multicomponent mixtures6 is n
3
n
i
( )
( )
{
}
βijk + Rij + Rik + Rjk ) ln
ln Mi +
i)1
3
xj ln νij + 6
n
∑
ln η )
(35)
n
∑∑
i)1
ln η )
j)i+1 k)j+1
∑
n
n
n
∑ ∑ ∑xxx A
xixjAij +
i j k ijk
i)1 j>i
i)1 j>i
k>j
n
∑ ∑ x x ln B i j
n-.1
+
ij
i)1 j>i
n
n
∑ ∑ ∑xxx
n
i j k
xixjxk ln νijk (28)
i)1 j>i
ln Bijk) (37)
k>j
j)i+1 k)j+1
Following the same procedure, this equation becomes
n
∑ x ln ν i
n
xi2 ln ηi + 2(
i)1
n
ln η )
∑ x ln η i
i)1
∑
n
xi ln ηi +
n
xixjxkMijk +
∑ ∑ ∑ i)1
j)1 j*i
i)1 j>i
(νiMiνjMjνkMk)2 for i * j * k * i (34)
while in the Heric equation Rij * 0 and R′ij ) 0. Also, equations exist to correlate the dynamic viscosity that are equivalent, such as the Nissan and Grunberg7 equation
n
∑ ∑ ∑ n-2
i
i)1 n
(33)
where η is the dynamic viscosity and the Frenkel8 equation
Following the previous analysis, an alternative expression of the above equation is ln ν )
(νijkMijk)6
n
∑ ∑x i)1
3 i
i)1 n-1
n-2
xjMij + 6
3
j)1 j*i n
n
(32)
(36)
∑x
ln νi - ln Mmix +
∑ ∑x i)1
3
3 i
i)1 n
)
n
∑x
ln ν )
(31)
νij3 Mij3 νji3 Mji3 ) νi Mi νj Mj
j)i+1 k)j+1 n-2
M*ji
and
n
∑ ∑ ∑ i)1
ij
(30)
Obviously, the Katti and Chaudhri3 equation is a special case of eqs 27 and 29. The condition is that i-j-k interactions are absent and i-j interactions equal j-i interactions without composition dependence. In the case of the McAllister equation
n
i)1
νiνj2
(
It is possible to reformulate the above equation as n
νji3
νijνji MijMji 3 ln 2 νiνj MiMj
j)i+1 k)j+1
βijk ′ (xi - xj)] (26)
M*ij
νij Mij νj Mj 1 3 ln + ln 2 νji Mji 2 νi Mi
n
∑ ∑ ∑
xixj[Rij + Rij′ (xi - xj)] +
νi2νj
where j is greater than i. The general solution is
+
i
( ) ( ) νij3
(Rij - Rij′ ) ) ln
n
∑ x ln M
- ln Mmix +
i)1
∑
Rij + Rij′ ) ln
n
i
- ln Mmix +
∑ x ln M i
i
[ ( ) ( )] ∑ ∑ ∑ { } n-2
νij3
νi2νj
n-1
M*ij + xj ln
n
xixjxk ln
i)1
j)i+1 k)j+1
νji3
νiνj2
n
+
n
∑ ∑ x x (2 ln B i j
M*ji
(νiMiνjMjνkMk)2
n
n
∑ ∑ ∑ x x x (2 ln B i j k
i)1 j>i
+
(νijkMijk)6
- ln ηi - ln ηj) +
ij
i)1 j>i
n
+
i)1
xixj xi ln
i
ijk)
(38)
k>j
Therefore eqs 36 and 38 are the same if
(29)
By comparing eq 27 to eq 29, the only difference is the composition dependence of the i-j-k term, otherwise the equations are identical. Heric6 mentions that an equivalence exists in his δij term, but the equivalence exists for the entire equation
Aij ) 2 ln Bij - ln ηi - ln ηj
(39)
Aijk ) 2 ln Bijk
(40)
and
Equations 36 and 38 should demonstrate equivalent performance when correlating dynamic viscosity data, using the same objective function and the same number of parameters. Also, similar results should occur for
Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010 n
∑xη
η)
i i
+
∑ ∑xxC
and the Hind et al. n
∑x
∑ ∑ ∑xxx C i)1 j>i
n
n
∑ ∑xxD i j
ij
+
(41)
k>j
n
n
∑ ∑ ∑xxx D i j k
i)1 j>i
i)1 j>i
i)1
i j k ijk
ijk)
k>j
(42) or equivalently n
η)
∑
n
xiηi +
i)1
n
∑ ∑ x x (2D i j
ij
- ηi - ηj) +
i)1 j>i
n
n
n
∑ ∑ ∑ x x x (2D i j k
i)1 j>i
ijk)
(43)
k>j
Therefore Cij ) 2Dij - ηi - ηj
(44)
Cijk ) 2Dijk
(45)
and
To verify the current assertions, this work uses three ternary systems each equation adjusting the parameters using a leastsquares method npts
∆)
∑ (Y
exp i
2 - Ycalc i )
6253
with his equation. The data are not weighted as suggested by Heric.6 Table 1 shows that the objective function is the same for the binary systems and also for the ternary systems when the ternary interaction parameter of the Heric equation is independent of the composition. We believe that Kalidas and Laddha5 probably did not find the global minimum of their objective function because their results differ substantially from the current ones and those of Heric. Another system is methyl butanoate + n-heptane + n-octane at 283.15 K. Matos et al.11 measure this system, and they use the equations considered here. They did not realize that the equations were the same, but if one looks at the average percentage error of the binary and ternary systems using both equations, they are the same as suggested here. Table 1 shows the current results agree with those of Matos et al.11 The final mixture is 1-heptanol-trichloroethylene-methylcyclohexane12 at 298.15 K. These viscosity data have been correlated to the McAllister, Heric, Nissan and Grunberg, and Frenkel equations. As shown in Table 1, the results from the first two equations are the same as are the results from the last two equations. For all the systems treated here, an extra ternary parameter, as suggested by Heric,6 does not improve the correlation of the data significantly. Also, the parameters when calculated using the expressions given here agree with the values obtained from the curve fits. Inconsistent values probably indicate a local minimum or use of different objective functions. Also as a note of proof, Belda Maximino13 uses the McAllister and Heric equations to correlate the kinematic viscosity data of 37 binary mixtures obtaining the same average percentage error.
n
equation n
ηi + 2(
2
i
+
i j ij
9,10
n
n
i)1 j>i
i)1
η)
n
n
(46)
i)1
Conclusions
where Yi is νi, or ηi. The first system is acetone + methanol + ethylene glycol at 303.15 K. This is the same mixture used by Kalidas and Laddha5 with the McAllister equation and by Heric6
The McAllister and Heric equations for correlating of the liquid kinematic viscosity of binary and multicomponent
Table 1. Binary and Ternary Constants and Percentage Deviations McAllister equation binary systems acetone + methanol acetone + ethylene glycol methanol + ethylene glycol methyl butanoate + n-heptane methyl butanoate + n-octane n-heptane + n-octane 1-heptanol + trichloroethylene 1-heptanol + methylcyclohexane trichloroethylene + methylcyclohexane
ν12
ν21
0.438247 0.563338 1.864998 0.637137 0.704208 0.728442 3.139257 3.893154 0.463228
Heric equation R12
R12 ′
-0.304156 -1.487923 0.634769 -0.322426 -0.266751 0.024744 0.093836 -0.363144 -0.124838
0.311744 -0.764087 -0.316207 0.026128 0.011809 -0.000296 0.361566 0.564939 -0.013776
∆ 4.4273 × 0.018700 0.073750 1.1778 × 1.4132 × 5.4554 × 0.000227 0.001569 4.5951 ×
0.430745 3.002715 6.106403 0.608003 0.740646 0.795392 0.903349 1.321809 0.631414
10
-5
10-5 10-5 10-7 10-6
Nissan-Grunberg equation 1-heptanol + trichloroethylene 1-heptanol + methylcyclohexane trichloroethylene + methylcyclohexane
∆ 4.4273 × 0.018700 0.073750 1.1778 × 1.4132 × 5.4554 × 0.000227 0.001569 4.5951 ×
10-5 10-5 10-5 10-7 10-6
Frenkel equation
A12
∆
B12
∆
0.096430 -0.212287 -0.195920
0.030976 0.077615 1.7627 × 10-5
1.872403 1.828037 0.544365
0.030976 0.077615 1.7627 × 10-5
McAllister equation ternary systems
ν123
Heric equation
∆
acetone + methanol + ethylene glycol
0.920109
0.004563
methyl butanoate + n-heptane + n-octane
0.673133
5.2635 × 10-5
1-heptanol + trichloroethylene + methylcyclohexane
1.084476
0.066178
β123 -1.334733 -1.364826 -0.038247 -0.015355 -0.819370 -0.819115
β123 ′ -0.475665 0.152733 0.152733
Nissan-Grunberg equation A123 1-heptanol + trichloroethylene + methylcyclohexane
-0.994972
∆ 0.165495
∆ 0.004563 0.004223 5.2633 × 10-5 3.4628 × 10-5 0.066179 0.066046
Frenkel equation B123
∆
0.608057
0.165495
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Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010
mixtures are equivalent. It is not necessary to include both equations when correlating viscosity data because the parameters are equivalent. The same conclusion applies for use of the Nissan and Grunberg equation and the Frenkel equation for correlating the dynamic viscosity. Acknowledgment The Texas Engineering Experiment Station and DGEST (Cuerpos Acade´micos Funding) have provided financial support for this work. Literature Cited (1) Eyring, H. Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J. Chem. Phys. 1936, 4, 283. ¨ ber die innere Reibung verdu¨nnter wa¨sseriger (2) Arrhenius, S. U Lo¨sungen. Z. Phys. Chem. 1887, 1, 285. (3) Katti, P. K.; Chaudhri, M. M. Viscosities of Binary Mixtures of Benzyl Acetate with Dioxane, Aniline, and m-Cresol. J. Chem. Eng. Data 1964, 9, 442. (4) McAllister, R. A. The Viscosity of Liquid Mixtures. AIChE J. 1960, 6, 427. (5) Kalidas, R.; Laddha, G. S. Viscosity of Ternary Liquid Mixtures. J. Chem. Eng. Data 1964, 9, 142. (6) Heric, E. L. On the Viscosity of Ternary Mixture. J. Chem. Eng. Data 1966, 11, 66.
(7) Grunberg, L.; Nissan, A. H. The Energies of Vaporisation, Viscosity and Cohesion and the Structure of Liquids. Trans. Faraday Soc. 1949, 45, 125. (8) Frenkel, Y. I. Kinematic Theory of Liquids; Oxford University Press: London, 1946. (9) Hind, R. K.; McLaughlin, E.; Ubbelohde, A. R. Structure and Viscosity of Liquids - Camphor+Pyrene Mixtures. Trans. Faraday Soc. 1960, 56, 328. (10) Hind, R. K.; McLaughlin, E.; Ubbelohde, A. R. Structure and Viscosity of Liquids - Viscosity-Temperature Relationships of Pyrrole and Pyrrolidine. Trans. Faraday Soc. 1960, 56, 331. (11) Matos, J. S.; Trenzado, J. L.; Gonzalez, E.; Alcalde, R. Volumetric Properties and Viscosities of the Methyl butanoate + n-Heptane + n-Octane Ternary System and its Binary Constituents in the Temperature Range from 283.15 to 313.15 K. Fluid Phase Equilib. 2001, 186, 207. (12) IIoukhani, H.; Samiey, B. Excess Molar Volumes, Viscosities, and Speeds of Sound of the Ternary Mixture {1-Heptanol (1) + Trichloroethylene (2) + Methylcyclohexane (3)} at T ) 298.15 K. J. Chem. Thermodyn. 2007, 39, 206. (13) Belda Maximino, R. Viscosity and density of binary mixtures of alcohols and polyols with three carbon atoms and water: equation for the correlation of viscosities of binary mixtures. Phys. Chem. Liq. 2009, 47, 515.
ReceiVed for reView February 3, 2010 ReVised manuscript receiVed April 12, 2010 Accepted April 28, 2010 IE1002763
