Weighted-Power-Mean Mixture Model: Application to Multicomponent

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Ind. Eng. Chem. Res. 2007, 46, 4660-4666

Weighted-Power-Mean Mixture Model: Application to Multicomponent Liquid Viscosity Walter W. Focke,*,† Carl Sandrock,† and Schalk Kok‡ Institute of Applied Materials, Departments of Chemical and Mechanical Engineering, UniVersity of Pretoria, Lynnwood Road, Pretoria 0001, South Africa

A multicomponent mixture may be viewed conceptually as a hypothetical collection of fluid clusters. In this context a mixture model is defined by prescriptions for (a) estimating fluid cluster properties and (b) combining them to yield an overall mixture property. A particularly flexible form is obtained using composition-weighted power means with the weighting based on global mole fractions. It predicts multicomponent properties from knowledge of pure component and binary mixture data. The classic quadratic Scheffe´ rational polynomials and the Wilson models are special cases. The model also generates a cubic Scheffe´ form for which ternary and higher coefficients can be expressed in terms of binary interactions. The binary-correlative and multicomponent-predictive capabilities of the model were evaluated for isothermal liquid viscosity data. A revised empirical mixing rule for liquid viscosity is proposed for systems where the constituent binaries show either convex or concave composition dependence. Introduction Physicists, chemists, engineers, and others continue to be involved in the development of empirical, semiempirical, and theoretical equations that express mixture properties in terms of composition and pure components attributes. Such mixture models or mixing rules are applied in chemical technology to assist with the design of process plants and the optimization of product formulations. Formulation chemists tend to employ empirical mixture models, for example, Scheffe´ polynomials,1,2 for mixture property data correlation. Process engineers prefer theoretical thermodynamic models, for example, Wilson,3 to deal with complex fluid mixtures. From the thermodynamic perspective, the behavior of multicomponent fluid mixtures is naturally affected by the interactions of unlike molecules.4 It is expedient to assume that only pairwise interactions between components need to be considered in mixtures.5,6 The implications of this assumption are twofold: First, it leads to a major reduction in the required experimental data gathering effort. Second, beyond the pure component data, information on the constituent binary systems suffices to fix model parameter values and thus to predict multicomponent behavior.4 Given these practical advantages, this study is limited to such binary predictive models. The mixing rule for the second virial coefficient is an interesting example. It has a sound theoretical basis,4 yet its composition dependence is equivalent to a quadratic Scheffe´ K-polynomial2,7 in the mole fractions xi. For a binary mixture it reads

B ) B11x12 + 2B12x1x2 + B22x22

(1)

The Bii’s in this expression represent the second virial coefficients of the pure components. The binary or cross-parameter B12 reflects the effect of interactions between pairs of unlike molecules. Binary mixture data are required to fix this parameter, * Tel. +27 12 420 2588. Fax +27 12 420 2516. E-mail: [email protected]. † Department of Chemical Engineering. ‡ Department of Mechanical and Aeronautical Engineering.

Figure 1. Schematic illustration of Scott’s model13 for a fluid mixture: In cluster 1 a molecule of component 1 is located at the center whereas cluster 2 has a molecule of component 2 at the center.

Figure 2. Contour plots of the error function (eq 27) for the binary data set.

but in a few instances it can be estimated from the pure component values using4

B12 ) xB11B22

(2)

Combining rules express the binary parameters of the mixture model in terms of pure component properties. The benefit is

10.1021/ie061465m CCC: $37.00 © 2007 American Chemical Society Published on Web 05/16/2007

Ind. Eng. Chem. Res., Vol. 46, No. 13, 2007 4661 Table 1. Isothermal Viscosity Data Sets Used for Model Testinga system components

a

T, K

data B: T: B: T: B: T: B: T: B: T: Q: B: T: B: T: B: T:

35 28 52 112 27 36 27 21 65 20 35 88 95 69 82 113 114

ref

1. acetone (1)-methanol (2)-water (3)

298.15

19

2. water (1)-trifluoroethanol (2)-tetraethylene glycol dimethyl ether (3)

303.15

3. water (1)-2-propanol (2)-diacetone alcohol (3)

303.15

4. benzyl alcohol (1)-benzaldehyde (2)-toluene (3)

293.15

5. n-heptane (1)-benzene (2)-cyclohexane (3)-ethanol (4)

298.15

6. water (1)-methanol (2)-ethanol (3)-1-propanol (4)

303.15

7. di-n-butyl ether (1)-1-propanol (2)-n-decane (3)-n-octane (4)

308.15

8. methyl butanoate (1)-n-heptane (2)-cyclo-octane (3)-1-octanol (4)-1-chloro-octane (5)-n-octane (6)

313.15

9. carbon tetrachloride (1)-ethanol (2)-acetone (3)-cyclohexane (4)-n-hexane (5)-2-propanol (6)

298.15

B: 101 T: 84

33, 34

10. methyl acetate (1)-methanol (2)-ethanol (3)-1-propanol (4)-2-propanol (5)-2-butanol (6)-2-methyl-2-butanol (7)

298.15

B: 124 T: 256

35-37

11. 1-chlorobutane (1)-1-butanol (2)-2-butanol (3)-1-butylamine (4)-n-hexane (5)-cyclohexane (6)-tetrahydrofuran (7)-1,4-dioxane (8)-1,3-dioxolane(9)-2-propanol(10)-chlorocyclohexane (11)

298.15

B: 263 T: 445

38-56

20, 21 22, 23 24 25, 26 27 28 29-32

B, binary data points; T, ternary data points, and Q, quaternary data points.

that they make binary mixture models completely predictive. Equation 2 provides a specific example of a geometric combining rule. Other types of combining rules, for example, linear, have been developed for other properties.4 Analogous empirical mixing rules for liquid viscosity have been proposed by Hind et al.8 and Grunberg and Nissan,9 respectively:

η ) η1x12 + 2η12x1x2 + η2x22

(3)

ln η ) x1 ln η1 + 2x1x2 ln η12 + x2 ln η2

(4)

Note that both expressions utilize Scheffe´ quadratic composition dependence. In the Hind model8 it is for the dynamic viscosity, but in the Grunberg-Nissan model9 it is for the natural logarithm of dynamic viscosity. McAllister10 expressed the composition dependence of the logarithm of the kinematic viscosity in terms of cubic Scheffe´ K-polynomials. This mixing rule is currently regarded as the best correlating technique for compositional effects in binary and ternary liquid mixtures of nonpolar components.11,12 However, owing to the adjustable ternary parameter, this equation is correlative rather than predictive for multicomponent mixtures. This communication outlines a facile method for deriving arbitrary binary predictive mixture models based on the thermodynamic approach suggested by Scott.13 A general binary mixture model is defined using composition-weighted power means. Its predictive utility is studied with regard to multicomponent liquid viscosity data for systems that include water and other associating components.

positive real numbers and Rn+ is its n product. The n-component mixture composition is quantified by the vector x ∈ Rn+ of normalized weights, for example, mole fractions. They are subject to the restrictions 0 e xi e 1 and ∑xi ) 1. The extremely short-range nature of the intermolecular interactions justifies the assumption that only binary interactions need to be considered. This allows the fluid to be viewed as an assembly of hypothetical fluid clusters.13 Figure 1 illustrates the concept. The cluster type is determined by the nature of the central molecule. Cluster properties are determined by the interaction of this molecule with its nearest neighbors. The molecular interactions are characterized by binary coefficients aij with i indicating the nature of the central molecule of the cluster and j a neighboring molecule. Thus it is postulated that there exists an unknown function y ) f(A, x) that links the physical property of interest y ∈ R+ with mixture composition via the coefficient matrix A ) [aij], aij ∈ R+. So, for an n-component system, the mixture is described by n2 adjustable coefficients: n characterizing the pure component properties aii and n(n - 1) corresponding to the binary interactions aij, i * j. Note that it is not assumed that aij ) aji. The function f is unknown a priori, but myriad candidate mixture models can be generated as follows: First postulate a procedure for computing cluster properties and then specify how the cluster properties are to be combined to determine the overall mixture property. Composition-weighted means, based on global mole fractions, provide a rational option. In the simplest case, both mixture and the cluster properties, ui, are specified as weighted arithmetic means: n

ui )

aijxj ∑ j)1

(5)

Model Development The following notation is used: Estimates are denoted by a circumflex; capital letters and bold lower case letters indicate matrices and vectors, respectively. R+ denotes the set of all

n

y)

uixi ∑ i)1

(6)

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Ind. Eng. Chem. Res., Vol. 46, No. 13, 2007

Substituting and simplifying yields the second-degree Scheffe´ K-polynomial:7 n

y)

n

∑ ∑ i)1 j)1

n

∑ i)1

aijxixj )

n

aiixi2 +

∑ ∑(aij + aji)xixj i)1 j>i

( )

n

n

∑ ∏ i)1 j)1

(7)

xi

aijxj

(8)

The power mean is a parametric form that includes many conventional means. See the appendix for details. Thus a very flexible form results if both the ui’s and the overall mixture property are defined as generalized weighted power means14 of orders r and s (r, s ∈ R), respectively:

(∑

)

n

y ) lim

pfr+

pfs+

(∑ )

1/p

(10)

k)1

y ) m[r,s](A, x) ) m[r,s]([aij], x) ) pfr+

(

)

p xi[M[s] ∑ i (ai, x)] i)1

y)

[∑ (∑ ) ] n

j)1

y ) lim

( [ ])

Equation 11 is homogeneous of order 1 in the coefficients aij, that is, for all λ,

m[r,s]([λaij], x) ) λm[r,s]([aij], x) ) λm[r,s](A, x)

p 1/p

(18)

With r ) s ) 1, eq 11 reduces to a homogeneous quadratic form, that is, the second-order Scheffe´ K-polynomial1,2,7 suggested by Hind et al.8 as the viscosity mixing rule. With r ) s ) 0, eq 11 also reduces to a Scheffe´ K-polynomial form but, in this case, for the logarithm of the response y, that is, the viscosity mixing rule suggested by Grunberg and Nissan,9

(11)

(12)

n

xi ∑aijxj ∑ i)1 j)1

n

j)1

(17)

When s ) 1, eq 11 reduces to the neural network model proposed by Focke:18

ln y )

r/s 1/r

xjasij

xi

i)1

yr

r/s-1

xjasij

Special Forms

1/p

For r, s * 0, it takes the form n

n

Because all coefficients are positive, the right-hand side is also positive, and the generalized weighted power mean is therefore a monotonic increasing function of the aij. Therefore, all CR(aij) g 0, and it immediately follows from eq 16 that the model is intrinsically well-conditioned with respect to all its adjustable coefficients aij.

pfr+

xkapik

n

(∑ )

xixjasij

n

Substituting eq 10 in eq 9 defines the mixture model. It is a generalized weighted double power mean of orders r and s:14-16

lim

CR(aij) )

(9)

n

ui(x) ) M[s] i (ai, x) ) lim

(16)

It can be shown that

1/p

xi[ui(x)]p

i)1

n

n

It is conventional to correct for the over-parametrization of this model by setting aij ) aji.1,2 If a weighted geometric mean of the ui’s is used instead, the exponential form of Wilson’s model3 is obtained:

y)

n

∑ ∑CR(aij) ) 1 i)1 j)1

n

∑ ∑ ln(aij) xixj i)1 j)1

(19)

In general, when r ) s and r * 0, the model reduces to a quadratic Scheffe´ polynomial for yr. As before, the overparametrization of the Scheffe´ polynomials is circumvented by setting aij ) aji. The exponential version of Wilson’s model3 (eq 8) is obtained with r ) 0 and s ) 1. The form for r ) 1 and s ) 0 is n

y)

n

xi∑axij ) x1ax11ax12...ax1n + ∑ i)1 j)1 j

1

2

n

(13)

x2ax211 ax222 ...ax2nn + ... + xnaxn11 axn22 ...axnnn (20)

According to Euler’s theorem on homogeneous functions of degree 1, it follows that

When aij ) a ∀ i, j, the values of r and s are immaterial as the response is just y ) a. With aij ) aii ∀ i, j, the response is

n

y)

n

∂y

∑ ∑aij ∂a i)1 j)1

n

(14)

yr )

ij

Higham17 defined a relative condition number that quantifies the sensitivity of a function with respect to small changes in a parameter value aij as follows:

aij ∂y CR(aij) ) y ∂aij

xiarii ∑ i)1

(21)

When aij ) ajj ∀ i, j, eq 11 simplifies to a similar expression: n

ys )

xiasii ∑ i)1

(22)

(15)

If |CR(aij)| , 1, the function is very well-conditioned; when |CR(aij)| ≈ 1, it is well-conditioned; but if |CR(aij)| . 1, it is considered to be ill-conditioned. Combining Higham’s definition17 (eq 15) with Euler’s result (eq 14) reveals that the relative condition numbers sum to unity:

These last two equations both simplify to the arithmetic mean of the pure component property values when, respectively, r ) 1 or s ) 1: n

y)

aiixi ∑ i)1

(23)

Ind. Eng. Chem. Res., Vol. 46, No. 13, 2007 4663 Table 3. Special Weighted Power Means p

T(z)

definition

weighted mean

min{ u1, u1, ..., un}

P f -∞ -1

1/z

H ) P-1 )

smallest element

[∑ ] n

xi

-1

harmonic

i)1 ui

n

0

ln(z)

∏u

G ) P0 )

xi

geometric

i

i)1 n

1

z

A ) P1 )

∑x u

arithmetic

i i

i)1

2

z2

Q ) P2 )

p

zp

Pp ) [

x∑x u

2

i i

quadratic

n

p 1/p

i i

power

i)1

pf∞ Figure 3. Contour plots of the error function (eq 27) for the ternary and quaternary data as predicted by the fit to the binary data set.

∑x u ]

max{ u1, u1, ..., un}

largest element

generate a subset of the higher order Scheffe´ models for which the ternary or higher coefficients are fully determined by binary data. For example, a cubic Scheffe´ K-polynomial is obtained if we set r ) 1 and s ) 1/2. Expansion of the formula, collecting terms, and comparing the coefficients with those in eq 24 reveals

ciii ) aii

(25a)

2ciij ) 2xaiiaij + aji

(25b)

6cijk ) 2(xaijaik + xajiajk + xakiakj)

(25c)

Equation 25c embodies a combining rule for the ternary constant, expressing it in terms of binary parameters. Consider a data set of N composition-property pairs {xk, yk}k)1,2...N. It is frequently assumed that composition is error free but that the scalar physical property, y, is the realized value of a random variable. In that case the regression model is described by Figure 4. Binary data fit and multicomponent viscosity predictions for the Hind et al.8 model (r ) s ) 1). Dots and open circles indicate binary (fitted) and multicomponent (predicted) data points, respectively.

Table 2. Overall Viscosity Correlating Performance of Equation 12 with r ) -5/6 and s ) 1/2 and Parameter Values Determined from Binary Data Only system

A

B

C

D

E

F

G

H

I

J

K

maximum 6.6 7.5 9.5 4.2 5.5 12.6 6.1 8.8 12.8 13.5 10.9 deviation, % AAD, % 2.7 1.6 1.8 1.0 1.5 1.8 0.9 0.9 1.5 2.2 2.1

The third-order Scheffe´ K-polynomial for a ternary mixture is

y ) c111x13 + c222x23 + c333x33 + 3c112x12x2 + 3c122x1x22 + 3c113x12x3 + 3c133x1x32 + 3c223x22x3 + 3c233x2x32 + 6c123x1x2x3 (24)

yk ) m[r,s](A, x) + k, k ) 1, 2, ..., N

Here yk ∈ R+ is the kth observation value of the relevant physical property and xk ∈Rn+ is the kth row of the N × n matrix of mixture compositions, while k ∈R is the kth error (residual). The n × n matrix A ) [aij], aij ∈ R+ represents the adjustable binary model coefficients. The family of parametrized functions m[r,s](A, x):A × x f R+ defines the regression model. It includes an adequate approximation of the unknown function f(A, x) if, for a particular set of values (r, s), it holds that m[r,s](A, xk) ) E(yk/xk). The residuals k in eq 26 include deviations reflecting the inability of a chosen model to represent the underlying data trends when this is not the case. The purpose of regression is to find the model that “best” fits the observations. This requires parameter estimates (rˆ, sˆ) and Aˆ such that some criterion is satisfied, for example, minimization of the sum of squares of the residuals: N

The requirement that multicomponent properties be predictable from knowledge of pure and binary data apparently disqualifies the higher order Scheffe´ polynomials from consideration. They include higher interaction parameters, for example, the ternary constant c123 that requires ternary or higher mixture data for their evaluation.1,2,7 This means that they cannot necessarily be determined from binary data alone. However, an interesting aspect of the current models is that it can actually

(26)

S(A) )

∑ k)1

N

k2 )

[yk - yˆ k(A, xk)]2 ∑ k)1

(27)

Preferred is the model form that provides the best predictions of multicomponent behavior given binary data. This suggests a two-stage regression where the model coefficients aij and the model parameters r and s are determined by consecutive leastsquares fits of the binary and multicomponent data, respectively.

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Figure 5. Binary data fit and multicomponent viscosity predictions for the Grunberg-Nissan9 model (r ) s ) 0). Dots and open circles indicate binary (fitted) and multicomponent (predicted) data points, respectively.

Figure 7. Binary data fit and multicomponent viscosity predictions for eq 28 with r ) -0.831 and s ) 0.504. Dots and open circles indicate binary (fitted) and multicomponent (predicted) data points, respectively.

Figure 6. Binary data fit and multicomponent viscosity predictions for the exponential Wilson3 model (r ) 0 and s ) 1). Dots and open circles indicate binary (fitted) and multicomponent (predicted) data points, respectively .

An advantage of the model is that it is possible to determine analytic derivatives of the error function (eq 27) in terms of the aij values for binary components. See eqs 15 and 17. Application to Isothermal Liquid Mixture Viscosity Chemical technology deals with a host of compositiondependent property variables. Liquid viscosity is used here as an example, but the approach should have general validity. Experimental observations are often made at isothermal conditions. Table 1 summarizes the isothermal viscosity data used in this study.19-56 It comprises 11 composite systems containing 36 ternary mixtures and one quaternary system. There were 36 components and a total of 2388 data points including 44 for the pure components and 964 binary data points. Four different optimization methods were employed, and several minimizations were conducted starting with different initial conditions to increase the likelihood of reaching a global optimum. The algorithms57,58 used (in order of derivative information requirements) were the Nelder-Mead57 simplex method using only error function values; steepest descent with line search;58 Newton’s method applied to the derivatives of

Figure 8. Dynamic viscosity of the system (methanol + triethyl amine + chloroform)33,34 at 298.15 K. Equation 28 is not able fit the convex-concave data trend of the methanol + chloroform binary data. This also leads to poor predictions for the multicomponent viscosity behavior.

the error function; and SQP,58 exploiting both first- and secondorder derivatives of the error function. It was also found that solving for the intermediate value Rij ) log aij improved convergence significantly. Where data was available at different temperatures, the components involved were treated as separate species. The diagonal elements of the coefficient matrix A were calculated using the arithmetic mean of the pure species data. The contour plots shown in Figures 2 and 3 were generated as follows. Values for (r, s) were fixed and the pairs of off-diagonal

Ind. Eng. Chem. Res., Vol. 46, No. 13, 2007 4665

elements aij determined by fitting the binary data. This procedure was repeated for all combinations in the range -1.5 e r, s e 1.5 using increments of 0.05. Figure 2 shows a contour plot of the error function (eq 27) determined for the binary data fit. Figure 3 shows the corresponding predictions for the ternary and quaternary data. Both plots show that good fits and predictions are possible with (r, s) values in the second and fourth quadrants. The best prediction lies in a shallow basin in the second quadrant (r ) -0.831 and s ) 0.504). Note that this point corresponds to a good fit but not the best fit to the binary data. For convenience we choose r ) -5/6 and s ) 1/2 to obtain the following mixing rule for liquid viscosity: n

ηmix ) [

n

xi(∑xjxηij) ∑ i)1 j)1

-5/3 -6/5

]

(28)

Figures 4-7 show the fit obtained using the model with the proposed r and s values and the fits obtained with the other empirical models. It is clear that the predictive performance of the mixture rules improves in the sequence

Hind < Grunberg-Nissan < Wilson < Equation (28) In general eq 28 provides excellent fits where the composition dependence for binary mixtures is strictly concave or convex. When this is not the case the data fits and consequently the predictions for multicomponent mixtures are poor. This structural deficiency of the model is illustrated in Figure 8 for the ternary system (methanol + triethyl amine + chloroform).34,35 The mixing rule defined by eq 28 should therefore be limited to systems where all the constituent binaries show either convex or concave composition dependence. Conclusion Mixing rules and mixture models express (isothermal) mixture properties in terms of composition and pure component attributes. Forms that allow predictions of multicomponent behavior from known binary data are preferred. Scott’s notion, that a fluid may be viewed as an assembly of fluid clusters, provides a suitable basis for generating such forms. Prescriptions for calculating cluster properties and their contribution to the overall property value define the mixture model. Choosing composition-weighted power means establishes eq 12, a family of functions parametrized by the exponent vector (r, s). The model includes well-known models as special cases, for example, (r, s) ) (0, 1) for Wilson, (1, 1) for Hind, and (0, 0) for Grunberg-Nissan. The model was applied to isothermal dynamic viscosity data. This resulted in a new mixing rule for multicomponent mixtures for systems where the binary composition dependence is strictly convex or concave. Acknowledgment Financial support for this research from the THRIP program of the Department of Trade and Industry and the National Research Foundation of South Africa as well as Xyris Technology is gratefully acknowledged. Appendix Generalized Weighted Means. Let u1, u1, .... un ∈ R+. Each ui is associated with a corresponding weight xi g 0 with ∑xi ) 1. Let T be a continuous and strictly increasing function that defines an invertible mapping T: R+ f R. The generalized

weighted quasi-arithmetic mean is defined by14

MT ) T -1(

n

n

xiT(∑ui)) ∑ i)1 j)1

(A1)

With T(z) ) zp, eq A1 reduces to the weighted power mean. However, a more precise definition of the weighted power mean also admits the special case with p ) 0: n

Pp(u, x, p) ) lim [ qfp+

xiuiq]1/q, ∑ i)1

p∈R

(A2)

Particular cases are presented in Table 3. The weighted power mean conforms to the following axiomatic characteristics: (i) Internal: min{u1, u2, ..., un} e Pp(u1, u2, ..., un) e max{u1, u2, ..., un}∀ n (ii) Reflexive: Pp(u, u, ... u) ) u ∀ n (iii) Symmetric: Pp(u1, u2, ..., ui, ..., uj, ..., un) ) Pp(u1, u2, ..., uj, ..., ui, ..., un) (Pp is independent of the way in which component indices are assigned.) (iv) Decomposable: Pp(u1, u2, ..., un) ) Pp(fk, fk, ..., fk, uk+1, ..., un) where fk ) Pp(u1, u2, ... uk) (The mean does not change when altering some values without altering their partial mean.) (v) Homogeneous of order 1, that is, ∀ λ: Pp(λu1, λu2, ..., λun) ) λPp(u1, u2, ..., un) The following fundamental inequality holds for weighted power means. Suppose r, s ∈ R/{0} with r > s. Then for any positive numbers u1, u2, ..., un n

[

n

xiuir]1/r g [∑xiuis]1/s ∑ i)1 i)1

Thus it holds that P-∞ e P-1 e P0 e P1 e Pq e P∞ where q > 1, that is, H e G e A e P. Literature Cited (1) Scheffe´, H. Experiments with Mixtures. J. Roy. Statist. Soc. B 1958, 20, 344. (2) Cornell, J. A. Experiments with Mixtures, 3rd ed.; John Wiley & Sons: New York, 2002. (3) Wilson, G. M. Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127. (4) Walas, S. M. Phase Equilibrium in Chemical Engineering; Butterworth: Boston, 1985. (5) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice Hall: Upper Saddle River, NJ, 1999. (6) Hamad, E. Z. Exact Limits of Mixture Properties and Excess Thermodynamic Functions. Fluid Phase Equilib. 1998, 142, 163. (7) Draper, N. R.; Pukelsheim, F. Mixture Models Based on Homogeneous Polynomials. J. Statist. Plan. Inference 1998, 71, 303. (8) Hind, R. K.; McLaughlin, E.; Ubbelohde, A. R. Structure and Viscosity of Liquids. Camphor + Pyrene Mixtures. Trans. Faraday Soc. 1960, 56, 328. (9) Grunberg, L.; Nissan, A. H. Mixture Law for Viscosity. Nature 1949, 164, 799. (10) McAllister, R. A. The Viscosity of Liquid Mixtures. AIChE J. 1960, 6, 427. (11) Nhaesi, A. H.; Asfour, A.-F. A. Prediction of the Viscosity of MultiComponent Liquid Mixtures: a Generalized McAllister Three-body Interaction Model. Chem. Eng. Sci. 2000, 55, 2861. (12) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (13) Scott, R. L. Corresponding States Treatment of Nonelectrolyte Solutions. J. Chem. Phys. 1956, 25, 193. (14) Bullen, P. S. Handbook of Means and their Inequalities; Kluwer Academic Publishers: Dortrecht, 2003.

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(15) Ku, H. T.; Ku, M. C.; Zhang, X. M. Generalized Power Means and Interpolating Inequalities. Proc. Amer. Math. Soc. 1999, 127, 145. (16) Tarnavas, C. D.; Tarnavas, D. D. An Inequality for Mixed Power Means. Math. Inequal. Appl. 1999, 2, 175. (17) Higham, N. J. Accuracy and Stability of Numerical Algorithms, 2nd ed.; SIAM: Philadelphia, 2002. (18) Focke, W. W. Neural Network Averaging. Neural Computat. 2005, 18, 1. (19) Noda, K.; Ohashi, M.; Ishida, K. Viscosities and Densities at 298.15 K for Mixtures of Methanol, Acetone and Water. J. Chem. Eng. Data 1982, 27, 326. (20) Olive´, F.; Patil, K. R.; Ferna´dez, J.; Coronas, A. Excess Volumes and Viscosities of the Ternary System Water-Trifluoroethanol-Tetraethylene Glycol Dimethyl Ether at 303.15 K. Thermochim. Acta 1995, 259, 57. (21) Olive´, F.; Patil, K. R.; Ferna´dez, J.; Coronas, A. Densities, Viscosities and Excess Properties of Trifluoroethanol-Water, Tetraethylene Glycol Dimethyl Ether-Water and Trifluoroethanol-Tetraethylene Glycol Dimethyl Ether. Int. J. Thermophys. 1994, 15, 661. (22) Boned, C.; Moha-Ouchane, M.; Allal, A. Viscosity and Density at High Pressures in an Associative Ternary. Int. J. Thermophys. 1998, 19, 1325. (23) Moha-Ouchane, M.; Boned, C.; Allal, A.; Benseddik, M. Viscosity and Excess Volume at High Pressures in Associative Binaries. Int. J. Thermophys. 1998, 19, 161. (24) Cruz, R. da, C.; Martins, R. J.; Cardoso, M. J. E. de M.; Barcia, O. E. Measurement of the Viscosity of Binary and Ternary Systems Containing Benzaldehyde, Toluene, and Benzyl Alcohol at 293.15 K and 0.1 MPa. J. Chem. Eng. Data 2002, 47, 927. (25) Kouris, S.; Panayiotou, C. Dynamic Viscosity of Mixtures of Benzene, Ethanol, and n-Heptane at 298.15 K. J. Chem. Eng. Data 1989, 34, 200. (26) Papaloannou, D.; Evangelou, T.; Panayiotou, C. Dynamic Viscosity of Multicomponent Liquid Mixtures. J. Chem. Eng. Data 1991, 36, 43. (27) Dizechi, M.; Marschall, E. Viscosity of some Binary and Ternary Liquid Mixtures. J. Chem. Eng. Data 1982, 20, 46. (28) Segade, L.; Casas, H.; Cabeza, O Ä .; Prego, M.; Garcıá-Garabal, S.; Franjo, C.; Jimeńez, E. Viscosities of di-n-Butyl Ether + 1-Propanol + n-Decane Mixture at 308.15 K. Fluid Phase Equilib. 2001, 182, 353. (29) Matos J. S.; Trenzado, J. L.; Alcalde R. Volumetric Properties and Viscosities of the Methyl Butanoate + n-Heptane + 1-Octane Ternary System and its Binary Constituents in the Temperature Range from 283.15 to 313.15 K. Fluid Phase Equilib. 2001, 201, 207. (30) Matos, J. S.; Trenzado, J. L.; Alcalde, R. Volumetric Properties and Viscosities of the Methyl Butanoate + n-Heptane + 1-Chlorooctane Ternary System and its Binary Constituents in the Temperature Range from 283.15 to 313.15 K. Fluid Phase Equilib. 2002, 202, 133. (31) Trenzado, J. L.; Matos, J. S.; Alcalde, R. Volumetric Properties and Viscosities of the Methyl Butanoate + n-Heptane + Cyclo-octane Ternary System and its Binary Constituents in the Temperature Range from 283.15 to 313.15 K. Fluid Phase Equilib. 2002, 200, 295. (32) Trenzado, J. L.; Matos J. S.; Alcalde R. Volumetric Properties and Viscosities of the Methyl Butanoate + n-Heptane + 1-Octanol Ternary System and its Binary Constituents in the Temperature Range from 283.15 to 313.15 K. Fluid Phase Equilib. 2003, 205, 171. (33) Wei, I.-C.; Rowley, R. L. Binary Liquid Mixture Viscosities and Densities. J. Chem. Eng. Data 1984, 29, 332. (34) Wei, I.-C.; Rowley, R. L. Ternary Liquid Mixture Viscosities and Densities. J. Chem. Eng. Data 1984, 29, 336. (35) Rodrigues, A.; Canosa, J.; Tojo, J. Dynamic Viscosities of the Ternary Liquid Mixture Involving CH3COOCH3, CH3OH, CH3CCH2CH2OH at 298.15 K. J. Chem. Thermodyn. 1998, 30, 1307. (36) Canosa, J.; Rodrı´gues, A.; Tojo, J. Dynamic Viscosity and Speed of Sound of the Ternary Mixture Methyl Acetate + Methanol + Ethanol at 298.15 K. J. Chem. Eng. Data 1998, 43, 961. (37) Canosa, J.; Rodrı´gues, A.; Tojo, J. Dynamic Viscosities of the Binary Mixtures (Methyl Acetate + 2-Methyl-2-butanol) and Ternary Mixtures (Methyl Acetate + Methanol + 2-Propanol, or 2-Butanol, or 2-Methyl-2-butanol) at 298.15 K. J. Chem. Thermodyn. 2001, 32, 551. (38) Acevedo, I. L.; Pedrosa, G. C.; Katz, M.; Mussari, L.; Postigo, M. A. Excess Molar Volumes and Excess Viscosities of the Propan-2-ol + Tetrahydrofuran + 1-Chlorobutane Ternary System at 25 °C. J. Solution Chem. 2000, 29, 1237. (39) Bardavid, S. M.; Pedrosa, G. C.; Katz, M.; Postigo, M. A.; Garcıá, P. Excess Molar Volume and Excess Viscosities for the n-Hexane + Dihloromethane + Tetrahydrofuran Ternary System at 25 °C. J. Solution Chem. 1996, 25, 1125. (40) Domıńguez, M.; Camacho, A.; Lo´pez, M. C.; Royo, F. M.; Urieta, J. S. Excess Molar Volumes and Excess Viscosities of Ternary Mixtures

(2-Butanol + 1-Chlorobutane + 1-Butylamine) and (2-Methyl-2-propanol + 1-Chlorobutane + 1-Butylamine) at 298.15 K. Can. J. Chem. 1995, 73, 896. (41) Domıńguez, M.; Rodrı´guez, S.; Lo´pez, M. C.; Royo, F. M.; Urieta, J. S. Densities and Viscosities of the Ternary Mixture 1-Butanol + 1-Chlorobutane + 1-Butylamine and 2-Methyl-1-propanol + 1-Chlorobutane + 1-Butylamine) at 298.15 K. J. Chem. Eng. Data 1996, 41, 37. (42) Domıńguez, M.; Pardo, J.; Lo´pez, M. C.; Royo, F. M.; Urieta, J. S. Viscosities of the Ternary Mixture (1-Butanol + n-Hexane + 1-Butylamine) at the Temperatures 298.15 and 313.15 K. Fluid Phase Equilib. 1996, 124, 147. (43) Domıńguez, M.; Santafe´, J.; Lo´pez, M. C.; Royo, F. M.; Urieta, J. S. Viscosities of the Ternary Mixture (1-Butanol + n-Hexane + 1-Chlorobutane) at 298.15 K and 313.15 K. Fluid Phase Equilib. 1998, 152, 133. (44) Domıńguez, M.; Pardo, J. I.; Gascoń, I.; Royo, F. M.; Urieta, J. S. Viscosities of the Ternary Mixture (2-Butanol+n-Hexane+1-Butylamine) at 298.15 and 313.15 K. Fluid Phase Equilib. 2000, 169, 277. (45) Domıńguez, M.; Langa, F.; Mainar, A. M.; Urieta, J. S. Measurements, Correlations, and Predictions of Viscosities for the Ternary Mixture (2-Butanol + Hexane + 1-Chlorobutane) at 298.15 and 313.15 K. J. Chem. Eng. Data 2003, 48, 302. (46) Haro, M.; Rodrı´guez, V.; Cea, P.; Lo´pez, M. C.; Lafuente, C. Viscosimetric Study of Multicomponent Liquid Mixtures Containing Oxygenated Compounds. Int. J. Thermophys. 2004, 25, 669. (47) Gascoń, I.; Lafuente, C.; Cea, P.; Royo, F. M.; Urieta, J. S. Viscosities of the Ternary Mixture (Cyclohexane + Tetrahydrofuran + Chlorocyclohexane) at the Temperatures 298.15 and 313.15 K. Fluid Phase Equilib. 1999, 164, 143. (48) Rodrı´guez, S.; Lafuente, C.; Cea, P.; Royo, F. M.; Urieta, J. S. Densities and Viscosities of Binary Mixtures of Some Cyclic Ethers + Chlorocyclohexane at 298.15 and 313.15 K. J. Chem. Eng. Data 1997, 42, 1285. (49) Gascoń, I.; Lo´pez, M. C.; Domıńguez, M.; Royo, F. M.; Urieta, J. S. Viscosities and Viscosity Predictions of the Ternary Mixture Cyclohexane + 1-3-Dioxolane + 1-Butanol at 298.15 and 313.15 K. J. Chem. Eng. Jpn. 2000, 33, 740. (50) Gascoń, I.; Pardo, J.; Santafe´, J.; Domıńguez, M.; Urieta, J. S. Experimental and Predicted Viscosities of the Binary System (n-Hexane + 1,3-Dioxolane) and for the Ternary System (n-Hexane + 1,3-Dioxolane + 1-Butanol) at 298.15 and 313.15 K. Fluid Phase Equilib. 2001, 180, 211. (51) Gascoń, I.; Mainar, A. M.; Royo, F. M.; Urieta, J. S.; AlvarezCerdeirinã, L. Experimental Viscosities and Viscosity Predictions of the Ternary Mixture (Cyclohexane+1,3-Dioxolane+2-Butanol) at 298.15 and 313.15 K. J. Chem. Eng. Data 2000, 45, 751. (52) Lafuente, C.; Artigas, H.; Pardo, J.; Royo, F. M.; Urieta, J. S. Viscosities of 1-Chlorobutane and 1,4-Dichlorobutane with Isomeric Butanols at 25 and 40 °C. J. Solution Chem. 1996, 25, 303. (53) Mariano, A.; Postigo, M. Excess Molar Volume and Viscosity Study for the Ternary System Tetrahydrofuran (1) + 1-Chlorobutane (2) + 2-Butanol (3) at 283.5, 298.15 and 313.15 K. Fluid Phase Equilib. 2006, 239, 146-155. (54) Mariano, A.; Camacho, A.; Postigo, M.; Valen, A.; Artigas, H.; Royo, F. M.; Urieta, J. S. Viscosities and Excess Energy of Activation for Viscous Flow for Binary Mixtures of Tetrahydrofuran with 1-Butanol, 2-Butanol and 1-Chlorobutane at 283.15, 298.15 and 313.15 K. Braz. J. Chem. Eng. 2000, 17, 459-469. (55) Mariano, A.; Mussari, L.; Moreno, R.; Camacho A.; Postigo, M. Viscosidad, energıá de flujo viscoso y densidad de los sitemas 1-butanol + 1-clorobutano y 2-butanol + 1-clorobutano a 283.15K, 298.15K y 313.15K. ReV. BoliV. Quı´m. 2001, 18, 85. (56) Postigo, M.; Mariano, A.; Mussari, L.; Camacho, A.; Urieta, J. S. Excess Molar Volume and Viscosity Study for the Ternary System Tetrahydrofuran (1) + 1-Chlorobutane (2) + 1-Butanol (3) at 283.15, 298.15 and 313.15 K. Fluid Phase Equilib. 2003, 207, 193. (57) Lagarias, J. C.; Reeds, J. A.; Wright, M. H.; Wright, P. E. Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM Journal of Optimization 1998, 9, 112. (58) Arora, J. S. Introduction to Optimum Design; McGraw-Hill: Singapore, 1989.

ReceiVed for reView November 15, 2006 ReVised manuscript receiVed March 30, 2007 Accepted April 2, 2007 IE061465M

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