correlating multicomponent mixture properties with homogeneous rational functions

Correlating Multicomponent Mixture Properties with HomogeneousRational Functions

Walter W. Focke* and Barend Du Plessis

Institute of Applied Materials, Department of Chemical Engineering, University of Pretoria, P.O. Box 35285,Menlo Park, Pretoria 0102, South Africa

Multivariate homogeneous rational functions, analogous to Pade approximants, are proposedfor the modeling of mixture properties. They are defined as the ratio of two homogeneous ScheffeK polynomials. This choice offers a flexible functional form, attractive symmetries, parameterparsimony, and consistent expressions for multicomponent mixtures. The proposed rationalfunctions satisfactorily correlate the nonlinear behavior of diverse multicomponent physicalproperty data. They also extend the application of classic binary activity coefficient models suchPorter, Margules, Scatchard, etc., to multicomponent mixtures.

Introduction

Accurate knowledge of physical properties is vital forthe reliable design and optimization of chemical plants.Relevant properties include density, thermal conductiv-ity, viscosity, heat of mixing, etc. Typically, they dependon the system pressure, temperature, and composition.Owing to the dominant position of distillation as aseparation technique, much research has focused ontheoretical and empirical models for the (excess) Gibbsfree energy. The historical development started in the19th century (van Laar and Margules) and culminatedin predictive versions of semitheoretical local composi-tion models.1 Although there are some outstandingissues, the success of the latter in industrial applicationsis incontrovertible. Progress in representing the heatof mixing has been less impressive.2

Formulation chemistry is a rudimentary form ofchemical product design. In essence, it entails themixing of compounds in order to get a product with therequired attributes. Desired characteristics are legionbut include effects such as adhesion, weather resistance,texture, shelf life, biodegradability, controlled biologicalactivity, etc. Key to chemical product formulation is theselection of appropriate ingredients and the optimiza-tion of their relative proportions.

In the past, chemical product formulation was re-garded as a “black art” because it relied on a trial-and-error approach.3 Once a basic formulation was estab-lished, it was refined by a process of successiveapproximation. Nowadays, a systematic experimentaldesign process is followed.4-6 Sound experimental mix-ture designs are employed to minimize the number offormulations that need to be tested.7,8 The factor spacedefines the range of possible compositions. Mixtureexperiments are conducted whereby predeterminedcompositions are prepared and their properties mea-sured. The resultant set of discrete data points is usedto generate a continuous and analytical representationof the variation of a given property F in factor space.Unfortunately, the theoretical knowledge availableabout the system is, more often than not, insufficient

to generate a sound mechanistic model for the responsesurface. This necessitates the use of empirical modelsthat, hopefully, will at least be locally satisfactory.Scheffe polynomials are the most common models ap-plied in the context of experimental mixture designs.7-9

When such second or higher order polynomials are fittedto data, they generate convoluted response surfaces thatfeature stationary points corresponding to “tops of hills”,“troughs of valleys”, and saddle points. If the model fitsthe data well, these should correspond to best or worstpossible formulations.

Clearly, algebraic correlating equations are moreconvenient to use than tabulations or graphs. Thequestion is, will advances in molecular simulation andmodeling ultimately supplant experimental data gen-eration and eliminate the need for correlating equa-tions? In Churchill and Zajic’s10 judgment, the depend-ability of current numerical solutions is still suspect andconditional on the inherent need for idealization andsimplification. Therefore, results obtained from molec-ular and computer simulations may not yet be asaccurate as good experimental data. Usually the com-puted values are discrete and differ from experimentaldata only in their regularity and in the precision of theirdetermination. Thus, for the moment, correlation equa-tions will remain convenient because they provide areliable and efficient data interpolation method for theoptimization of chemical processes and product formula-tions.

This paper addresses the common need of formulationchemists and chemical engineers to correlate physicalproperties with mixture composition. It combines con-cepts from experimental mixture designs with thosedeveloped for solution thermodynamics. Rational func-tions based on revised canonical Scheffe polynomials arepresented that should facilitate empirical data correla-tion and the prediction of multicomponent mixture data.

Ideal Solutions and the “Blending” Rule

The concentrations of the components are conve-niently expressed in terms of fractions (e.g., by mass,volume, or mole). No matter what is chosen as the basis,all proportions have to be nonnegative and their sum

* To whom correspondence should be addressed. Tel.:+27 12 420 2588. Fax: +27 12 420 2516. E-mail:[email protected].

8369Ind. Eng. Chem. Res. 2004, 43, 8369-8377

10.1021/ie049415+ CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 11/25/2004

must equal unity. Mathematically, these requirementsare expressed as follows:

and

In formulations, the “blending rule” states the generalexpectation that mixture properties vary linearly withcomposition:

In eqs 1-3, q is the number of components in themixture, Fi denotes the pure-component property value,and xi is the mole fraction of component i in the mixture.

The mixing function (property) is defined as thedifference in the actual mixture property value and theexpected value according to the blending rule:

In thermodynamics, the ideal solution concept servesas a standard to which the properties of real mixturescan be compared. For ideal solutions, some properties,e.g., the entropy of mixing, include an additional termthat is proportional to ∑xi ln xi. Excess properties aredefined as the difference between the actual mixtureproperty and the corresponding value for an idealsolution:

With the exception of entropy- and entropy-dependentproperties, e.g., the Gibbs and Helmholtz free energies,eqs 3 and 4 also express the properties of the idealsolution. Thus, the heat of mixing and volume of mixingare equivalent to the corresponding excess properties.

Although the “blending rule” and the ideal solutionmodel provide useful first approximations, deviationsare often significant. Both the excess and mixingproperties equal zero in the two concentration limits forbinary mixtures. This implies a leading behavior of theform FE ∼ fx1x2 and justifies the following definition ofa peeled property for binary mixtures:

Plotting f ) f(x1) reveals the complexity of the systembehavior, making it a useful vehicle for model discrimi-nation.11

Polynomial Models for Excess Properties

Taking eq 2 into consideration, a single compositionvariable is sufficient for a binary blend. Redlich andKister12 proposed z ) x1 - x2 as the compositionvariable. Note that this maps the composition range intothe finite interval z ∈ [-1, 1]. They next proposedpolynomial expansions for the peeled property of theform

Equation 7 is equivalent to a truncated Maclaurin

(Taylor) series. The coefficients Ck are related to the firstderivatives of the function f evaluated at z ) 0:

The use of polynomials to approximate functions in asmall interval is justified by Taylor’s theorem.13 Taylorpolynomials are partial sums or truncated versions ofa Taylor series expansion. They tend to provide goodapproximations for the value of the function near thepoint of expansion. However, the approximation errorof a given Taylor expansion may increase rapidly atpoints further away: The errors tend to “bunch up” nearthe ends of the interval.14 Approximations based onorthogonal polynomials, e.g., Chebyshev polynomials,are generally more efficient: They can “distribute” theapproximation error more uniformly over the intervalof interest.

Scheffe Polynomials

The ordinary second-order polynomial with threeindependent composition variables is

Scheffe polynomials emerge from ordinary polynomialswhen the mixture constraint, eq 2, is taken into account.Thus, when x1 + x2 + x3 ) 1 is utilized, eq 9 simplifiesto7,9

This is the canonical second-order Scheffe form for aternary mixture. The coefficients âi and âij are combina-tions of the corresponding Ri and Rij in eq 9. Thecoefficient âi is the property value assumed by purecomponent i. The values that âij assume quantify thenonlinear mixing effects between components i and j.

Comparing eqs 9 and 10 reveals that Scheffe polyno-mials are trimmed-down versions of Taylor polynomials.They have fewer adjustable parameters and featureunambiguous multivariate forms.7-9 The first-orderScheffe model is equivalent to the “blending rule”described by eq 3. The canonical form of the second-order Scheffe polynomial for multicomponent mixturesis7,8

The third-order Scheffe is probably the most complexpolynomial commonly used in experimental design.Written in canonical form, it is given by7,8

xi g 0 for i ) 1, 2, ..., q (1)

∑i

xi ) x1 + x2 + ... + xq ) 1 (2)

Fideal ≡ ∑i

Fixi ) F1x1 + F2x2 + ... + Fqxq (3)

∆F ≡ F - Fideal (4)

FE ≡ F - FIS (5)

f ≡ FE/x1x2 (6)

f = C0 + C1z + C2z2 + ... + Cnzn (7)

Ck ) 1k! (dkf

dzk)z)0

(8)

S3(2) ) R0 + R1x1 + R2x2 + R3x3 + R12x1x2 +

R13x1x3 + R23x2x3 + R11x12 + R22x2

2 + R33x32 (9)

S3(2) ) â1x1 + â2x2 + â3x3 + â12x1x2 + â13x1x3 +â23x2x3 (10)

Sq(2) ) ∑i)1

q

âixi + ∑∑i<j

q

âijxixj (11)

Sn(3) ) ∑i)1

n

âixi + ∑∑i<j

n

âijxixj + ∑∑ ∑i<j<k

n

âijkxixjxk +

∑∑i<j

n

γijxixj(xi - xj) (12)

8370 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004

Note that in the canonical form there are first-orderterms, second-order terms, and higher (third) orderterms.

The Scheffe forms defined by eqs 3, 11, and 12 arealso called S models or S polynomials.15 The K modelsor K-polynomial forms are alternative but equivalenthomogeneous representations: All of the terms have thesame order as the highest order term in the canonicalScheffe S form.15 For example, the K-model version ofthe quadratic three-component system, equivalent to eq10, is

The coefficients of the quadratic Scheffe S and K modelsare interrelated:

Table 1 lists the first five Scheffe K-polynomial formsfor binary mixtures. Note the notation used: Theconstants ci ) cii ) ciii ) ciiii ) ... are pure-componentproperty values. The cij cross terms are adjustableparameters. They characterize the nonlinear mixingeffects between components i and j.

The general notation for the q-component nth-orderK polynomial is

In eq 15, each term is nth order in composition; the cijk...’sare adjustable parameters, and the Mijk...’s are multi-nomial coefficients defined by

where n ) ∑k)1q mk and mk denotes the number of times

the label k occurs in the subscript of the multinomialMij... or parameter cij....

Rational Functions

A rational function is simply the ratio of two polyno-mials. Pade approximations are rational function ana-

logues of Taylor series expansions. They are expressedas the quotient of the following two polynomials:

It is conventional to set λ0 ) 1 without any loss ofgenerality. The other parameters are determined on thebasis of the Taylor expansion of the approximatedfunction f(z). The Pade approach provides very goodapproximations for a function inside a closed interval.14

Marsh16 advocated their use for representing excessproperties, i.e., the excess Gibbs free energy. Like theRedlich-Kister and orthogonal polynomials, these Padeapproximation methods were at first developed forbinary data reduction. However, extensions to multi-component mixtures have been proposed.2,17-19

Dimitrov and Kamenski19 used rational functions tomodel the nonideal property behavior of binary mix-tures. They used polynomials based on Redlich-Kisterexpansions. Instead, we propose homogeneous mixturemodels based on Scheffe rational functions:

Here Kq(n) denotes an nth-order Scheffe K polynomialfor a q-component system. Scatchard’s model20 is anexample of a Scheffe rational function. For a ternarymixture, it is described by

Tables 2-4 provide details of the first few canonicalScheffe rational functions for binary mixtures.

Consider the pure-component limits of these forms.The pure-component property of component i is obtainedas a ratio:

Because the ci’s represent (known) pure-componentproperties, eq 20 reveals that ai and bi pairs (or aii andbii or aiii and bi, etc.) are not independent variables. Thisproblem can be solved if, for example, the bi ) bii ) biii

Table 1. Scheffe K Polynomials for Binary Mixtures

m model K2(m) ) P2(m,0) ∆K2(m)/x1x2 notes

1 “ideal” c1x1 + c2x2 02 Porter31 c11x1

2 + 2c12x1x2 + c22x22 A12 A12 ) 2c12 - c11 - c22

3 Margules2 c111x13 + 3c112x1

2x2 +3c122x1x2

2 + c222x23

A112x1 + A122x2A112 ) 3c112 - 2c111 - c222

A122 ) 3c122 - c111 - 2c222

4 c1111x14 + 4c1112x1

3x2 +6c1122x1

2x22 +

4c1222x1x23 + c2222x2

4A1112x1

2 + A1122x1x2 + A1222x22

A1112 ) 4c1112 - 3c1111 - c2222A1122 ) 6c112 - 3c1111 - 3c2222

A1222 ) 4c122 - c1111 - 3c2222

5 c11111x15 + 5c11112x1

4x2 +10c11122x3x2

2 +10c11222x1

2x23 +

5c12222x1x24 + c22222x2

4

A11112x13 + A11122x1

2x2 + A11222x1x22 +

A12222x23

A11112 ) 5c11112 - 4c11111 - c22222A11122 ) 10c11122 - 6c11111 - 4c22222

A11222 ) 10c11222 - 4c11111 - 6c22222

A12222 ) 5c12222 - c11111 - 4c22222

K3(2) ) c11x12 + c22x2

2 + c33 x32 + 2c12x1x2 +2c13x1x3 + 2c23x2x3 (13)

cii ) âi and cij ) (âij + âi + âj)/2 (14)

Kq(n) ) ∑iejek...

q

Mijk...cijk...xixjxk... (15)

Mijk... ) ( n!m1!m2!...mq! ) (16)

f =

∑n)0

N

κnzn

∑n)0

N

λnzn

(17)

Pq(m,n) ≡ Kq(m)/Kq(n) (18)

F3(2,1) )

a11x12 +a22x2

2 +a33x32 + 2a12x1x2 + 2a13x1x3 +2a23x2x3

b1x1 + b2x2 + b3x3

(19)

Fi ≡ ci ) ai/bi ) aii/bii ) aiii/bi ) aii/biiii ) ... (20)

Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 8371

) biiii ... values are identified with another unrelatedand independent pure-component property. It is expedi-ent to choose as the denominator a relevant weighingcharacteristic. Preferably, this attribute would be re-lated to the molecular structure and have a determiningeffect or proven influence on the property being studied.Scatchard20 used eq 19 to describe the internal energyof a liquid mixture. He associated the constant bi with

the molar volume of the pure liquid. More recently, itwas shown that the excess Gibbs free energy of a liquidmixture can be expressed as a Pq(2,2) Scheffe rationalfunction, with the bi’s corresponding to the covolumeparameters of the van der Waals equation of state.21

Here this parameter is chosen for illustration purposesbecause it features the added advantage of being tem-perature-independent. Further investigation may reveal

Table 2. Pade P2(p,1) K Polynomials for Binary Mixtures

m model (#)a P2(m,2)

∆P2(m,1)

x1x2 notes

0 (0)a0

b1x1 + b2x2

A12

b1x1 + b2x2A12 ) a0(2 -

b2

b1-

b1

b2)

1 (0)a1x1 + a2x2

b1x1 + b2x2

A12

b1x1 + b2x2A12 ) a1(1 -

b2

b1) + a2(1 -

b1

b2)

2Scatchard20

(1)

a11x12 + 2a12x1x2 + a22x2

2

b1x1 + b2x2

A12

b1x1 + b2x2A12 ) 2a12 - a11(b2

b1) - a22(b1

b2)

3 Sigma35 a111x13 + 3a112x1

2x2 + 3a122x1x22 + a222x2

3

b1x1 + b2x2

A112x1 + A122x2

b1x1 + b2x2

A112 ) 3a112 - a111(1 +b2

b1) - a222(b1

b2)

(2) A122 ) 3a122 - a111(b2

b1) - a222(1 +

b1

b2)

4 (3)a1111x1

4 + 4a1112x13x2 + 6a1122x1

2x22 +

4a1222x1x23 + a2222x2

4

b1x1 + b2x2

A1112x12 + A1122x1x2 + A1222x2

2

b1x1 + b2x2

A1112 ) 4a1112 - a1111(2 +b2

b1) - a1111(b1

b2)

A1122 ) 6a1122 - a1111(1 + 2b2

b1) - a2222(1 + 2

b1

b2)

A1222 ) 4a1222 - a2222(2 +b1

b2) - a1111(b2

b1)

a # ) number of adjustable binary parameters.


m model (#)a P2(m,2)∆P2(m,2)

x1x2 notes

0

(1)

a0

b11x12 + 2b12x1x2 + b22x2

2

A12x1 + A12x2

b11x12 + 2b12x1x2 + b22x2

2

A12 ) a0(3 - 2b12

b11-

b11

b22)

A21 ) a0(3 - 2b12

b22-

b22

b11)

1(1)

a1x1 + a2x2

b11x12 + 2b12x1x2 + b22x2

2

A12x1 + A12x2

b11x12 + 2b12x1x2 + b22x2

2

A12 ) 2a1(1 -b12

b11) + a2(1 -

b11

b22)

A21 ) a1(1 -b22

b11) + 2a2(1 -

b12

b22)

2van der Waals21 a11x1

2 + 2a12x1x2 + a22x22

b11x12 + 2b12x1x2 + b22x2

2

A12x1 + A12x2

b11x12 + 2b12x1x2 + b22x2

2

A12 ) 2a12 + a11(1 -2b12

b11) - a22(b11

b22)

(2) A21 ) 2a12 + a11(b22

b11) - a22(1 -

2b12

b22)

3F-G21 a111x1

3 + 3a112x12x2 + 3a122x1x2

2 + a222x23

b11x12 + 2b12x1x2 + b22x2

2

A112x1 + A122x2

b11x12 + 2b12x1x2 + b22x2

2

A112 ) 3a112 - a111(2b12

b11) - a222(b11

b22)

(3) A112 ) 3a122 - a111(b22

b11) - a222(2b12

b22)

a # ) number of adjustable binary constants.


superior alternatives. It should be stressed that the bijk’swill be interpreted as adjustable interaction volumesrelevant to the main physical property to be correlated.In this context, all bijk > 0 and the denominatorpolynomial has no roots in the composition domain.

Parameter Estimation

Model parameters can be determined by a direct fitof the full model to all available data. Scheffe rationalfunctions also allow a sequential approach: The modelparameters can, in principle at least, be evaluated inthe following sequence: (i) the constants ai, bi, and cifrom pure-component data; (ii) aij, bij, and cij from binarymixture data; (iii) aijk, bijk, and cijk from ternary data,etc.

This approach may offer advantages when, for in-stance, the quality of the experimental data deteriorateswith the number of components in the mixture. Mostexperimentalists would agree that the difficulties en-countered, in determining physical properties to anydegree of precision, increase considerably with thenumber of components in the mixture. This is especiallytrue for heat of mixing and vapor-liquid and liquid-liquid equilibrium measurements.

Model parameter values should be determined suchthat the response surface best fits the known data.Several methods are available and are discussed indetail by Montgomery22 and Lawson and Erjavec.23 Themethod of maximum likelihood selects parameter esti-mates that, for a specified model and error distribution,maximize the probability of occurrence of the sampleresults. Rubio et al.24 have applied the maximumlikelihood principle to the determination of Pade pa-rameters. They also used it to derive selection criteriafor the optimal Pade approximants.

The least-squares method was used for data reductionin the present study. It seeks parameter values thatminimize the sum of squares of the deviations betweendata and the model. Marsh et al.16 used this method toobtain parameter values for rational functions, i.e., Padeapproximants. Hernandez-Pacheco and Mann25 sug-gested that the model parameters be evaluated by alinear least-squares fit. To this effect, they recast eq 18in a linear-in-the-coefficients form by multiplyingthroughout with the denominator polynomial. Whileuseful for obtaining initial estimates for model param-eters, the method is not a valid substitute for thenonlinear least-squares fit: Rational functions are, afterall, nonlinear models.

The square of the correlation coefficient is a measureof the amount of reduction in the variability obtainedby using the regression variables in the model.22 Addingan additional variable to the model will always increaser2, regardless of whether the additional variable isstatistically significant or not. Thus, a large value of r2

does not necessarily imply that the regression model isa good one. Because r2 always increases as variables areadded, it is preferable to use an adjusted r2 statisticdefined as22

In this equation, N is the number of data points and pdenotes the number of adjustable parameters in themodel. Because this adjusted r2 statistic does notnecessarily increase as variables are added to the model,it can be used as a primitive model discriminator. Toillustrate this, consider the viscosity of the systemacetone (1)-methanol (2)-water (3) at 25 °C. Modelcoefficients were determined simultaneously by fitting


m model (#)a P2(m,3) ∆P2(m,3)/x1x2 notes

1

(2)

a1x1 + a2x2

b111x13 + 3b112x1

2x2 + 3b122x1x22 + b222x2

3

A1112x12 + A1122x1x2 + A1222x2

2

b111x13 + 3b112x1

2x2 + 3b122x1x22 + b222x2

3

A1112 ) 3a1(1 -b112

b111) + a2(1 -

b111

b222)

A1122 ) 3a1(1 -b122

b111) + 3a2(1 -

b112

b222)

A1222 ) a1(1 -b222

b111) + 3a2(1 -

b122

b222)

2

(3)

a11x2 + 2a12x1x2 + a22x2

2

b111x13 + 3b112x1

2x2 + 3b122x1x22 + b222x2

3

A1112x12 + A1122x1x2 + A1222x2

2

b111x13 + 3b112x1

2x2 + 3b122x1x22 + b222x2

3

A1112 ) a11(2 - 3b112

b111) + 2a12 - a22(b111

b222)

A1122 ) a11(1 - 3b122

b111) + 4a12 + a22(1 - 3

b112

b222)

A1222 ) -a11(b222

b111) + 2a12 + a22(2 - 3

b122

b222)

3

(4)

a111x13 + 3a112x1

2x2 + 3a122x1x22 + a222x2

3

b111x13 + 3b112x1

2x2 + 3b122x1x22 + b222x2

3

A1112x12 + A1122x1x2 + A1222x2

2

b111x13 + 3b112x1

2x2 + 3b122x1x22 + b222x2

3

A1112 ) a111(1 - 3b112

b111)+ 3a112 - a222(b111

b222)

A1122 ) -3a111(b122

b111)+ 3a112 + 3a122 - 3a222(b112

b222)

A1222 ) -a111(b222

b111) + 3a122 + a222(1 - 3

b122

b222)

a # ) number of adjustable binary parameters. Note: There is no advantage in plotting in terms of Aijkl because there are fewer adjustableparameters!

radj2 ) 1 - (N - 1

N - p)(1 - r2) (21)


the full model to all data points. Table 5 comparescalculated values for the r2 statistics. It shows that theP3(2,3) and P3(3,3) models perform significantly betterthan the lower Scheffe forms. However, the differencebetween the adjusted r2 values for theses two models issmall. This raises the question of whether this differenceis statistically significant. Models with the fewestadjustable parameters should be given preference indata correlation. Thus, it is necessary to determinewhether the use of the higher order model is justified.

Assume that the performances of models 1 and 2, withp1 and p2 (with p2 g p1) adjustable parameters, respec-tively, are to be compared. Statistical significancetesting can be used for such model discrimination. Anull hypothesis is posed that states that there is nosignificant difference between the model performances.It cannot be rejected unless the test statistic employedexceeds a critical value for a predetermined significancelevel.

Residuals are the differences between measured andpredicted property values. Thus, the residual for datapoint i and model k is denoted by εki ) Fi - Fki. Thisgenerates the two vectors (ε11, ε12, ..., ε1N) and (ε21, ε22,..., ε2N). If the residuals are normally distributed, thestatistic ∑i(εik - εjk)2/σk

2 ∼ øn-pk-12 will have a ø2

distribution with N - pk - 1 degrees of freedom.22 Thus,an F test22 can be used to compare the null hypothesisH0, σ2

2/σ12 ) σ2/σ1 ) 1, against the alternative H1, σ2

2/σ1

2 ) σ2/σ1 < 1:

In the viscosity example mentioned above, there are atotal of 62 binary and ternary data points. The twomodels have p1 ) 10 and p2 )14 adjustable parameters,respectively. Analysis yields a value for the F statisticof F ) 2.41. This exceeds the critical value of 1.6 (atthe 5% confidence level), and the null hypothesis isrejected. It is concluded that use of the P3(3,3) modelinstead of the P3(2,3) model is justified because it leadsto a reduction in the “error” standard deviation. Thedata fit of this model is illustrated in Figures 1 and 2.

Discussion

The superior ability of the rational functions to fitbinary mixture physical property data is well-estab-lished and is therefore not reconsidered here.16,22,24 Thefigures exemplify the power of Scheffe rational functionsto correlate diverse multicomponent mixture data: Vis-cosity data26-30 for ternary and quaternary liquidmixtures are presented in Figures 1-4. Figure 5 showsthe performance of the Scatchard P5(2,1) model incorrelating the heat-of-mixing data for a set of nonpolarbinary, ternary, quaternary, and quinary mixtures.31

The performance of the Scheffe S3(3) model is examinedin Figure 6 with vapor-liquid equilibrium data for themethyl acetate-chloroform-benzene system.32

The proposed Scheffe rational functions have a num-ber of interesting and useful features.

Table 5. Viscosity of Acetone (1)-Methanol (2)-Water (3)at 25 °C: Correlation Coefficients for Scheffe RationalFunctions

N ) 62 P3(2,1) P3(2,2) P3(3,2) P3(2,3) P3(3,3)

p 3 6 10 10 14r2 0.896 96 0.966 82 0.994 27 0.999 54 0.999 82radj

2 0.893 47 0.963 85 0.993 28 0.999 46 0.999 78

F ) ∑(ε1i - εj1)2

σ12(N - p1 - 1)

/∑(ε2i - εj2)

2

σ22(N - p2 - 1)

∼ FN-p1-1;N-p2-1

(22)

Figure 1. Binary viscosity data26 for mixtures of methanol,acetone, and water at 25 °C correlated with the P2(3,3) Pade model.

Figure 2. Ternary and binary viscosity data26 for mixtures ofmethanol, acetone, and water at 25 °C correlated with the P3(3,3)Pade model.

Figure 3. Viscosity data27,28 for quaternary mixtures of benzene,cyclohexane, ethanol, and heptane correlated globally using theP4(2,3) model.


Consistent Expressions for Multicomponent Mix-tures. Other expressions,2 previously proposed for mul-ticomponent mixtures, suffer from one or more of thefollowing flaws:

(i) The properties of multicomponent mixtures aredependent on how the component indices are assigned.2

(ii) In a ternary or higher system, the relations do notreduce to the binary or lower form in the limit of infinitedilution of a third component.17

(iii) The expressions are not invariant with respectto dividing one component into two or more identicalsubcomponents; i.e., they suffer from the Michelsen-Kistenmacher syndrome.33

Scheffe polynomials and rational functions are freefrom all of these shortcomings.

Extension and Prediction. Parameter values donot change when additional components are added tothe mixture. However, it may be necessary to introducenew parameters, e.g., ternary constants. Note that Sq-(2) and Pq(1,2), Pq(2,1), and Pq(2,2) are predictivemodels: ternary and higher data are fully determinedby knowledge of the binaries. Higher order modelsrequire the determination of ternary and higher coef-ficients.

Parameter Economy and Scalability. The ScheffeK-form polynomials and rational functions defined aboveform a self-consistent nested set; i.e., they “allow mixingand matching”. This concept is illustrated by way of thefollowing example. The viscosity data at 25 °C forternary mixtures of acetone (1)-n-hexane (2)-ethanol(3) were correlated using Scheffe rational functions. The“best” models were identified according to the adjustedr2 statistic described in eq 21. The optimum models werefound to be

Inspection shows that it should be possible to describethe ternary behavior by a P3(2,2) Scheffe rationalfunction:

The quadratic nature of this model makes it possible topredict ternary properties directly from knowledge ofthe binary behavior. The conversion of the differentbinary forms to the P2(2,2) (quadratic-over-quadratic)form is readily accomplished: The expressions for η-(x1,x2) and η(x2,x3) are multiplied by 1/(x1 + x2) and 1/(x2+ x3), respectively. This promotes the denominator ineach case to the quadratic form. Similarly, η(x1,x3) is

Figure 4. Predictions for ternary viscosity data29,30 in theacetone-hexane-ethanol system. P4(2,2) model parameters ob-tained from binary data only.

Figure 5. Heat-of-mixing data31 for binary, ternary, quaternary,and quinary mixtures of hexane-benzene-cyclohexane-heptane-toluene globally correlated with the Scatchard P5(2,1) model.

Figure 6. Vapor-liquid equilibrium data at 50 °C for the methylacetate-chloroform-benzene system. The binary data were cor-related with the Margules model except for the methyl acetate-benzene binary, for which the Porter model proved adequate. Theternary data were correlated with the Scheffe S3(3) model, i.e.,with a ternary constant.

Binary 1-2: η(x1,x2) )a11x1

2 + 2a12x2x3 + a22x22

b1x1 + b2x2

(adjustable binary parameter a12)

Binary 1-3: η(x1,x3) )a1x1 + a3x3

b11x12 + 2b13x2x3 + b33x3

2

(adjustable binary parameter b13)

Binary 2-3: η(x2,x3) )a22x2

2 + 2a23x2x3 + a33x32

b2x2 + b3x3

(adjustable binary parameter a23)

η(x1,x2,x3) )

a11x12 + a22x2

2 + a33x32 + 2(a12x1x2 + a13x1x3 + a13x1x3)

b11x12 + b22x2

2 + b33x32 + 2(b12x1x2 + b13x1x3 + b13x1x3)

(24)


multiplied by x1 + x3 to change the numerator to asecond-order polynomial. These procedures yield

Figure 4 indicates that the predicted ternary behavioragrees reasonably well with experimental data. Theaverage and maximum absolute deviations for ternaryviscosity values were 0.45 and 3.43%, respectively. Thus,by considering the data sequentially, it was revealedthat three adjustable coefficients are adequate for thissystem and not all six parameters inherent in the P3-(2,2) model.

The concepts illustrated in this example are readilyextended to higher order Scheffe polynomials andScheffe rational functions. However, in general it maybecome necessary to determine higher order parametersfrom appropriate multicomponent data, i.e., ternaryconstants from ternary data if cubic polynomials appearin the multicomponent expressions.

Links to Well-Established Classical Models. Tables1-3 reveal that well-established traditional models forbinary mixtures such as Porter,34 Margules,1 Scat-chard,20 Sigma,35 etc., are, in fact, equivalent to eithera particular Scheffe K-form polynomial or rationalfunction. Conversely, the corresponding Scheffe modelor rational function provides consistent multicomponentexpressions for these models. For instance, Table 1shows that the Margules model is, in fact, equivalentto a cubic Scheffe K-form polynomial. Thus, for a ternarymixture, the proper excess Gibbs free energy expressionis

The first three terms on the right embody binarycontributions shown in Table 1, while the last termrepresents the ternary contribution with coefficient A123) 6c123 - 2(c111 + c222 + c333). Figure 6 shows that theMargules model is able to correlate the vapor-liquidequilibrium data for the methyl acetate-chloroform-benzene system. In fact, the Porter model (quadraticScheffe) was used for the methyl acetate-benzenebinary. Thus, only five binary constants and one ternaryconstant were required to correlate the full data set.

In thermodynamic practice, it is common practice tocorrelate excess properties of binary mixtures. Tables2-4 indicate that there are some dangers associatedwith this approach. The functional form for the binarymixing functions ∆P (and by analogy also excess proper-ties) is not unique. According to Table 3, ∆P has theexact same mathematical form for 0 e m e 3:

Note the nature of parameters A112 and A122. For m )0 or 1, they are constants. For m ) 2, only one of themis freely adjustable, whereas for m ) 3, both areindependently adjustable constants. This demonstratesthat care should be taken when fitting excess and peeledmodel forms to experimental data.

Conclusion

Previous use of rational functions and Pade approxi-mants was largely limited to fitting physical property

data for pure components or binary mixtures. Homo-geneous Scheffe rational functions can be defined as theratio of two Scheffe K polynomials. They allow consis-tent correlation of physical properties for mixtures thatcontain any number of components. Parameters can beevaluated globally or sequentially. In the latter ap-proach, the binary coefficients are determined frombinary data, ternary constants from ternary data, etc.The Scheffe rational functions also form an ordered andself-consistent set. Lower order forms that, for example,adequately correlate a given binary data set can beintegrated seamlessly into a higher form without in-creasing the number of adjustable parameters. Suchparameter parsimony is an attractive feature.

In thermodynamics, it is customary to correlate theexcess (FE) or even peeled properties (f ) FE/x1x2) ofbinary mixtures. Inspection shows that their math-ematical form is not unique. Different Scheffe ap-proximations for the property F can have the samefunctional form for f, although the number of adjustableparameters may differ. It is recommended that actualproperty values be modeled where possible.

The analysis presented here also revealed consistentmulticomponent forms for classic and more recentactivity coefficient models such as Porter, Margules, andSigma. These can now be considered further for cor-relating multicomponent behavior, e.g., liquid-liquidequilibrium.

Acknowledgment

Financial support for this research from the THRIPprogram of the Department of Trade and Industry andthe National Research Foundation of South Africa aswell as Xyris Technology is gratefully acknowledged.

Notation

A ) constant in the mixing or excess functiona, b ) constants, Scheffe K-form rational functionsC ) constant, truncated Taylor (Maclaurin) series, eq 7c ) constant, Scheffe K polynomialF ) arbitrary physical property or F statisticF ) predicted value for the physical propertyf ) peeled physical property defined by eq 6Gm

E ) excess Gibbs free energy [J/mol]Hm ) enthalpy [J/mol]K ) Scheffe K-polynomial formM ) multinomial coefficient in Scheffe K polynomialm, n ) order of the polynomialN ) number of data pointsP ) Scheffe K-polynomial rational function, eq 18p ) degrees of freedom, number of adjustable parameters

in a modelq ) number of components in the mixtureR ) gas constant (R ) 8.314 321 4) [J/mol‚K]r ) correlation coefficientS ) Scheffe S polynomialT ) absolute temperature (K)xi ) (liquid) mole fraction of component iyi ) vapor mole fraction of component iz ) composition variable for binary mixtures (z ) x1 - x2)

Greek Letters

R ) constant, standard polynomial form, eq 9â ) constant, Scheffe S polynomial, eqs 10-12γ ) constant, Scheffe S polynomial, eq 12∆ ) differenceε, εj ) residual, mean residual

b12 ) (b1 + b2)/2 b23 ) (b2 + b3)/2 anda13 ) (a1 + a3)/2

GmE/RT ) x1x2[A112x1 + A122x2] + x1x3[A113x1 +

A133x3] + x2x3[A223x2 + A233x3] + x1x2x3A123 (25)

∆P2(m,2)x1x2

)A112x1 + A122x2

b11x12 + 2b12x1x2 + b22x2

2(26)


η ) viscosity [mPa‚s]ø ) chi-square distributionκ ) constant, Pade polynomial standard form, eq 17λ ) constant, Pade polynomial standard form, eq 17σ ) sample variance

Subscripts

adj ) adjustedideal ) linear combination of pure-component properties,

i.e., blending rulei, j, k ) component indicesM ) multinomial coefficientm ) number of times an index appearsn ) order of the polynomialIS ) ideal solution

Superscripts

E ) excess

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Received for review July 4, 2004Revised manuscript received October 5, 2004

Accepted October 11, 2004

IE049415+


correlating multicomponent mixture properties with homogeneous rational functions

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